3-D flow of a compressible viscous micropolar fluid with spherical symmetry: a local existence theorem

  • Ivan Dražić1Email author and

    Affiliated with

    • Nermina Mujaković2

      Affiliated with

      Boundary Value Problems20122012:69

      DOI: 10.1186/1687-2770-2012-69

      Received: 26 December 2011

      Accepted: 12 June 2012

      Published: 2 July 2012

      Abstract

      We consider nonstationary 3-D flow of a compressible viscous heat-conducting micropolar fluid in the domain to be the subset of R 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq1_HTML.gif bounded with two concentric spheres that present solid thermoinsulated walls. In thermodynamical sense fluid is perfect and polytropic. Assuming that the initial density and temperature are strictly positive we will prove that for smooth enough spherically symmetric initial data there exists a spherically symmetric generalized solution locally in time.

      Keywords

      micropolar fluid generalized solution spherical symmetry weak and strong convergence

      1 Introduction

      The theory of micropolar fluids is introduced in [1] by Eringen. Various problems with different initial and boundary conditions for incompressible micropolar fluid are presented in [2], but the theory of compressible micropolar fluid is still in the beginning. N. Mujakovic in [3] developed the model for one-dimensional isotropic, viscous, compressible micropolar fluid which is in thermodynamical sense perfect and polytropic. In the same work, the local existence of the solution for homogeneous boundary conditions is proved. N. Mujakovic in [4] and in the references cited therein proved the local and global existence of inhomogeneous boundary conditions for velocity and microrotation as well as stabilization and regularity. In [5] the Cauchy problem for the described problem was also considered. In the last years we find some interesting works with different kind of problems concerning micropolar fluid, e.g., [6, 7], but till now the described model of compressible micropolar fluid in three-dimensional case has not been considered.

      In this work we consider the three-dimensional model with spherical symmetry. The first article in which the problem of spherical symmetry was described is [8], but for classical fluid. The spherical symmetry for classical fluid is also considered in articles [912].

      In the setting of the field equations we use the Eulerian description.

      In what follows we use the notation:

      ρ - mass density

      v - velocity

      p - pressure

      T - stress tensor

      T x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq2_HTML.gif - an axial vector with the Cartesian components ( T x ) i = ε i j k T j k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq3_HTML.gif, where ε i j k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq4_HTML.gif is Levi-Civita alternating tensora

      sym T = 1 2 ( T + T T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq5_HTML.gif, skw T = 1 2 ( T T T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq6_HTML.gif

      ω - microrotation velocity

      ω skw http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq7_HTML.gif - skew tensor with Cartesian components ( ω skw ) i j = ε k i j ω k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq8_HTML.gif

      j I http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq9_HTML.gif - microinertia density (a positive scalar field)

      C - couple stress tensor

      θ - absolute temperature

      E - internal energy density

      q - heat flux density vector

      f - body force density

      g - body couple density

      δ - body heat density

      The problem we consider here is based on local forms of the conservation laws for mass, momentum, momentum moment and energy, which are stated respectively as follows:
      ρ ˙ + ρ div v = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ1_HTML.gif
      (1)
      ρ v ˙ = div T + ρ f , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ2_HTML.gif
      (2)
      ρ j I ω ˙ = div C + T x + ρ g , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ3_HTML.gif
      (3)
      ρ E ˙ = T : v + C : ω T x ω div q + ρ δ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ4_HTML.gif
      (4)
      where a ˙ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq10_HTML.gif denotes material derivative of a field a:
      a ˙ = a t + ( a ) v . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equa_HTML.gif
      The scalar product of tensors A and B is defined by
      A : B = tr ( A T B ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equb_HTML.gif
      The linear constitutive equations for stress tensor, couple stress tensor and heat flux density vector are respectively in the forms:
      T = ( p + λ div v ) I + 2 μ sym v 2 μ r skw v 2 μ r ω skw , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ5_HTML.gif
      (5)
      C = c 0 ( div ω ) I + 2 c d sym ω 2 c a skw ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ6_HTML.gif
      (6)
      q = k θ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ7_HTML.gif
      (7)

      where

      λ, μ - coefficients of viscosity,

      μ r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq11_HTML.gif, c 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq12_HTML.gif, c d http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq13_HTML.gif, c a http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq14_HTML.gif - coefficients of microviscosity,

      k - heat conduction coefficient

      are constants with the properties
      μ 0 , 3 λ + 2 μ 0 , μ r 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ8_HTML.gif
      (8)
      c d 0 , 3 c 0 + 2 c d 0 , | c d c a | c d + c a , k 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ9_HTML.gif
      (9)
      Assuming that the fluid is perfect and polytropic, for pressure and internal energy we have the equations
      p = R ρ θ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ10_HTML.gif
      (10)
      E = c v θ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ11_HTML.gif
      (11)

      where R and c v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq15_HTML.gif are positive constants.

      Let Ω = { x R 3 , a < | x | < b } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq16_HTML.gif, a > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq17_HTML.gif, denote the domain bounded by two concentric spheres with radii a and b. The boundary of the described domain is Ω = { x R 3 , | x | = a  or  | x | = b } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq18_HTML.gif. We shall consider the problem (1)-(11) in the region Q T = Ω × ] 0 , T [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq19_HTML.gif (where T > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq20_HTML.gif is arbitrary) with the following initial conditions:
      ρ ( x , 0 ) = ρ 0 ( x ) , v ( x , 0 ) = v 0 ( x ) , ω ( x , 0 ) = ω 0 ( x ) , θ ( x , 0 ) = θ 0 ( x ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ12_HTML.gif
      (12)
      for x Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq21_HTML.gif and boundary conditions
      v | Ω = 0 , ω | Ω = 0 , θ ν | Ω = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ13_HTML.gif
      (13)

      for 0 < t < T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq22_HTML.gif; the vector ν is an exterior unit normal vector.

      For simplicity we also assume that f = g = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq23_HTML.gif and δ = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq24_HTML.gif.

      The initial boundary problems for the system (1)-(13) so far have not been considered in three-dimensional case. The same and similar models in one-dimensional case were considered in [3, 5, 13] and [4]. In [2] the three-dimensional model was considered but for an incompressible micropolar fluid.

      In this paper we prove the local existence of generalized spherically symmetric solution to the problem (1)-(13) in the domain Ω, assuming that the initial functions are also spherically symmetric. In the proof we use the Faedo-Galerkin method. We follow some ideas of [14] where this method was applied to a classical fluid (where microrotation is equal to zero) in one-dimensional case as well as the ideas from [3] and [13] where the same result as here was provided for one-dimensional case.

      The paper is organized as follows. In the second section, we derive a spherically symmetric form of (1)-(4), introduce Lagrangian description, and present the main result. In the third section, we consider an approximate problem and get an approximate solution for each n N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq25_HTML.gif. In the forth section, we prove uniform a priori estimates for the approximate solutions. The proof of the main result is given in the fifth section.

