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Exterior problem for the spherically symmetric isentropic compressible Navier-Stokes equations with density-dependent viscosity
Boundary Value Problems volume 2016, Article number: 49 (2016)
Abstract
In this paper, we study the exterior problem for the spherically symmetric isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficients. Under certain assumptions imposed on the initial data, we show that there exists a unique global strong solution to the exterior problem and obtain the regularity of the strong solution. Some ideas and more delicate estimates are introduced to prove these results.
1 Introduction
In general, the N-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficients read
where \(t\in(0,+\infty)\) is the time and \(\mathbf{x}\in R^{N}\), \(\rho >0\) and u denote the density and velocity, respectively. The pressure function is taken as \(P(\rho)=\rho^{\gamma}\) with \(\gamma>1\), and
is the strain tensor and \(\mu(\rho)\), \(\lambda(\rho)\) are the Lamé viscosity coefficients satisfying
There is a huge literature on the studies of the compressible Navier-Stokes equations with density-dependent viscosity coefficients. For example, as \(\mu(\rho)=1\), \(\lambda(\rho)=\rho^{\beta}\), and \(\beta>3\), Vaigant and Kazhikhov [1] established the existence and uniqueness of global strong solution to the two-dimensional Navier-Stokes system of equations for a barotropic compressible viscous fluid in the square. Ducomet and Nec̆asová [2] proved the existence and uniqueness of global strong solution to the two-dimensional compressible Navier-Stokes-Fourier system with vorticity-type boundary conditions and density-dependent viscosities in any smooth bounded region of \(R^{2}\). The mathematical derivations of the viscous Saint-Venant system were addressed in the simulation of flow surface in shallow region [3, 4]. The physical model of the viscous Saint-Venant system is the prototype model (corresponding to (1.1) with \(P(\rho)=\rho^{2}\), \(\mu(\rho)=\rho\), and \(\lambda(\rho )=0\)), and Bresch and Desjardins proved the existence of solutions for the 2D shallow water equations [5, 6]. The well-posedness of solutions to the free boundary value problem with initial finite mass and the flow density being connected with the infinite vacuum either continuously or via jump discontinuity was investigated by many authors, refer to [7–17] and references therein. Mellet and Vasseur [18] considered barotropic compressible Navier-Stokes equations with density-dependent viscosity coefficients that vanish on the vacuum and proved the stability of weak solutions in periodic domain and whole space. The global existence of strong solutions for one-dimensional compressible Navier-Stokes equations was shown by Mellet and Vasseur [19]. Ducomet et al. [20] investigated the Cauchy problem for the equations of selfgravitating motions of a barotropic gas with density-dependent viscosities, where the pressure \(P(\rho)\) is not necessarily a monotone function of the density and proved that the Cauchy problem admits a global weak solution. The Cauchy problem for the equations of spherically symmetric motions in \(R^{3}\) of a selfgravitating barotropic gas, with possibly non-monotone pressure law, was considered by Ducomet et al. [21], and they also proved the global existence of weak solution. The qualitative behaviors of global solutions and dynamical asymptotics of vacuum states were also considered, for instance, the finite time vanishing of finite vacuum or asymptotical formation of vacuum in large time, the dynamical behaviors of vacuum boundary, the large time convergence to rarefaction wave with vacuum, and the stability of shock profile with large shock strength, refer to [22–27] and references therein.
In this present paper, we consider the exterior problem for the spherically symmetric isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficients and focus on the global existence, uniqueness and regularity of the strong solution, etc. As \(P(\rho)=\rho^{\gamma}\) (\(\gamma\geq2\)), \(\mu(\rho)=\rho\), and \(\lambda(\rho)=0\), we show that the exterior problem admits a unique global strong solution.
The rest part of the paper is arranged as follows. In Section 2, the main results as regards the global existence of strong solution for the spherically symmetric isentropic compressible Navier-Stokes equations are stated. In Section 3, the a priori estimates for strong solution to the exterior problem are established, and in Section 4 the main results are proved.
