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Abstract elliptic operators appearing in atmospheric dispersion
Boundary Value Problems volume 2014, Article number: 43 (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.
It is well known that many classes of PDEs, pseudo DEs and integro DEs can be expressed as a differential-operator equation (DOE). DOEs have been studied extensively in the literature (see [1–22] and the references therein). Note the regularity results for the PDE studied e.g. in [11, 23–25]. The main goal of the present paper is to discuss the maximal regularity properties of nonlocal boundary value problems (BVPs) for the following DOE:
Afterwards, the well-posedness of initial and BVP (IBVP) for the following abstract parabolic equation:
is derived, where are complex-valued functions, A and are linear operators in a Banach space E, and , respectively, are an E-valued unknown and data function. By using this, we obtain the existence and uniqueness result of IBVP for the following nonlinear parabolic equation:
Finally, we discuss the application of the above result to systems of parabolic PDEs. Particularly, we consider the system that serves as a model of systems used to describe photochemical generation and atmospheric dispersion of ozone and other pollutants. The model of the process is given by the atmospheric reaction-advection-diffusion system having the form
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
Let E be a Banach space. denotes the space of strongly measurable E-valued functions that are defined on the measurable subset with the norm
The Banach space E is called an UMD-space if the Hilbert operator
is bounded in , (see, e.g., ). UMD-spaces include e.g. , spaces and Lorentz spaces , .
A linear operator A is said to be ψ-positive in a Banach space E with bound if is dense on E and for any , , where I is the identity operator in E, and is the space of bounded linear operators in E. It is well known [, §1.15.1] that there exist fractional powers of a positive operator A. Let denote the space endowed with the norm
Let and be two Banach spaces. By , , , will be denoted the interpolation spaces obtained from by the K-method [, §1.3.2].
Let ℕ denote the set of natural numbers. A set is called R-bounded (see, e.g., ) if there is a positive constant C such that for all and , ,
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 .
Since we will consider the problem with spectral parameter, we need the concept of the uniform R-boudedness of a parameter-dependent family of operators. A set is called the uniform R-bounded with respect to the parameter if there is a constant M independent on h such that
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.
Let and E be two Banach spaces. is continuously and densely embedded into E. Let Ω be a domain in and m be a positive integer. denotes the space of all functions that have generalized derivatives with the norm
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 .
Theorem A 1 Suppose the following conditions are satisfied:
E is a UMD-space and A is an R-positive operator in E;
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 .
Then the embedding is continuous and for the following uniform estimate holds:
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 .
Theorem A 2 Suppose all conditions of Theorem A1 are satisfied and . Moreover, let Ω be a bounded region and . Then the embedding
Theorem A 3 Suppose all conditions of Theorem A1 are satisfied. Let . Then the embedding
is continuous and there exists a positive constant such that for all the uniform estimate holds:
From [, Theorem 2.1] we obtain the following.
Theorem A 4 Let E be a Banach space, A be a φ-positive operator in E with bound M, . Let m be a positive integer, and . Then, for an operator generates a semigroup which is holomorphic for . Moreover, there exists a positive constant C (depending only on M, φ, m, α, and p) such that for every and ,
3 Boundary value problems for abstract elliptic equations with constant coefficients
Consider first the BVP for the constant coefficients DOE
here A is a linear operator in a Banach space E, are complex numbers, and λ is a complex parameter.
Let and . Let , denote the roots of the equations
Condition 3.1 Assume;
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.
Theorem 3.1 Assume Condition 3.1 is satisfied. Then problem (3.1)-(3.2) has a unique solution for , , with sufficiently large and the following uniform coercive estimate holds:
For proving Theorem 3.1, we consider the BVP for the ordinary DOE
where , , , , , , are complex numbers, a is a complex number, λ is a complex parameter, and A is a linear operator in E. Let us first consider the corresponding homogeneous problem:
Let , be roots of equations . We put , and
Condition 3.2 Assume the following conditions are satisfied:
, , for , ;
A is a R-positive operator in a UMD-space E, m is a nonnegative integer.