      2 Spherically symmetric form and the main result

      We first derive the spherically symmetric form of (1)-(7) and (10)-(11). A spherically symmetric solution of (1)-(7) has the form:
      v i ( x , t ) = x i r v ( r , t ) , ω i ( x , t ) = x i r ω ( r , t ) , i = 1 , 2 , 3 , ρ ( x , t ) = ρ ( r , t ) , θ ( x , t ) = θ ( r , t ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ14_HTML.gif
      (14)
      where x = ( x 1 , x 2 , x 3 ) R 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq26_HTML.gif, r = | x | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq27_HTML.gif, v = ( v 1 , v 2 , v 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq28_HTML.gif and ω = ( ω 1 , ω 2 , ω 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq29_HTML.gif. We assume that
      ρ 0 ( x ) = ρ 0 ( r ) , v 0 ( x ) = x r v 0 ( r ) , ω 0 ( x ) = x r ω 0 ( r ) , θ 0 ( x ) = θ 0 ( r ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ15_HTML.gif
      (15)
      where ρ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq30_HTML.gif, v 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq31_HTML.gif, ω 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq32_HTML.gif and θ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq33_HTML.gif are known real functions defined on ] a , b [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq34_HTML.gif, and thus we reduce system (1)-(7) and conditions (10)-(13) to the following equations for ρ ( r , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq35_HTML.gif, v ( r , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq36_HTML.gif, ω ( r , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq37_HTML.gif and θ ( r , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq38_HTML.gif of the form:
      ρ t + r ( v ρ ) + 2 ρ r v = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ16_HTML.gif
      (16)
      ρ ( v t + v v r ) = R r ( ρ θ ) + ( λ + 2 μ ) r ( v r + 2 v r ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ17_HTML.gif
      (17)
      ρ j I ( ω t + v ω r ) = 4 μ r ω + ( c 0 + 2 c d ) r ( ω r + 2 ω r ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ18_HTML.gif
      (18)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ19_HTML.gif
      (19)
      with the following initial and boundary conditions
      ρ ( r , 0 ) = ρ 0 ( r ) , v ( r , 0 ) = v 0 ( r ) , ω ( r , 0 ) = ω 0 ( r ) , θ ( r , 0 ) = θ 0 ( r ) , r ] a , b [ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ20_HTML.gif
      (20)
      v ( a , t ) = v ( b , t ) = 0 , ω ( a , t ) = ω ( b , t ) = 0 , θ r ( a , t ) = θ r ( b , t ) = 0 , 0 < t < T . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ21_HTML.gif
      (21)
      To investigate the local existence, it is convenient to transform the system (16)-(19) to that in Lagrangian coordinates. The Eulerian coordinates ( r , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq39_HTML.gif are connected to the Lagrangian coordinates ( ξ , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq40_HTML.gif by the relation
      r ( ξ , t ) = r 0 ( ξ ) + 0 t v ˜ ( ξ , t ) d τ , r 0 ( ξ ) = r ( ξ , 0 ) = ξ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ22_HTML.gif
      (22)
      where v ˜ ( ξ , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq41_HTML.gif is defined by
      v ˜ ( ξ , t ) = v ( r ( ξ , t ) , t ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ23_HTML.gif
      (23)
      We introduce the new function η by
      η ( ξ ) = a ξ s 2 ρ 0 ( s ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ24_HTML.gif
      (24)
      Note that if ρ 0 ( s ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq42_HTML.gif for s ] a , b [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq43_HTML.gif (which is assumed in Theorem 2.1 later), then there exists an inverse function η 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq44_HTML.gif. Let the constant L be defined as
      η ( b ) = a b s 2 ρ 0 ( s ) d s = L . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ25_HTML.gif
      (25)
      From (16) we can easily get the equation
      ρ r ξ r 2 = ρ 0 ( ξ ) ξ 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ26_HTML.gif
      (26)
      i.e.,
      a r ( ξ , t ) ρ ( s , t ) s 2 d s = a ξ ρ 0 ( s ) s 2 d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ27_HTML.gif
      (27)
      It is useful to introduce the next coordinate
      x = L 1 η ( ξ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ28_HTML.gif
      (28)
      and the following functions
      ρ ( x , t ) = ρ ˜ ( η 1 ( x L ) , t ) , v ( x , t ) = v ˜ ( η 1 ( x L ) , t ) , ω ( x , t ) = ω ˜ ( η 1 ( x L ) , t ) , θ ( x , t ) = θ ˜ ( η 1 ( x L ) , t ) , r ( x , t ) = r ( η 1 ( x L ) , t ) , ρ 0 ( x ) = ρ 0 ( η 1 ( x L ) ) , v 0 ( x ) = v 0 ( η 1 ( x L ) ) , ω 0 ( x ) = ω 0 ( η 1 ( x L ) ) , θ 0 ( x ) = θ 0 ( η 1 ( x L ) ) , r 0 ( x ) = r 0 ( η 1 ( x L ) ) = η 1 ( x L ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equc_HTML.gif
      Similarly as in [15], for a new coordinate x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq45_HTML.gif we get
      0 = L 1 η ( a ) x L 1 η ( b ) = 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ29_HTML.gif
      (29)
      Taking into account (26) and (24), we obtain that the functions ρ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq46_HTML.gif, v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq47_HTML.gif, ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq48_HTML.gif, θ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq49_HTML.gif and r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq50_HTML.gif satisfy the system that we write omitting the primes for simplicity:
      ρ t = 1 L ρ 2 x ( r 2 v ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ30_HTML.gif
      (30)
      v t = R L r 2 x ( ρ θ ) + λ + 2 μ L 2 r 2 x ( ρ x ( r 2 v ) ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ31_HTML.gif
      (31)
      ρ ω t = 4 μ r j I ω + c 0 + 2 c d j I L 2 r 2 ρ x ( ρ x ( r 2 ω ) ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ32_HTML.gif
      (32)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ33_HTML.gif
      (33)
      in ] 0 , 1 [ × ] 0 , T [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq51_HTML.gif, where T > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq20_HTML.gif is arbitrary. Now we have the following boundary and initial conditions
      v ( 0 , t ) = v ( 1 , t ) = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ34_HTML.gif
      (34)
      ω ( 0 , t ) = ω ( 1 , t ) = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ35_HTML.gif
      (35)
      θ x ( 0 , t ) = θ x ( 1 , t ) = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ36_HTML.gif
      (36)
      for t ] 0 , T [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq52_HTML.gif,
      ρ ( x , 0 ) = ρ 0 ( x ) , v ( x , 0 ) = v 0 ( x ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ37_HTML.gif
      (37)
      ω ( x , 0 ) = ω 0 ( x ) , θ ( x , 0 ) = θ 0 ( x ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ38_HTML.gif
      (38)
      for x ] 0 , 1 [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq53_HTML.gif. We also have
      r ( x , t ) = r 0 ( x ) + 0 t v ( x , τ ) d τ , ( x , t ) ] 0 , 1 [ × ] 0 , T [ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ39_HTML.gif
      (39)
      From
      r x ( x , t ) = L ρ ( x , t ) r 2 ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equd_HTML.gif
      putting t = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq54_HTML.gif and integrating over ] 0 , x [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq55_HTML.gif, we get
      r 0 ( x ) = ( a 3 + 3 L 0 x 1 ρ 0 ( y ) d y ) 1 3 , x ] 0 , 1 [ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ40_HTML.gif
      (40)

      where a > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq17_HTML.gif is a radius of the smaller boundary sphere.

      We assume the inequalities
      ρ 0 ( x ) m , θ 0 ( x ) m for  x ] 0 , 1 [ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ41_HTML.gif
      (41)

      where m R + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq56_HTML.gif.

      Before stating the main result, we introduce the following definition.

      Definition 2.1 A generalized solution of the problem (30)-(38) in the domain Q T = ] 0 , 1 [ × ] 0 , T [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq57_HTML.gif is a function
      ( x , t ) ( ρ , v , ω , θ ) ( x , t ) , ( x , t ) Q T , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ42_HTML.gif
      (42)
      where
      ρ L ( 0 , T ; H 1 ( ] 0 , 1 [ ) ) H 1 ( Q T ) , inf Q T ρ > 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ43_HTML.gif
      (43)
      v , ω , θ L ( 0 , T ; H 1 ( ] 0 , 1 [ ) ) H 1 ( Q T ) L 2 ( 0 , T ; H 2 ( ] 0 , 1 [ ) ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ44_HTML.gif
      (44)

      that satisfies Equations (30)-(33) a.e. in Q T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq58_HTML.gif and conditions (34)-(38) in the sense of traces.