2 Notations and main results
In this present paper, the viscosity terms are assumed to satisfy \(\mu (\rho)=\rho\) and \(\lambda(\rho)=0\) in (1.1) and the strain tensor is taken as \(D(\mathbf{U})=\nabla\mathbf{U}\). The isentropic compressible Navier-Stokes equations become
The initial data and boundary conditions of (2.1) are imposed as
where \(\Omega:=R^{3}/\Omega_{r_{-}}\), \(\Omega_{r_{-}}\) is a ball of radius \(r_{-}\) centered at the origin in \(R^{3}\), and \(\bar{\rho}>0\) is a constant.
We will investigate the spherically symmetric solution of the system (2.1) in the spherically symmetric exterior domain Ω in the present paper, so we denote
which gives the following system of equations for \(r>0\):
with the initial data and boundary conditions
and the initial data satisfies for some constant \(\underline{\rho}>0\)
Next, we give the definition of a weak solution to the exterior problem (2.1)-(2.2).
Definition 2.1
(weak solution)
For any \(T>0\), \((\rho,u)\) is said to be a weak solution of the exterior problem (2.1)-(2.2), if \((\rho,u)\) has the following regularities:
and equations (2.1) are satisfied in the sense of a distribution. Namely, for all \(\varphi\in C_{0}^{\infty}(\bar{\Omega}\times[0,T])\)
and for all \(\psi=(\psi_{1},\psi_{2},\psi_{3})\in C_{0}^{\infty}(\bar {\Omega}\times[0,T])\)
where the diffusion term makes sense as
Then we can give the main results as follows.
Theorem 2.1
Let \(\gamma\geq2\). Assume that the initial data satisfies (2.6). Then there exists a unique global strong solution \((\rho,u)\) to the exterior problem (2.4)-(2.5) satisfying for \(T>0\)
here and below \(C(T)>0\) denotes the constant dependent on time and \(C>0\) denotes the constant independent of time.
If further \(r^{2}u_{0}\in H^{2}([r_{-},+\infty))\), then \((\rho,u)\) satisfies
Remark 2.1
Theorem 2.1 holds for the Saint-Venant model for shallow water, i.e., \(P(\rho)=\rho^{2}\), \(\mu(\rho)=\rho\), \(\lambda(\rho)=0\).
Remark 2.2
In this paper, we can obtain several estimates in (3.80) and (3.81) which are not uniformly on time, these estimates can be used to get the compactness results for the exterior problem (2.4)-(2.5), but they not be applied to investigate the large time behaviors of the strong solution.
3 The a priori estimates
It is convenient to prove Theorem 2.1 in terms of Lagrange coordinates, and the key step is to establish several useful a priori estimates. Take the Lagrange coordinates to transform
which maps \((r,t)\in[r_{-},+\infty)\times R^{+}\) into \((x,\tau)\in [0,+\infty)\times R^{+}\). The relation between Lagrangian coordinates and Eulerian coordinates is satisfied by
Under the Lagrangian coordinates transform, the exterior problem (2.4)-(2.5) is reformulated to
where the initial data satisfies
First, we are ready to establish the a priori estimates for the solution \((\rho,u)\) to the exterior problem (3.3). First of all, we can establish the following a priori estimates.
Lemma 3.1
Let \(T>0\). Under the conditions in Theorem 2.1, we have for the strong solution \((\rho,u)\) to the exterior problem (3.3)
Proof
Multiplying (3.3)2 by \(r^{2} u\) and integrating the result with respect to x over \([0,+\infty)\), making use of (3.3)1, we have
integrating (3.6) with respect to τ, we obtain (3.5). □
Lemma 3.2
Let \(T>0\). Under the conditions in Theorem 2.1, we have for the strong solution \((\rho,u)\) to the exterior problem (3.3)
Proof
Differentiating (3.3)1 with respect to x, we have
Summing (3.8) and (3.3)2, we have
Note that
and so
which together with (3.9) yields
Multiplying (3.12) by \((u+r^{2}\rho_{x})r^{2}\), and integrating the result with respect to x and τ, we have (3.7). □
Lemma 3.3
Let \(T>0\). Under the conditions in Theorem 2.1, we have for the strong solution \((\rho, u)\) to the exterior problem (3.3)
where C is a positive constant independent of time.