Theorem 3.2 Let Condition 3.2 hold. Then, problem (3.5)-(3.6) has a unique solution for , with sufficiently large and the following coercive uniform estimate holds:
Proof In a similar way as in [, Theorem 3.1] we obtain the representation of the solution of (3.4):
where and are uniformly bounded operators. Then in view of the positivity of A we obtain from (3.8)
By virtue of Theorem A4 we obtain
Moreover, due to the positivity of the operator A and the estimate (3.11), in view of Theorem A4 we get the uniform estimate
Hence, from (3.9)-(3.12) we obtain (3.7). □
Theorem 3.3 Assume Condition 3.2 to hold. Then the operator for and for sufficiently large is an isomorphism from
Moreover, the following uniform coercive estimate holds:
Proof The uniqueness of solution of problem (3.4) is obtained from Theorem 3.3. Let us define
We now show that problem (3.4) has a solution for all , and , where is the restriction on of the solution of the equation
and is a solution of the problem
A solution of (3.14) is given by
It follows from the above expression that
It is sufficient to show that the operator functions
are Fourier multipliers in uniformly in λ. Actually, due to and the positivity of A we have
It is clear that
In a similar way we see that the sets
are R-bounded. Then, in view of Kahane’s contraction principle and from the product properties of the collection of R-bounded operators (see e.g. , Lemma 3.5, Proposition 3.4) we obtain
Namely, the R-bound of the set is independent of λ. Moreover, it is clear that
Hence, by using the well-known inequality , , for and we get the estimate
From (3.17) and (3.18) we have the uniform estimate
Due to R-positivity of the operator A, the set
is R-bounded. Then, by estimate (3.17) and by Kahane’s contraction principle we obtain the R-boundedness of the set . In a similar way we obtain the uniform estimates
Consider the set
Due to the R-positivity of the operator A, in view of estimate (3.17), by virtue of Kahane’s contraction principle, from the additional and product properties of the collection of R-bounded operators, for , , and the independent symmetric -valued random variables , , we obtain the uniform estimate
In a similar way, the above estimate is obtained for . So, by [, Theorem 3.4] it follows that and are the uniform collection of multipliers in . Then, by using the equality (3.16) we see that problem (3.14) has a solution and the following uniform estimate holds:
Let be the restriction of u on . Then the estimate (3.17) implies that . By virtue of the trace theorem (see e.g. [, §1.8.2]) we get
Hence, . Thus, by virtue of Theorem 3.2, problem (3.15) has a unique solution that belongs to the space . Moreover, we have
From (3.19) we obtain
Therefore, by Theorem A3 and by estimate (3.21) we obtain
Hence, in view of Theorem 3.2 and estimates (3.20)-(3.22) we get
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.
Proof of Theorem 3.1 Let . It is clear that
Let us consider the BVP
where are defined by equalities (3.6). Problem (3.24) can be expressed as the following BVP for the ordinary DOE:
where is the operator in X defined by
respectively. Since and X are UMD-spaces (see e.g. [, Theorem 4.5.2]), by virtue of Theorem 3.3 we obtain the result that problem (3.25) has a unique solution , for and , respectively. Moreover, for and sufficiently large the following coercive uniform estimates hold:
From (3.26) we find that problem (3.24) has a unique solution,
and the following uniform coercive estimates hold:
By applying Theorem 3.3, for and we get the following uniform estimate:
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
Consider the BVP for DOE 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).
Condition 4.1 Assume:
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.
Let and . Consider the operator O in X generated by problem (4.1), i.e.,
The main result is the following.