      Remark 2.1 From the embedding and interpolation theorems (e.g., [16] and [17]) one can conclude that from (43) and (44) it follows:
      ρ L ( 0 , T ; C ( [ 0 , 1 ] ) ) C ( [ 0 , T ] , L 2 ( ] 0 , 1 [ ) ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ45_HTML.gif
      (45)
      v , ω , θ L 2 ( 0 , T ; C ( 1 ) ( [ 0 , 1 ] ) ) C ( [ 0 , T ] , H 1 ( ] 0 , 1 [ ) ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ46_HTML.gif
      (46)
      v , ω , θ C ( Q ¯ T ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ47_HTML.gif
      (47)

      It is easy to check that the solution (42) with properties (43)-(44) satisfies the condition for a strong solution of the described problem.

      The aim of this paper is to prove the following statements.

      Theorem 2.1 Let the functions
      ρ 0 , θ 0 H 1 ( ] 0 , 1 [ ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ48_HTML.gif
      (48)
      v 0 , ω 0 H 0 1 ( ] 0 , 1 [ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ49_HTML.gif
      (49)
      satisfy conditions (41). Then there exists T 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq59_HTML.gif, 0 < T 0 T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq60_HTML.gif, such that the problem (30)-(38) has a generalized solution in Q 0 = Q T 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq61_HTML.gif, having the property
      θ > 0 in  Q ¯ 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ50_HTML.gif
      (50)
      For the function r, it holds
      r L ( 0 , T ; H 2 ( ] 0 , 1 [ ) ) H 2 ( Q 0 ) C ( Q ¯ 0 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ51_HTML.gif
      (51)
      a 2 r 2 M in  Q ¯ 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ52_HTML.gif
      (52)
      Remark 2.2 Notice that the function r 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq62_HTML.gif introduced by (40) belongs to H 2 ( ] 0 , 1 [ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq63_HTML.gif. Because of the embedding H 1 ( ] 0 , 1 [ ) C ( [ 0 , 1 ] ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq64_HTML.gif we can conclude that there exists M R + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq65_HTML.gif such that
      ρ 0 ( x ) , v 0 ( x ) , ω 0 ( x ) , θ 0 ( x ) M , x [ 0 , 1 ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ53_HTML.gif
      (53)
      From (40) and (41) we get
      r 0 C ( 1 ) ( [ 0 , 1 ] ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ54_HTML.gif
      (54)
      0 < a r 0 ( x ) M , 0 < a 1 r 0 ( x ) M 1 , x [ 0 , 1 ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ55_HTML.gif
      (55)

      where a 1 = L M 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq66_HTML.gif and M 1 = L ( m a 2 ) 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq67_HTML.gif.

      The proof of Theorem 2.1 is essentially based on a careful examination of a priori estimates and a limit procedure. We first study, for each n N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq25_HTML.gif, an approximate problem and derive the a priori estimates for approximate solutions independent of n by utilizing a technique of Kazhikov [14, 18] and Mujakovic [3, 13] for one-dimensional case. Using the obtained a priori estimates and results of weak compactness, we extract the subsequence of approximate solutions, which, when n tends to infinity, has limit in the same weak sense on ] 0 , 1 [ × ] 0 , T 0 [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq68_HTML.gif for sufficiently small T 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq59_HTML.gif, 0 < T 0 T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq60_HTML.gif. Finally, we show that this limit is the solution to our problem.

      3 Approximate solutions

      We shall find a local generalized solution to the problem (30)-(38) as a limit of approximate solutions
      ( ρ n , v n , ω n , θ n ) , n N , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ56_HTML.gif
      (56)
      obtained in what follows. First, we introduce the approximations v n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq69_HTML.gif and r n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq70_HTML.gif of the functions v and r by
      v n ( x , t ) = i = 1 n v i n ( t ) sin ( π i x ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ57_HTML.gif
      (57)
      r n ( x , t ) = r 0 ( x ) + 0 t v n ( x , τ ) d τ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ58_HTML.gif
      (58)

      where r 0 ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq71_HTML.gif is defined by (40) and v i n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq72_HTML.gif, i = 1 , 2 , , n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq73_HTML.gif are unknown smooth functions defined on an interval [ 0 , T n ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq74_HTML.gif, T n T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq75_HTML.gif.

      Then, we can write the solution ρ n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq76_HTML.gif to the problem
      ρ n t + L 1 ( ρ n ) 2 x ( ( r n ) 2 v n ) = 0 , ρ n ( x , 0 ) = ρ 0 ( x ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ59_HTML.gif
      (59)
      in the similar way as in [3] and [13] in the form
      ρ n ( x , t ) = L ρ 0 ( x ) L + ρ 0 ( x ) x 0 t ( r n ) 2 v n d τ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ60_HTML.gif
      (60)
      Since r n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq70_HTML.gif and v n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq69_HTML.gif are sufficiently smooth functions, we can conclude that the function ρ n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq76_HTML.gif is continuous on the rectangle [ 0 , 1 ] × [ 0 , T n ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq77_HTML.gif with the property ρ n ( x , 0 ) = ρ 0 ( x ) m > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq78_HTML.gif. Because of aforementioned, we can conclude that there exists such T n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq79_HTML.gif, 0 < T n T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq80_HTML.gif that
      ρ n ( x , t ) > 0 , for  ( x , t ) [ 0 , 1 ] × [ 0 , T n ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ61_HTML.gif
      (61)
      We also introduce the approximations ω n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq81_HTML.gif and θ n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq82_HTML.gif of the functions ω and θ respectively by
      ω n ( x , t ) = j = 1 n ω j n ( t ) sin ( π j x ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ62_HTML.gif
      (62)
      θ n ( x , t ) = k = 0 n θ k n ( t ) cos ( π k x ) ; http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ63_HTML.gif
      (63)

      where ω j n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq83_HTML.gif and θ k n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq84_HTML.gif are again unknown smooth functions defined on an interval [ 0 , T n ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq74_HTML.gif, T n T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq75_HTML.gif.

      Evidently, the boundary conditions
      v n ( 0 , t ) = v n ( 1 , t ) = ω n ( 0 , t ) = ω n ( 1 , t ) = θ n x ( 0 , t ) = θ n x ( 1 , t ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ64_HTML.gif
      (64)

      for t ] 0 , T n [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq85_HTML.gif are satisfied.

      According to the Faedo-Galerkin method, we take the following approximate conditions:
      0 1 [ v n t + R L ( r n ) 2 x ( ρ n θ n ) λ + 2 μ L 2 ( r n ) 2 x ( ρ n x ( ( r n ) 2 v n ) ) ] sin ( π i x ) d x = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ65_HTML.gif
      (65)
      0 1 [ ω n t + 4 μ r j I ω n ρ n c 0 + 2 c d j I L 2 ( r n ) 2 x ( ρ n x ( ( r n ) 2 ω n ) ) ] sin ( π j x ) d x = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ66_HTML.gif
      (66)
      0 1 [ θ n t k c v L 2 x ( ( r n ) 4 ρ n θ n x ) + R c v L ρ n θ n x ( ( r n ) 2 v n ) λ + 2 μ c v L 2 ρ n [ x ( ( r n ) 2 v n ) ] 2 + 4 μ c v L x ( r n ( v n ) 2 ) c 0 + 2 c d c v L 2 ρ n [ x ( ( r n ) 2 ω n ) ] 2 + 4 c d c v L x ( r n ( ω n ) 2 ) 4 μ r c v ( ω n ) 2 ρ n ] cos ( π k x ) d x = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ67_HTML.gif
      (67)

      for i , j = 1 , , n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq86_HTML.gif, k = 0 , 1 , , n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq87_HTML.gif.