Proof
Let
and
It follows from (3.5) and (3.15) that
As \(\rho\to+\infty\), we have for some \(\theta\in(0,1)\), if \(1<\gamma\leq3\),
and if \(\gamma>3\), we have
which with (3.16) yields
 □
Next, the Lagrangian structure of the particle transport for this exterior problem (3.3) will be shown as follows. Without loss of generality, we define two particle paths \(r_{1}(t)\), \(r_{2}(t)\) in Eulerian coordinates as
where \(r_{10}\) and \(r_{20}\) satisfy
Since we have the conservation of total mass,
and the two paths \(r_{1}(t)\), \(r_{2}(t)\) are transformed into \(x=a\), \(x=b\), furthermore, the domain \([r_{1}(t),r_{2}(t)]\) is transformed into \([a,b]\), where \(0\leq a< b<+\infty\).
Lemma 3.4
Let \(T>0\). Under the conditions in Theorem 2.1, we have for the strong solution \((\rho, u)\) to the exterior problem (3.3)
where C is a positive constant independent of time.
Proof
By the Lagrangian coordinates transform (3.1), for any \(x\in [a,b]\) and \(r_{1}(t)\leq r(x,\tau)\leq r_{2}(t)\), where \(r_{-}\leq r_{10}\leq r(x,0)\leq r_{20}<+\infty\), we can find that
which implies for \((x,\tau)\in[0,+\infty)\times[0,T]\) that
For any \(0\leq a< b<+\infty\), we have
which together with
implies
 □
Lemma 3.5
Let \(T>0\). Under the conditions in Theorem 2.1, we have for the strong solution \((\rho, u)\) to the exterior problem (3.3)
where \(C(T)\) is a positive constant dependent on time T.
Proof
From (3.3) we have
From (3.33)
By means of \(\gamma\geq2\), we have
which with Gronwall’s inequality yields the lemma. □
Lemma 3.6
Let \(T>0\). Under the conditions in Theorem 2.1, for a small constant \(\delta_{1}\in(0,\frac{b-a}{2})\)
where \(C(\delta_{1},T)\) is a positive constant dependent on \(\delta_{1}\) and time T. Furthermore, for a small constant \(\delta_{2}\in(0,\frac{b}{2})\)
where \(C(\delta_{2},T)\) is a positive constant dependent on \(\delta_{2}\) and time T.
Proof
For a small positive constant \(\eta\in(0,\frac{b-a}{8})\), we can find two points \(x_{1}\in[a+\eta,a+2\eta]\), \(x_{2}\in[a+3\eta,a+4\eta]\), and define, for \(x_{i}\), \(i=1,2\),
meanwhile we have
Define two particle paths \(r_{x_{i}}(t)\) as
and we have from the conservation of total mass
Then we can find a curve in Eulerian coordinates
defined by
such that we have
Furthermore, in Lagrangian coordinates, there exists \(x_{*}\in[x_{1},x_{2}]\):
such that
In the same way, as η is small enough, we can find another two points \(x_{3}\in[b-4\eta,b-3\eta]\), \(x_{4}\in[b-2\eta,b-\eta]\) and define, for \(x_{i}\), \(i=3,4\),
which implies
Define two particle paths
and we have from the conservation of total mass
As by the argument above, there exists a curve in Eulerian coordinates
defined by
meanwhile, in Lagrangian coordinates there exists \(x^{*}\in[x_{3},x_{4}]\):
such that
Set \(v(x,\tau)=\frac{1}{\rho(x,\tau)r^{2}(x,\tau)}\) and define
From (3.3)1, we have
For any \(\beta>1\), multiplying (3.58) by \(\beta v^{\beta -1}\), and integrating the equation over \([x_{*},x^{*}]\times[0,\tau]\), we have
which yields
where
and
Namely
A complicated computation gives
and
Next, we will give the estimate of the term \(J_{5}\). We have
In the same way, we can obtain
meanwhile, we know that
and
Finally, we obtain
Applying Gronwall’s inequality, for \(\tau\in[0,T]\)
Let \(\delta_{1}:=4\eta\), for \(x\in[a+\delta_{1},b-\delta_{1}]\)
By Young’s inequality, we have
which yields
Furthermore, repeating the above arguments with few modifications on the domain \([0,b]\), we can prove (3.37). The details are omitted here. The proof is completed. □
Lemma 3.7
Let \(T>0\). Under the conditions in Theorem 2.1, we have for the strong solution \((\rho, u)\) to the exterior problem (3.3)
where \(C(T)\) is a positive constant dependent on time T.
