Abstract elliptic operators appearing in atmospheric dispersion
© Shakhmurov and Sahmurova; licensee Springer. 2014
Received: 4 December 2013
Accepted: 31 January 2014
Published: 19 February 2014
The Erratum to this article has been published in Boundary Value Problems 2014 2014:116
In this paper, the boundary value problem for the differential-operator equation with principal variable coefficients is studied. Several conditions for the separability and regularity in abstract -spaces are given. Moreover, sharp uniform estimates for the resolvent of the corresponding elliptic differential operator are shown. It is implies that this operator is positive and also is a generator of an analytic semigroup. Then the existence and uniqueness of maximal regular solution to nonlinear abstract parabolic problem is derived. In an application, maximal regularity properties of the abstract parabolic equation with variable coefficients and systems of parabolic equations are derived in mixed -spaces.
MSC:34G10, 34B10, 35J25.
and the state variables represent concentration densities of the chemical species involved in the photochemical reaction. The relevant chemistry of the chemical species involved in the photochemical reaction appears in the nonlinear functions with the terms , representing elevated point sources, and where , are real-valued functions. The advection terms describe transport of the velocity vector field of atmospheric currents or wind; see  and references therein.
2 Definitions, notations, and background
is bounded in , (see, e.g., ). UMD-spaces include e.g. , spaces and Lorentz spaces , .
Let and be two Banach spaces. By , , , will be denoted the interpolation spaces obtained from by the K-method [, §1.3.2].
where is a sequence of independent symmetric -valued random variables on Ω. The smallest C for which the above estimate holds is called a R-bound of the collection Φ and denoted by .
for all and , and . It is implied that .
The ψ-positive operator A is said to be R-positive in a Banach space E if the set , , is R-bounded.
The operator is said to be ψ-positive in E uniformly with respect to t with bound if is independent of t, is dense in E and for all , , where M does not depend on t and λ.
For , , , the space will be denoted by . For the space is denoted by .
Sometimes we use one and the same symbol C without distinction in order to denote positive constants which may differ from each other even in a single context. When we want to specify the dependence of such a constant on a parameter, say α, we write .
The embedding theorems in vector-valued spaces play a key role in the theory of DOEs. For estimating lower order derivatives we use following embedding theorems from .
E is a UMD-space and A is an R-positive operator in E;
- (2)is an n-tuple of nonnegative integer numbers and m is a positive integer such that
h is a positive parameter with , where is a fixed positive number;
is a region such that there exists a bounded linear extension operator from to .
Remark 2.1 If is a region satisfying the strong m-horn condition (see [, §7]), , , then for there exists a bounded linear extension operator from to .
From [, Theorem 2.1] we obtain the following.
3 Boundary value problems for abstract elliptic equations with constant coefficients
here A is a linear operator in a Banach space E, are complex numbers, and λ is a complex parameter.
E is a UMD-space and A is a uniformly R-positive operator in E for ;
, , ;
, , , .
The main result of this section is the following.
, , for , ;
A is a R-positive operator in a UMD-space E, m is a nonnegative integer.
Hence, from (3.9)-(3.12) we obtain (3.7). □
Finally, from (3.21) and (3.23) we obtain (3.13). □
Now, by using of Theorems 3.2, 3.3 we can prove the main result of this section.
From estimates (3.27)-(3.28) we obtain the assertion for problem (3.24). Then by continuing this process n times we obtain the conclusion. □
4 Boundary value problems for abstract elliptic equations with variable coefficients
where G, , , , are defined as in (3.1)-(3.2), are complex-valued continuous functions, and are linear operators in a Banach space E for , and and , respectively, are E-valued unknown and data functions. We will derive in this section the maximal regularity properties of problem (4.1).
E is a UMD-space and is a uniformly R-positive operator in E for ;
, , , for all , ;
, , , , ;
, , , , .
Remark 4.1 Let , and , where are real-valued positive functions and are natural numbers. Then Condition 4.1 is satisfied.
Remark 4.2 The periodicity conditions are given due to nonlocality of boundary conditions. For local boundary conditions these assumptions are not required.
The main result is the following.
for and .
5 Abstract Cauchy problem for parabolic equation
where and are linear operator functions in a Banach space E, are complex-valued functions, λ is a complex parameter, , and G, , , are the domains defined in (3.1)-(3.2).
Analogously, denotes the Sobolev space with the corresponding mixed norm (see  for the scalar case).
In this section, we obtain the existence and uniqueness of the maximal regular solution of problem (5.1)-(5.2) in mixed norms. Let O denote the differential operator in generated by (4.1) for .
the operator O is an R-positive in ;
the operator O is a generator of an analytic semigroup.
where are local operators generated by BVPs with constant coefficients of type (3.1)-(3.2) and and are uniformly bounded operators defined in the proof of Theorem 4.1. By virtue of [, Theorem 5.1] the operators are R-positive. Then by using the above representation and by virtue of Kahane’s contraction principle, and the product and additional properties of the collection of R-bounded operators (see e.g. [, Lemma 3.5, Proposition 3.4]) we obtain the assertions. □
Since , by Theorem 4.1 we have . This relation and the estimate (5.4) implies the assertion. □
6 Nonlinear abstract parabolic problem
where are complex-valued functions, , are complex numbers, and G, , , are domains defined in (3.1)-(3.2).
E is an UMD-space;
are continuous functions on , , , , , where ;
- (3)there exist , such that the operator for is R-positive in E uniformly with respect to and ; moreover,
- (4): is continuous; moreover, for each positive r there is a positive constant such that
the function F: such that is measurable for each and is continuous for a.a. , ; moreover, for a.a. , , and ; .
By reasoning as in [, Theorem 5.1] we obtain the following result.
Theorem 6.1 Let Condition 6.1 be satisfied. Then there are and such that problem (6.1)-(6.2) has a unique solution belonging to .
7 The mixed value problem for system of parabolic equations
From Theorem 6.1 we obtain the following result.
are continuous functions on , ;
, , ;
- (3), ; the eigenvalues of the matrix and are positive for all , ; there is a positive constant C such that
- (4)the function is measurable for each and the function for a.a. is continuous and ; for each there is a function such that
Then problem (7.1)-(7.3) has a unique solution that belongs to the space .
Proof By virtue of  the space is a UMD-space. It is easy to see that the operator A is R-positive in . Then by using conditions (1)-(3) we see that condition (5) of Theorem 6.1 holds. So in view of Theorem 6.1 we obtain the conclusion. □
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