      Let v 0 i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq88_HTML.gif, ω 0 j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq89_HTML.gif, and θ 0 k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq90_HTML.gif be the Fourier coefficients of the functions v 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq31_HTML.gif, ω 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq32_HTML.gif, and θ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq33_HTML.gif respectively:
      v 0 i = 2 0 1 v 0 ( x ) sin ( π i x ) d x , i = 1 , , n , ω 0 j = 2 0 1 ω 0 ( x ) sin ( π j x ) d x , j = 1 , , n , θ 00 = 0 1 θ 0 ( x ) d x , θ 0 k = 2 0 1 θ 0 ( x ) cos ( π k x ) d x , k = 1 , , n . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Eque_HTML.gif
      Let v 0 n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq91_HTML.gif, ω 0 n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq92_HTML.gif and θ 0 n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq93_HTML.gif be
      v 0 n ( x ) = i = 1 n v 0 i sin ( π i x ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ68_HTML.gif
      (68)
      ω 0 n ( x ) = j = 1 n ω 0 j sin ( π j x ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ69_HTML.gif
      (69)
      θ 0 n ( x ) = k = 0 n θ 0 k cos ( π k x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ70_HTML.gif
      (70)
      We take the initial conditions for v n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq69_HTML.gif, ω n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq81_HTML.gif and θ n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq82_HTML.gif in the form
      v n ( x , 0 ) = v 0 n ( x ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ71_HTML.gif
      (71)
      ω n ( x , 0 ) = ω 0 n ( x ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ72_HTML.gif
      (72)
      θ n ( x , 0 ) = θ 0 n ( x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ73_HTML.gif
      (73)
      Let z m n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq94_HTML.gif, λ p q n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq95_HTML.gif and μ s l g n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq96_HTML.gif be
      z m n ( t ) = 0 t v m n ( τ ) d τ , m = 1 , , n , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ74_HTML.gif
      (74)
      λ p q n ( t ) = 0 t z p n ( τ ) v q n ( τ ) d τ , p , q = 1 , , n , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ75_HTML.gif
      (75)
      μ s l g n ( t ) = 0 t z l n ( τ ) z s n ( τ ) v g n ( τ ) d τ , s , l , g = 1 , , n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ76_HTML.gif
      (76)
      then we have
      r n ( x , t ) = r 0 ( x ) + m = 1 n z m n ( t ) sin ( π m x ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ77_HTML.gif
      (77)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ78_HTML.gif
      (78)
      where r 0 ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq71_HTML.gif and ρ 0 ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq97_HTML.gif are known functions. Taking into account (57), (62), (63), (74)-(78), from (65)-(67) we obtain for { ( v i n , ω j n , θ k n , z m n , λ p q n , μ s l g n ) : i , j , m , p , q , s , l , g = 1 , , n , k = 0 , 1 , , n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq98_HTML.gif, the following Cauchy problem:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ79_HTML.gif
      (79)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ80_HTML.gif
      (80)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ81_HTML.gif
      (81)
      z ˙ m n ( t ) = v m n , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ82_HTML.gif
      (82)
      λ ˙ p q n ( t ) = z p n v q n , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ83_HTML.gif
      (83)
      μ ˙ s l g n ( t ) = z s n z l n v g n , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ84_HTML.gif
      (84)
      v i n ( 0 ) = v 0 i , ω j n ( 0 ) = ω 0 j , θ k n ( 0 ) = θ 0 k , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ85_HTML.gif
      (85)
      z m n ( 0 ) = 0 , λ p q n ( 0 ) = 0 , μ s l g n ( 0 ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ86_HTML.gif
      (86)
      Here we have λ 0 = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq99_HTML.gif, λ k = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq100_HTML.gif for k = 1 , 2 , , n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq101_HTML.gif and
      Φ i n = 2 0 1 [ λ + 2 μ L 2 ( r n ) 2 x ( ρ n x ( ( r n ) 2 v n ) ) R L ( r n ) 2 x ( ρ n θ n ) ] sin ( π i x ) d x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ87_HTML.gif
      (87)
      Ψ j n = 2 0 1 [ c 0 + 2 c d j I L 2 ( r n ) 2 x ( ρ n x ( ( r n ) 2 ω n ) ) 4 μ r j I ω n ρ n ] sin ( π j x ) d x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ88_HTML.gif
      (88)
      Π k n = 0 1 [ k c v L 2 x ( ( r n ) 4 ρ n θ n x ) R c v L ρ n θ n x ( ( r n ) 2 v n ) + λ + 2 μ c v L 2 ρ n [ x ( ( r n ) 2 v n ) ] 2 4 μ c v L x ( r n ( v n ) 2 ) + c 0 + 2 c d c v L 2 ρ n [ x ( ( r n ) 2 ω n ) ] 2 4 c d c v L x ( r n ( ω n ) 2 ) + 4 μ r c v ( ω n ) 2 ρ n ] cos ( π k x ) d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ89_HTML.gif
      (89)

      Notice that the functions on the right-hand side of (79)-(84) satisfy the conditions of the Cauchy-Picard theorem [19, 20] and we can easily conclude that the following statements are valid.

      Lemma 3.1 For each n N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq102_HTML.gif there exists such T n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq79_HTML.gif, 0 < T n T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq80_HTML.gif that the Cauchy problem (79)-(86) has a unique solution defined on [ 0 , T n ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq74_HTML.gif. The functions v n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq69_HTML.gif, ω n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq81_HTML.gif and θ n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq82_HTML.gif defined by the formulas (57), (62) and (63) belong to the class C ( Q ¯ n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq103_HTML.gif, Q n = ] 0 , 1 [ × ] 0 , T n [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq104_HTML.gif and satisfy conditions (71)-(73).

      From (77) and (78) we can also easily conclude that
      ρ n C ( Q ¯ n ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ90_HTML.gif
      (90)
      r n C ( 1 ) ( Q ¯ n ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ91_HTML.gif
      (91)
      Lemma 3.2 There exists T n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq79_HTML.gif, 0 < T n T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq80_HTML.gif, such that the functions ρ n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq76_HTML.gif, r n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq70_HTML.gif and r n x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq105_HTML.gif satisfy the conditions
      m 2 ρ n ( x , t ) 2 M , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ92_HTML.gif
      (92)
      a 2 r n ( x , t ) 2 M , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ93_HTML.gif
      (93)
      a 1 2 r n x ( x , t ) 2 M 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ94_HTML.gif
      (94)

      on Q ¯ n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq106_HTML.gif. The constants m, a, a 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq107_HTML.gif, M and M 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq108_HTML.gif are introduced by (40), (41), (53) and (55).

      Proof The statements follow from (90)-(91), (41), (53) and (55). □

      4 A priori estimates

      Our purpose is to find T 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq109_HTML.gif, 0 < T 0 T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq60_HTML.gif such that for each n N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq102_HTML.gif there exists a solution to the problem (79)-(86), defined on [ 0 , T 0 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq110_HTML.gif. It will be sufficient to find uniform (in n N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq102_HTML.gif) a priori estimates for the solution ( ρ n , v n , ω n , θ n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq111_HTML.gif defined through Lemmas 3.1 and 3.2.

      In what follows we denote by C > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq112_HTML.gif or C i > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq113_HTML.gif ( i = 1 , 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq114_HTML.gif) a generic constant, not depending on n N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq25_HTML.gif, which may have different values at different places.

      We also use the notation
      f = f L 2 ( ] 0 , 1 [ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equf_HTML.gif

      Some of our considerations are very similar or identical to that of [3] or [13]. In these cases we omit proofs or details of proofs making references to corresponding pages of the articles [3] or [13].