Proof
By means of \(\rho\rightarrow\bar{\rho}\) as \(x\rightarrow+\infty\), we know that \(\exists M>0\) such that
where \(C_{1}\) is a positive constant independent of T. We apply Lemma 3.6 on the domain \([0,M+\delta_{2}]\times[0, T ]\) with \(\delta_{2}\in(0,\frac{M}{2})\) a constant small enough, and we can obtain
where \(C_{2}(T)\) is a positive constant dependent on T. The proof is completed. □
Lemma 3.8
Let \(T>0\). Under the conditions in Theorem 2.1, we have for the strong solution \((\rho,u)\) to the exterior problem (3.3)
where \(C(T)>0\) denotes a constant dependent on time.
Proof
Multiplying (3.3)2 by \(\rho^{-2}(r^{2} u)_{\tau}\) and integrating the result with respect to x over \([0,+\infty)\), making use of (3.4), we obtain
which implies
From (3.3)2, (3.5), (3.7), (3.13), and (3.77), we can deduce that for some small \(\epsilon\in(0,1)\)
using (3.83)-(3.85), we can obtain
Differentiating (3.3)2 with respect to Ï„, multiplying the result by \((r^{2}u)_{\tau}\) and integrating the result with respect to x over \([0,+\infty)\), we have
A complicated computation gives
and by means of Gronwall’s inequality, (3.3)2, (3.5), (3.7), (3.13), (3.77), and (3.86), we have
we can complete the proof of Lemma 3.8. □
Remark 3.9
By Lemmas 3.1-3.8, the following inequality holds:
4 Proof of the main results
Proof
Let \((\rho_{0}, u_{0})\) be the initial data as described in the theorem, and define \(\rho_{0}^{\delta}:=j_{\delta}\ast\rho_{0}\), \(u_{0}^{\delta}:=j_{\delta}\ast u_{0}\), where \(j_{\delta}=\delta^{-1}j(x/\delta)\) is the standard mollifier. Then, for any \(0<\beta<1\), \(\rho_{0}^{\delta}\in C^{1+\beta}([0,+\infty))\) and \(u_{0}^{\delta}\in C^{2+\beta}([0,+\infty))\), which implies that as \(\delta\rightarrow 0\), \(\rho_{0}^{\delta}\rightarrow\rho_{0}\) in \(W^{1,2}([0,+\infty))\), \(u_{0}^{\delta}\rightarrow u_{0}\) in \(L^{2}([0,+\infty))\).