Theorem 4.1 Assume Condition 4.1 is satisfied. Then problem (4.1) has a unique solution for , and the following uniform coercive estimate holds:
Proof First we will show the uniqueness of solution. For this aim we use microlocal analysis. Let be rectangular regions with sides parallel to the coordinate planes covering G and let be a corresponding partition of unity, i.e., , and , where denotes the space of all infinitely differentiable functions on G with compact support. Now for being a solution of (4.1) and we get
Freezing the coefficients of (4.3), extending outside of up to , we obtain the BVP
and are the usual binomial coefficients. It is clear that . By applying Theorem 3.1 we obtain the following a priori estimate:
for the solution of (4.5) on the domains containing the boundary points. In a similar way we obtain the same estimates for the domains . In view of , by Theorem A1, in view of the continuity of coefficients, choosing diameters of supp sufficiently small we see that for all small δ there is a positive continuous function so that
Consequently, from (4.6)-(4.8) we have
Choosing from (4.9) we obtain
Since and by (4.10) we find that the solution of (4.1) satisfies the estimate (4.2). It is clear that
Hence, by using the definition of Y and applying Theorem A1 we obtain
From the above estimate we have
The estimate (4.11) implies that uniqueness of the solution of problem (4.1). It implies that the operator has a bounded inverse in its rank space. We need to show that this rank space coincides with the space X, i.e., we have to show that for all there is a unique solution of problem (4.1). We consider the smooth functions with respect to on that equal 1 on supp , where supp and . Let us construct for all j the functions that are defined on the regions and satisfying problem (4.1). Problem (4.1) can be expressed as
Consider operators in that are generated by the problem
By virtue of Theorem 3.1, the local operators have bounded inverses from to and for all we have the following uniform estimate:
Extending the solutions of (4.13) to zero on the outside of and using the substitutions we obtain the equations
where are bounded linear operators in defined by
In fact, due to the smoothness of the coefficients of the expression and in view of the estimate (4.14) for sufficiently large there is a sufficiently small such that
Moreover, by Theorem A1 we find that for all there is a constant such that
Hence, for with sufficiently large there is a such that . Consequently, (4.15) for all j have a unique solution . Moreover,
Thus, are bounded linear operators from X to . Thus, the functions are solutions of (4.12). Consider the linear operator U in defined by
It is clear from the constructions and from the estimate (4.14) that the operators are bounded linear from X to and for and sufficiently large we have
Therefore, U is a bounded linear operator in X. By the construction of the solution operators of the local problems (4.12), we get
where are bounded linear operators defined by
Indeed, by Theorem A1, estimate (4.16), and from the expression we find that the operators are bounded linear from X to X and for with sufficiently large there is an such that . Therefore, there exists a bounded linear invertible operator , i.e., we infer for all that the BVP (3.1) has a unique solution
Remark 4.3 Theorem 4.1 implies that the resolvent satisfies the sharp uniform estimate
for and .
5 Abstract Cauchy problem for parabolic equation
Consider now the initial BVP for the following parabolic equation with variable coefficients, i.e.,
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).
For , , will denote the space of all E-valued p-summable functions with mixed norm (see e.g. ), i.e., the space of all measurable functions f defined on , for which
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 .
Theorem 5.1 Let all conditions of Theorem 4.1 hold for and . Then:
the operator O is an R-positive in ;
the operator O is a generator of an analytic semigroup.
Proof In fact, by virtue of Theorem 4.1 we see that for the BVP (4.1) have a unique solution expressed in the form
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. □
Theorem 5.2 Let all conditions of Theorem 5.1 hold. Then for problem (5.1)-(5.2) has a unique solution and for sufficiently large the following coercive estimate holds:
Proof Problem (5.1)-(5.2) can be expressed as the following Cauchy problem:
Theorem 5.1 implies that the operator O is R-positive and also is a generator of an analytic semigroup in . Then by virtue of  or [, Theorem 4.2] we see that for problem (5.3) has a unique solution and the following uniform estimate holds:
Since , by Theorem 4.1 we have . This relation and the estimate (5.4) implies the assertion. □
6 Nonlinear abstract parabolic problem
Consider the following nonlinear parabolic problem:
where are complex-valued functions, , are complex numbers, and G, , , are domains defined in (3.1)-(3.2).
Let . Moreover, we let
Remark 6.1 By virtue of [, §1.8] the operators are continuous from onto and there are the constants and such that for , , , , ,
Condition 6.1 Suppose the following hold:
E is an UMD-space;
are continuous functions on , , , , , where ;
there exist , such that the operator for is R-positive in E uniformly with respect to and ; moreover,
: is continuous; moreover, for each positive r there is a positive constant such that
for , , , , , , ;
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
Consider the initial and BVP for the system of nonlinear parabolic equations
where , , , are complex numbers, are complex-valued functions, G, , are defined as in (3.1)-(3.2), and
Let A be the operator in defined by
From Theorem 6.1 we obtain the following result.
Theorem 7.1 Let the following condition hold:
are continuous functions on , ;
, , ;
, ; the eigenvalues of the matrix and are positive for all , ; there is a positive constant C such that
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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The authors declare that they have no competing interests.
All abstract results belong to VS; the application part belongs to AS.
An erratum to this article is available at http://dx.doi.org/10.1186/1687-2770-2014-116.
About this article
- separable boundary value problems
- equations with variable coefficients
- estimates of the resolvent
- differential-operator equations
- well-posedness for parabolic problems
- reaction-diffusion equations