      Lemma 4.1 For t [ 0 , T n ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq115_HTML.gif it holds
      2 r n x 2 ( t ) 2 C ( 1 + 0 t 2 v n x 2 ( τ ) 2 d τ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ95_HTML.gif
      (95)
      Proof From (58) follows
      2 r n x 2 = r 0 + 0 t 2 v n x 2 d τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equg_HTML.gif

      and using Remark 2.2 we get (95) immediately. □

      Lemma 4.2 For t [ 0 , T n ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq115_HTML.gif, the following inequality holds:
      ω n ( t ) 2 + 0 t ( ω n ( τ ) 2 + x ( r n 2 ω n ) ( τ ) 2 d τ ) C . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ96_HTML.gif
      (96)
      Proof Multiplying (66) by ω j n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq116_HTML.gif and summing over j = 1 , , n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq117_HTML.gif, after integration by parts, we obtain
      1 2 d d t ω n ( t ) 2 + 4 μ r j I 0 1 ( ω n ) 2 ρ n d x + c 0 + 2 c d j I L 2 0 1 ρ n [ x ( ( r n ) 2 ω n ) ] 2 d x = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equh_HTML.gif
      Integrating over [ 0 , t ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq118_HTML.gif, 0 < t T n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq119_HTML.gif, and taking into account (72), we obtain
      1 2 ω n ( t ) 2 + 4 μ r j I 0 t 0 1 ( ω n ) 2 ρ n d x d τ + c 0 + 2 c d j I L 2 0 t 0 1 ρ n [ x ( ( r n ) 2 ω n ) ] 2 d x d τ = 1 2 ω 0 n 2 1 2 ω 0 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equi_HTML.gif

      Using (92), we get (96). □

      In what follows, we use the inequalities
      | f | 2 2 f f x , | f x | 2 2 f x 2 f x 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ97_HTML.gif
      (97)

      (for a function f vanishing at x = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq120_HTML.gif and x = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq121_HTML.gif and with the first derivative vanishing at some point x [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq122_HTML.gif) that satisfy the functions v n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq69_HTML.gif, ω n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq81_HTML.gif and θ n x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq123_HTML.gif.

      Lemma 4.3 For t [ 0 , T n ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq124_HTML.gif, the following inequality holds:
      | 0 1 θ n ( x , t ) d x | C ( 1 + v n x ( t ) 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ98_HTML.gif
      (98)
      Proof Multiplying (65) by c v 1 v i n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq125_HTML.gif and summing over i = 1 , , n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq126_HTML.gif, after integration by parts and adding to (67) for k = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq127_HTML.gif, we have
      d d t ( 1 2 c v v n ( t ) 2 + 0 1 θ n ( x , t ) d x ) = c 0 + 2 c d c v L 2 0 1 ρ n [ x ( ( r n ) 2 ω n ) ] 2 d x + 4 μ r c v 0 1 ( ω n ) 2 ρ n d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equj_HTML.gif
      Integrating over [ 0 , t ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq118_HTML.gif, 0 < t T n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq119_HTML.gif and using (92) we get
      ( 1 2 c v v n ( t ) 2 + 0 1 θ n ( x , t ) d x ) C 0 t ( ω n ( t ) 2 + x ( ( r n ) 2 ω n ) ( τ ) 2 ) d τ + 1 2 c v v 0 n ( t ) 2 + θ 0 n ( t ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equk_HTML.gif

      Taking into account (96), (71), (73), and (97) we obtain (98). □

      Lemma 4.4 ([3], Lemma 5.3)

      For ( x , t ) Q ¯ n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq128_HTML.gif, the following inequality holds:
      | θ n ( x , t ) | C ( 1 + θ n x ( t ) + v n x ( t ) 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ99_HTML.gif
      (99)
      Lemma 4.5 For t [ 0 , T n ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq115_HTML.gif, the following inequality holds:
      ρ n x ( t ) 2 C ( 1 + ( 0 t 2 v n x 2 ( τ ) 2 d τ ) 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ100_HTML.gif
      (100)
      Proof Taking the derivative of the function ρ n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq76_HTML.gif with respect to x and using the estimates (92)-(94), we obtain
      | ρ n x | C ( 1 + 0 t ( | v n | + | v n | | 2 r n x 2 | + | v n x | + | 2 v n x 2 | ) d τ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equl_HTML.gif

      With the help of (97) applied to the function v n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq69_HTML.gif, the Hoelder and Young inequalities as well as (95), we get (100). □