Next, we consider the Cauchy problem (3.3) with the initial data \((\rho_{0},u_{0})\) replaced by \((\rho_{0}^{\delta},u_{0}^{\delta})\), using the energy estimates and the contraction mapping theorem, we can obtain the existence of a unique local solution \((\rho^{\delta},u^{\delta})\) with \(\rho^{\delta}\), \(\rho_{x}^{\delta}\), \(\rho_{\tau}^{\delta}\), \(\rho_{\tau x}^{\delta}\), \(u^{\delta}\), \(u_{x}^{\delta}\), \(u_{\tau}^{\delta}\), \(u_{xx}^{\delta}\in C^{\beta,\beta/2}([0,+\infty)\times[0,T^{*}])\) for some \(T^{*}>0\). Furthermore, from Lemmas 3.1-3.8, we see that \(\rho^{\delta}\) is pointwise bounded from below and above, \(u^{\delta}\), \(\rho_{x}^{\delta}\in L^{\infty}([0,T];L^{2}([0,+\infty)))\), \(u_{x}^{\delta}\in L^{2}([0,T];L^{2}([0,+\infty)))\), \(\rho^{\delta}\), \(\rho_{x}^{\delta}\), \(\rho_{\tau}^{\delta}\), \(\rho_{\tau x}^{\delta}\), \(u^{\delta}\), \(u_{x}^{\delta}\), \(u_{\tau}^{\delta}\), \(u_{xx}^{\delta}\in C^{\beta,\beta /2}([0,+\infty)\times[0,T])\) for any \(T>0\). Therefore, we can continue the local solution globally in time and deduce that there exists a unique global solution \((\rho^{\delta},u^{\delta})\) of the Cauchy problem (3.3) with \((\rho_{0},u_{0})\) replaced by \((\rho_{0}^{\delta},u_{0}^{\delta})\), which is carried out as in [9].
Thus, extracting a subsequence of \((\rho^{\delta},u^{\delta})\), still denoted by \((\rho^{\delta},u^{\delta})\), such that as \(\delta\rightarrow0\), we have
Moreover, from (3.5), (3.7), (3.13), and (3.77), the global existence of weak solutions of the Cauchy problem (3.3) can be directly proved. As a matter of fact, because of (3.80) and (3.81), \((\rho,u)\) is also a global strong solution.
Next, we will prove the uniqueness of global strong solution as follows: let \((\rho_{1}(x,t), u_{1}(x,t))\) and \((\rho_{2}(x,t),u_{2}(x,t))\) be two global strong solutions of the exterior problem (3.3) on the time interval \([0,T]\). For convenience, we set
and we have from (3.3)1
Then, from (3.3)2, we obtain
Multiplying the above equation by \((u_{1}-u_{2})r^{2}\) and integrating the result with respect to x over \([0,+\infty)\), we have
where \(C_{0}\) and C are positive constants independent of T and \(a(x,\tau)\) is defined as follows:
which has a positive lower bound on \([0,+\infty)\times[0,T]\). Furthermore, we have
and
where C is a positive constant independent of T. Then, applying (4.10) and integrating (4.7) over \([0,\tau]\), we obtain
which together with \(a(x,\tau)\geq C>0\) gives
From \((r^{2}u_{2})_{x}\in L^{2}([0,T],H^{1}([0,+\infty)))\) and Sobolev’s embedding theorem, we have \((r^{2}u_{2})_{x}\in L^{2}([0,T],L^{\infty}([0,+\infty)))\), then using Gronwall’s inequality, we can prove that
The proof of the uniqueness is complete. □
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Acknowledgements
The authors thank the referee for the helpful comments and suggestions on the paper. The research of RX Lian is supported by NNSFC No. 11101145, China Postdoctoral Science Foundation No. 2012M520360, Doctoral Foundation of North China University of Water Sources and Electric Power No. 201032, Innovation Scientists and Technicians Troop Construction Projects of Henan Province. The research of J Liu is supported by NNSFC No. 11326140, the Doctoral Starting up Foundation of QuZhou University No. BSYJ201314.
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RL proved and checked the theorem, and wrote the paper, JL rechecked the proofs. All authors read and approved the final manuscript.
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Lian, R., Liu, J. Exterior problem for the spherically symmetric isentropic compressible Navier-Stokes equations with density-dependent viscosity. Bound Value Probl 2016, 49 (2016). https://doi.org/10.1186/s13661-016-0535-5
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DOI: https://doi.org/10.1186/s13661-016-0535-5