      Lemma 4.6 For t [ 0 , T n ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq115_HTML.gif it holds
      d d t ( v n x ( t ) 2 + ω n x ( t ) 2 + θ n x ( t ) 2 ) + B ( 2 v n x 2 ( t ) 2 + 2 ω n x 2 ( t ) 2 + 2 θ n x 2 ( t ) 2 ) C ( 1 + v n x ( t ) 16 + ω n x ( t ) 16 + θ n x ( t ) 16 + ( B 0 t 2 v n x 2 ( τ ) 2 d τ ) 8 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ101_HTML.gif
      (101)
      where
      B = m a 4 32 min { λ + 2 μ L 2 , c 0 + 2 c d j I L 2 , k c v L 2 } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equm_HTML.gif
      Proof As in [3] Lemma 5.5, [14] pp.63-66 and in [13] Lemma 5.6, multiplying (65), (66) and (67) respectively by ( π i ) 2 v i n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq129_HTML.gif, ( π j ) 2 ω j n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq130_HTML.gif and ( π k ) 2 θ k n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq131_HTML.gif and taking into account (57), (62) and (63), after summation over i , j , k = 1 , 2 , , n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq132_HTML.gif and addition of the obtained equations, we get
      1 2 d d t ( v n x 2 + ω n x 2 + θ n x 2 ) + λ + 2 μ L 2 0 1 ρ n ( r n ) 4 ( 2 v n x 2 ) 2 d x + c 0 + 2 c d j I L 2 0 1 ρ n ( r n ) 4 ( 2 ω n x 2 ) 2 d x + k c v L 2 0 1 ρ n ( r n ) 4 ( 2 θ n x 2 ) 2 d x = p = 1 28 I p ( t ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ102_HTML.gif
      (102)
      where
      I 1 = 2 ( λ + 2 μ ) L 2 0 1 ( r n ) 3 ρ n x r n x v n 2 v n x 2 d x , I 2 = 2 ( λ + 2 μ ) L 2 0 1 ρ n ( r n ) 2 ( r n x ) 2 v n 2 v n x 2 d x , I 3 = 2 ( λ + 2 μ ) L 2 0 1 ρ n ( r n ) 3 2 r n x 2 v n 2 v n x 2 d x , I 4 = 4 ( λ + 2 μ ) L 2 0 1 ρ n ( r n ) 3 r n x v n x 2 v n x 2 d x , I 5 = λ + 2 μ L 2 0 1 ( r n ) 4 ρ n x v n x 2 v n x 2 d x , I 6 = R L 0 1 ( r n ) 2 θ n ρ n x 2 v n x 2 d x , I 7 = R L 0 1 ( r n ) 2 ρ n θ n x 2 v n x 2 d x , I 8 = 4 μ r j I 0 1 ω n ρ n 2 ω n x 2 d x , I 9 = 2 ( c 0 + 2 c d ) j I L 2 0 1 ( r n ) 3 ρ n x r n x ω n 2 ω n x 2 d x , I 10 = 2 ( c 0 + 2 c d ) j I L 2 0 1 ρ n ( r n ) 2 ( r n x ) 2 ω n 2 ω n x 2 d x , I 11 = 2 ( c 0 + 2 c d ) j I L 2 0 1 ρ n ( r n ) 3 2 r n x 2 ω n 2 ω n x 2 d x , I 12 = 4 ( c 0 + 2 c d ) j I L 2 0 1 ρ n ( r n ) 3 r n x ω n x 2 ω n x 2 d x , I 13 = c 0 + 2 c d j I L 2 0 1 ( r n ) 4 ρ n x ω n x 2 ω n x 2 d x , I 14 = 4 k c v L 2 0 1 ρ n ( r n ) 3 r n x θ n x 2 θ n x 2 d x , I 15 = k c v L 2 0 1 ( r n ) 4 ρ n x θ n x 2 θ n x 2 d x , I 16 = 2 R c v L 0 1 ρ n r n r n x v n θ n 2 θ n x 2 d x , I 17 = R c v L 0 1 ρ n ( r n ) 2 θ n v n x 2 θ n x 2 d x , I 18 = 4 ( λ + 2 μ ) c v L 2 0 1 ρ n ( r n ) 2 ( r n x ) 2 ( v n ) 2 2 θ n x 2 d x , I 19 = 4 ( λ + 2 μ ) c v L 2 0 1 ρ n ( r n ) 3 r n x v n v n x 2 θ n x 2 d x , I 20 = λ + 2 μ c v L 2 0 1 ρ n ( r n ) 4 ( v n x ) 2 2 θ n x 2 d x , I 21 = 4 μ c v L 0 1 ( v n ) 2 r n x 2 θ n x 2 d x , I 22 = 8 μ c v L 0 1 r n v n v n x 2 θ n x 2 d x , I 23 = 4 ( c 0 + 2 c d ) c v L 2 0 1 ρ n ( r n ) 2 ( r n x ) 2 ( ω n ) 2 2 θ n x 2 d x , I 24 = 4 ( c 0 + 2 c d ) c v L 2 0 1 ρ n ( r n ) 3 r n x ω n x ω n 2 θ n x 2 d x , I 25 = c 0 + 2 c d c v L 2 0 1 ρ n ( r n ) 4 ( ω n x ) 2 2 θ n x 2 d x , I 26 = 4 c d c v L 0 1 ( ω n ) 2 r n x 2 θ n x 2 d x , I 27 = 8 c d c v L 0 1 r n ω n ω n x 2 θ n x 2 d x , I 28 = 4 μ r c v 0 1 ( ω n ) 2 ρ n 2 θ n x 2 d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equn_HTML.gif
      Taking into account (92)-(94) and (95)-(100), we estimate the terms on the right-hand side of (102). For instance,
      | I 1 | = 2 ( λ + 2 μ ) L 2 | 0 1 ( r n ) 3 ρ n x r n x v n 2 v n x 2 d x | C max x [ 0 , 1 ] | v n ( x , t ) | ρ n x ( t ) 2 v n x 2 ( t ) C v n x ( t ) ρ n x ( t ) 2 v n x 2 ( t ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equo_HTML.gif
      Applying the Young inequality, we get
      | I 1 | ε 2 v n x 2 ( t ) 2 + C ( v n x ( t ) 4 + ρ n x ( t ) 4 ) ε 2 v n x 2 ( t ) 2 + C ( 1 + v n x ( t ) 16 + ( 0 t 2 v n x 2 ( τ ) 2 d τ ) 8 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equp_HTML.gif
      where ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq133_HTML.gif is arbitrary. In the analogous way, we obtain the following inequalities:
      | I 2 | ε 2 v n x 2 ( t ) 2 + C ( 1 + v n x ( t ) 16 ) , | I 3 | ε 2 v n x 2 ( t ) 2 + C ( 1 + v n x ( t ) 16 + ( 0 t 2 v n x 2 ( τ ) 2 d τ ) 8 ) , | I 4 | ε 2 v n x 2 ( t ) 2 + C ( 1 + v n x ( t ) 16 ) , | I 5 | ε 2 v n x 2 ( t ) 2 + C ( 1 + v n x ( t ) 16 + ( 0 t 2 v n x 2 ( τ ) d τ ) 8 ) , | I 6 | ε 2 v n x 2 ( t ) 2 + C ( 1 + θ n x ( t ) 16 + v n x ( t ) 16 + ( 0 t 2 v n x 2 ( τ ) 2 d τ ) 8 ) , | I 7 | ε 2 v n x 2 ( t ) 2 + C ( 1 + θ n x ( t ) 16 ) , | I 8 | ε 2 ω n x 2 ( t ) 2 + C , | I 9 | ε 2 ω n x 2 ( t ) 2 + C ( 1 + ω n x ( t ) 16 + ( 0 t 2 v n x 2 ( τ ) 2 d τ ) 8 ) , | I 10 | ε 2 ω n x 2 ( t ) 2 + C , | I 11 | ε 2 ω n x 2 ( t ) 2 + C ( 1 + ω n x ( t ) 16 + ( 0 t 2 v n x 2 ( τ ) 2 d τ ) 8 ) , | I 12 | ε 2 ω n x 2 ( t ) 2 + C ( 1 + ω n x ( t ) 16 ) , | I 13 | ε 2 ω n x 2 ( t ) 2 + C ( 1 + ω n x ( t ) 16 + ( 0 t 2 v n x 2 ( τ ) 2 d τ ) 8 ) , | I 14 | ε 2 θ n x 2 ( t ) 2 + C ( 1 + θ n x ( t ) 16 ) , | I 15 | ε 2 θ n x 2 ( t ) 2 + C ( 1 + θ n x ( t ) 16 + ( 0 t 2 v n x 2 ( τ ) 2 d τ ) 8 ) , | I 16 | ε 2 θ n x 2 ( t ) 2 + C ( 1 + θ n x ( t ) 16 + v n x ( t ) 16 ) , | I 17 | ε 2 θ n x 2 ( t ) 2 + C ( 1 + θ n x ( t ) 16 + v n x ( t ) 16 ) , | I 18 | ε 2 θ n x 2 ( t ) 2 + C ( 1 + v n x ( t ) 16 ) , | I 19 | ε 2 θ n x 2 ( t ) 2 + C ( 1 + v n x ( t ) 16 ) , | I 20 | ε 2 θ n x 2 ( t ) 2 + ε 2 v n x 2 ( t ) 2 + C ( 1 + v n x ( t ) 16 ) , | I 21 | ε 2 θ n x 2 ( t ) 2 + C ( 1 + v n x ( t ) 16 ) , | I 22 | ε 2 θ n x 2 ( t ) 2 + C ( 1 + v n x ( t ) 16 ) , | I 23 | ε 2 θ n x 2 ( t ) 2 + C ( 1 + ω n x ( t ) 16 ) , | I 24 | ε 2 θ n x 2 ( t ) 2 + C ( 1 + ω n x ( t ) 16 ) , | I 25 | ε 2 θ n x 2 ( t ) 2 + ε 2 ω n x 2 ( t ) 2 + C ( 1 + ω n x ( t ) 16 ) , | I 26 | ε 2 θ n x 2 ( t ) 2 + C ( 1 + ω n x ( t ) 16 ) , | I 27 | ε 2 θ n x 2 ( t ) 2 + C ( 1 + ω n x ( t ) 16 ) , | I 28 | ε 2 θ n x 2 ( t ) 2 + C ( 1 + ω n x ( t ) 16 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equq_HTML.gif

      Using these inequalities with sufficiently small ε and estimates (92)-(94), from (102) we get (101). □

      Lemma 4.7 There exists T 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq59_HTML.gif, ( 0 < T 0 T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq60_HTML.gif) such that for each n N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq25_HTML.gif the Cauchy problem (79)-(86) has a unique solution defined on [ 0 , T 0 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq110_HTML.gif. Moreover, the functions v n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq69_HTML.gif, ω n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq81_HTML.gif, θ n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq82_HTML.gif, ρ n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq76_HTML.gif and r n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq70_HTML.gif satisfy the inequalities
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ103_HTML.gif
      (103)
      a 2 r n ( x , t ) 2 M , a 1 2 r n x ( x , t ) 2 M 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ104_HTML.gif
      (104)
      m 2 ρ n ( x , t ) 2 M , ( x , t ) Q ¯ 0 , Q 0 = Q T 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ105_HTML.gif
      (105)
      max t [ 0 , T 0 ] ρ n x ( t ) C , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ106_HTML.gif
      (106)
      max t [ 0 , T 0 ] 2 r n x 2 ( t ) C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ107_HTML.gif
      (107)

      (a, a 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq107_HTML.gif, m and M are defined by (41) and (53)-(55)).

      Proof To get the estimate (103) we use an approach similar to that in [3] (Lemma 5.6) and [14] (pp.64-67). First, we introduce the function
      y n ( t ) = v n x ( t ) 2 + ω n x ( t ) 2 + θ n x ( t ) 2 + B 0 t 2 v n x 2 ( τ ) 2 d τ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ108_HTML.gif
      (108)
      Using Lemma 4.6, we find that the function y n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq134_HTML.gif satisfies the differential inequality
      y ˙ n ( t ) C ( 1 + y n 8 ( t ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ109_HTML.gif
      (109)
      There exists a constant C ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq135_HTML.gif such that
      C ¯ = d v 0 d x 2 + d ω 0 d x 2 + d θ 0 d x 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equr_HTML.gif
      and we can conclude that
      y n ( 0 ) = d v 0 n d x 2 + d ω 0 n d x 2 + d θ 0 n d x 2 C ¯ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ110_HTML.gif
      (110)
      We compare the solution of the problem (109)-(110) with the solution of the Cauchy problem
      y ˙ ( t ) = C ( 1 + y 8 ( t ) ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ111_HTML.gif
      (111)
      y ( 0 ) = C ¯ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ112_HTML.gif
      (112)
      Let [ 0 , T [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq136_HTML.gif, 0 < T T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq137_HTML.gif be an existence interval of the solution of the problem (111)-(112). From (109)-(112) we conclude that
      y n ( t ) y ( t ) , t [ 0 , T [ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ113_HTML.gif
      (113)
      Let T 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq59_HTML.gif be such that 0 < T 0 T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq138_HTML.gif. From (113) and (108) we obtain
      max t [ 0 , T 0 ] ( v n x ( t ) 2 + ω n x ( t ) 2 + θ n x ( t ) 2 ) + 0 T 0 2 v n x 2 ( τ ) 2 d τ C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ114_HTML.gif
      (114)
      and from (101) it follows
      d d t ( v n x ( t ) 2 + ω n x ( t ) 2 + θ n x ( t ) 2 ) + B ( 2 v n x 2 ( t ) 2 + 2 ω n x 2 ( t ) 2 + 2 θ n x 2 ( t ) 2 ) C . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ115_HTML.gif
      (115)

      Integrating (101) over [ 0 , t ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq118_HTML.gif, 0 < t T 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq139_HTML.gif and using estimates (110) and (115), we immediately get (103).

      Now, using the inequalities (97) for the function v, we easily get
      | v n ( x , t ) | 2 v n x ( t ) , v n x ( t ) 2 2 v n x 2 ( t ) , | v n x ( t ) | 2 2 v n x 2 ( t ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ116_HTML.gif
      (116)
      Using (116), we derive the following estimates:
      0 T 0 | v n ( x , t ) | d τ 4 0 T 0 2 v n x 2 ( t ) d τ 4 ( C B 1 ) 1 2 T 0 1 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ117_HTML.gif
      (117)
      0 T 0 | v n x ( t ) | d τ 2 0 T 0 2 v n x 2 ( t ) d τ 2 ( C B 1 ) 1 2 T 0 1 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ118_HTML.gif
      (118)
      where C and B are from (103). Assuming that
      T 0 < min { T , a 2 B 64 C , a 1 2 B 16 C , ( L B 1 2 16 M 2 ( 4 M 1 + M ) C 1 2 ) 2 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equs_HTML.gif

      and using (117) and (118) from (58) and (60), we get (104)-(105).

      Because of (57), (62) and (63), from (103) and (98), we easily get that for t [ 0 , T 0 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq140_HTML.gif hold
      i = 1 n ( | v i n ( t ) | + | ω i n ( t ) | + | θ i n ( t ) | ) C , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ119_HTML.gif
      (119)
      | θ 0 n ( t ) | C , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ120_HTML.gif
      (120)

      and we can conclude that the solution of the problem (79)-(86) is defined on [ 0 , T 0 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq110_HTML.gif. □

      From (119) and (120), we can easily conclude that
      max t [ 0 , T 0 ] ( v n ( t ) 2 + ω n ( t ) 2 + θ n ( t ) 2 ) C , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ121_HTML.gif
      (121)
      and from (95), (100) and (99) it follows
      max t [ 0 , T 0 ] 2 r n x 2 ( t ) C , max t [ 0 , T 0 ] 2 ρ n x 2 ( t ) C , max ( x , t ) Q ¯ 0 | θ n ( t ) | C . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ122_HTML.gif
      (122)
      Lemma 4.8 Let T 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq59_HTML.gif be defined by Lemma  4.7. Then for each n N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq141_HTML.gif the inequalities
      0 T 0 ( v n t ( τ ) 2 + ω n t ( τ ) 2 + θ n t ( τ ) 2 ) d τ C , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ123_HTML.gif
      (123)
      max t [ 0 , T 0 ] ρ n t ( t ) C , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ124_HTML.gif
      (124)
      max t [ 0 , T 0 ] r n t ( t ) C , max t [ 0 , T 0 ] 2 r n x t ( t ) C , 0 T 0 2 r n t 2 ( τ ) 2 d τ C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ125_HTML.gif
      (125)

      hold true.

      Proof Multiplying (65) by d v i n d t ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq142_HTML.gif, summing over i = 1 , 2 , , n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq73_HTML.gif and using (104)-(105), we obtain
      v n t ( t ) 2 = R L 0 1 ( r n ) 2 x ( ρ n θ n ) v n t d x + λ + 2 μ L 2 0 1 ( r n ) 2 x ( ρ n x ( ( r n ) 2 v n ) ) v n t d x C ( max ( x , t ) Q ¯ 0 | θ n ( x , t ) | ρ n x ( t ) v n t ( t ) + θ n x ( t ) v n t ( t ) + max ( x , t ) Q ¯ 0 | v n ( x , t ) | ρ n x ( t ) v n t ( t ) + v n ( t ) v n t ( t ) + max ( x , t ) Q ¯ 0 | v n ( x , t ) | 2 r n x 2 ( t ) v n t ( t ) + v n t ( t ) v n x ( t ) + max ( x , t ) Q ¯ 0 | v n x ( x , t ) | ρ n x ( t ) v n t ( t ) + 2 v n x 2 ( t ) v n x ( t ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ126_HTML.gif
      (126)
      Using (121), (122), (103), (116) and applying the Young inequality, we get
      v n t ( t ) 2 C ( 1 + 2 v n x 2 2 ) + ε v n t ( t ) 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ127_HTML.gif
      (127)
      Taking into account (103) for sufficiently small ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq133_HTML.gif from (127), we obtain
      0 T 0 v n t ( τ ) 2 d τ C . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ128_HTML.gif
      (128)

      In the same way, from (66) and (67) we obtain the estimates for ω n t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq143_HTML.gif and θ n t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq144_HTML.gif respectively. The estimates (124) and (125) follow directly from (59) and (58). □

      From Lemmas 4.7 and 4.8 we easily derive the following statements.

      Proposition 4.1 Let T 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq59_HTML.gif be defined by Lemma  4.7. Then for the sequence { ( r n , ρ n , v n , ω n , θ n ) : n N } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq145_HTML.gif the following properties are satisfied:
      1. (i)

        { r n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq146_HTML.gif is bounded in L ( Q 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq147_HTML.gif, L ( 0 , T 0 ; H 2 ( ] 0 , 1 [ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq148_HTML.gif and H 2 ( Q 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq149_HTML.gif;

         
      2. (ii)

        { r n x } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq150_HTML.gif is bounded in L ( Q 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq147_HTML.gif;

         
      3. (iii)

        { ρ n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq151_HTML.gif is bounded in L ( Q 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq147_HTML.gif, L ( 0 , T 0 ; H 1 ( ] 0 , 1 [ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq152_HTML.gif and H 1 ( Q 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq153_HTML.gif;

         
      4. (iv)

        { v n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq154_HTML.gif, { ω n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq155_HTML.gif, { θ n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq156_HTML.gif are bounded in L ( 0 , T 0 ; H 1 ( ] 0 , 1 [ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq152_HTML.gif, H 1 ( Q 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq153_HTML.gif and L 2 ( 0 , T 0 ; H 2 ( ] 0 , 1 [ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq157_HTML.gif.

         

      5 The proof of Theorem 2.1

      In the proofs we use some well-known facts of functional analysis (e.g., [21]). Let T 0 R + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq158_HTML.gif be defined by Lemma 4.7. Theorem 2.1 is a consequence of the following lemmas.

      Lemma 5.1 There exists a function
      r L ( 0 , T 0 ; H 2 ( ] 0 , 1 [ ) ) H 2 ( Q 0 ) C ( Q ¯ 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ129_HTML.gif
      (129)
      and the subsequence of { r n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq146_HTML.gif (for simplicity reasons denoted again as { r n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq146_HTML.gif) such that
      r n r weakly-  in  L ( 0 , T 0 ; H 2 ( ] 0 , 1 [ ) ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ130_HTML.gif
      (130)
      r n r weakly in  H 2 ( Q 0 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ131_HTML.gif
      (131)
      r n r strongly in  C ( Q ¯ 0 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ132_HTML.gif
      (132)
      r n x r x strongly in  C ( Q ¯ 0 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ133_HTML.gif
      (133)
      The function r satisfies the conditions
      a 2 r 2 M in  Q ¯ 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ134_HTML.gif
      (134)
      r ( x , 0 ) = r 0 ( x ) , x [ 0 , 1 ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ135_HTML.gif
      (135)

      where r 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq62_HTML.gif is defined by (40).

      Proof The conclusions (130) and (131) follow immediately from Proposition 4.1. Let ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq159_HTML.gif, ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq160_HTML.gif belong to Q ¯ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq161_HTML.gif. Then we have
      | r n ( x , t ) r n ( x , t ) | | r n ( x , t ) r n ( x , t ) | + | r n ( x , t ) r n ( x , t ) | , | r n x ( x , t ) r n x ( x , t ) | | r n x ( x , t ) r n x ( x , t ) | + | r n x ( x , t ) r n x ( x , t ) | . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equt_HTML.gif
      Using (104), (58), (116), (103) and (107), we obtain
      | r n ( x , t ) r n ( x , t ) | x x | r n x ( ξ , t ) | d ξ C | x x | , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ136_HTML.gif
      (136)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ137_HTML.gif
      (137)
      | r n x ( x , t ) r n x ( x , t ) | 2 r n x 2 ( t ) | x x | 1 / 2 C | x x | 1 / 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ138_HTML.gif
      (138)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ139_HTML.gif
      (139)
      and we can conclude that the sequences { r n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq162_HTML.gif and { r n x } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq163_HTML.gif satisfy the conditions of Arzelà-Ascoli theorem. Applying that theorem, we get the strong convergence in (132) and (133). Because of (132) and (104) we have
      a 2 ε < r n ( x , t ) ε < r ( x , t ) < r n ( x , t ) + ε < 2 M + ε , ( x , t ) Q ¯ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ140_HTML.gif
      (140)
      for each ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq133_HTML.gif and sufficiently big n N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq25_HTML.gif. From (140) we can easily conclude that (134) is satisfied. From (132) it follows
      lim n max x [ 0 , 1 ] | r n ( x , 0 ) r ( x , 0 ) | = lim n max x [ 0 , 1 ] | r 0 ( x ) r ( x , 0 ) | = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ141_HTML.gif
      (141)

      and because of that we have (135). □

      Lemma 5.2 There exists a function
      ρ L ( 0 , T 0 ; H 1 ( ] 0 , 1 [ ) ) H 1 ( Q 0 ) C ( Q ¯ 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ142_HTML.gif
      (142)
      and the subsequence of { ρ n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq151_HTML.gif (denoted again as { ρ n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq151_HTML.gif) such that
      ρ n ρ  weakly-  in  L ( 0 , T 0 ; H 1 ( ] 0 , 1 [ ) ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ143_HTML.gif
      (143)
      ρ n ρ weakly in  H 1 ( Q 0 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ144_HTML.gif
      (144)
      ρ n ρ strongly in  C ( Q ¯ 0 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ145_HTML.gif
      (145)
      The function ρ satisfies the conditions
      m 2 ρ ( x , t ) 2 M in  Q ¯ 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ146_HTML.gif
      (146)
      ρ ( x , 0 ) = ρ 0 ( x ) , x [ 0 , 1 ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ147_HTML.gif
      (147)

      Proof Taking into account Proposition 4.1, estimates (103)-(106) and the Arzelà-Ascoli theorem, we prove in the same way as in the previous lemma the properties (143)-(147). □

      Lemma 5.3 There exist functions
      v , ω , θ L ( 0 , T 0 ; H 1 ( ] 0 , 1 [ ) ) H 1 ( Q 0 ) L 2 ( 0 , T 0 ; H 2 ( ] 0 , 1 [ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equu_HTML.gif
      and the subsequence of { v n , ω n , θ n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq164_HTML.gif (denoted again as { v n , ω n , θ n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq164_HTML.gif) such that
      ( v n , ω n , θ n ) ( v , ω , θ ) weakly-  in  ( L ( 0 , T 0 ; H 1 ( ] 0 , 1 [ ) ) ) 3 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ148_HTML.gif
      (148)
      ( v n , ω n , θ n ) ( v , ω , θ ) weakly in  ( H 1 ( Q 0 ) ) 3 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ149_HTML.gif
      (149)
      ( v n , ω n , θ n ) ( v , ω , θ ) weakly in  ( L 2 ( 0 , T 0 ; H 2 ( ] 0 , 1 [ ) ) ) 3 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ150_HTML.gif
      (150)
      ( v n , ω n , θ n ) ( v , ω , θ ) strongly in  ( L 2 ( Q 0 ) ) 3 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ151_HTML.gif
      (151)
      The functions v, ω and θ satisfy the conditions
      v ( 0 , t ) = v ( 1 , t ) = ω ( 0 , t ) = ω ( 1 , t ) = 0 , t [ 0 , T 0 ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ152_HTML.gif
      (152)
      θ x ( 0 , t ) = θ x ( 1 , t ) = 0 , a.e. in  ] 0 , T 0 [ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ153_HTML.gif
      (153)
      v ( x , 0 ) = v 0 ( x ) , ω ( x , 0 ) = ω 0 ( x ) = 0 , θ ( x , 0 ) = θ 0 ( x ) , x [ 0 , 1 ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_Equ154_HTML.gif
      (154)

      Proof The conclusions (148)-(151) follow from Proposition 4.1 and embedding properties (see Remark 2.1). For verification of the boundary and initial conditions (152), (153) and (154), we use the Green formula as follows.

      Let φ be a function of C ( [ 0 , T 0 ] ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq165_HTML.gif equal to zero in a neighborhood of T 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq59_HTML.gif, with φ ( 0 ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq166_HTML.gif and u H 1 ( ] 0 , 1 [ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq167_HTML.gif. Using the integration by parts we have for v n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-69/MediaObjects/13661_2011_Article_167_IEq69_HTML.gif and v (e.g.) the following equalities:
      0 T 0 0 1 v n t ( x , t ) u ( x ) φ ( t ) d x