Degenerate parabolic equations appearing in atmospheric dispersion of pollutants

Linear and nonlinear degenerate abstract parabolic equations with variable coefficients are studied. Here the equation and boundary conditions are degenerated on all boundary and contain some parameters. The linear problem is considered on the moving domain. The separability properties of elliptic and parabolic problems in mixed Lp spaces are obtained. Moreover, the existence and uniqueness of optimal regular solution of mixed problem for nonlinear parabolic equation is established. Note that, these problems arise in fluid mechanics and environmental engineering.

The main objective of the present paper is to discusse the initial and BVP for the following nonlinear degenerate parabolic equation ∂u ∂t ∂ [2] u ∂x 2 k + B t, x, u, D [1] u u = F t, x, u, D [1] u , ( where a k (x) are complex valued functions, B and F are nonlinear operators in a Banach space E and D [1] u = ∂ [1] u ∂x 1 , ∂ [1] u ∂x 2 , . . ., ∂ [1] u ∂x n , x = (x 1 , x 2 , . . . , First, we consider the BVP for the degenerate elliptic DOE with small parameters n ∑ k=1 ε k a k (x k ) ∂ [2] u ∂x ∂ [1] u where a k are complex-valued functions, ε k are small parameters, A(x) and A k (x) are linear operators, λ is a complex parameter.Namely we prove that, for f ∈ L p (G; E), |arg λ| ≤ ϕ, 0 < ϕ ≤ π and sufficiently large |λ|, problem (1.2) has a unique solution u ∈ W [2] p (G; E(A), E) and the following coercive uniform estimate holds Especially, it is shown that the corresponding differential operator is positive and also is a generator of an analytic semigroup.Then by using this result, we prove the well-posedeness in L p (G; E) to initial and BVP for the following degenerate abstract parabolic equation with parameters ∂u ∂t Finally, via maximal regularity properties of (0.3) and contaction mapping argument we de- rive the existence and uniqueness of solution of the problem (1.1).Note that, the equation and boundary conditions are degenerated on all edges of boundary G.Moreover, it happened with the different rate at both boundary edges.
In application, the system of degenerate nonlinear parabolic equations is presented.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 initial and BVP for the atmospheric reaction-advection-diffusion system having the form where and the state variables u i represent concentration densities of the chemical species involved in the photochemical reaction.The relevant chemistry of the chemical species involved in the photochemical reaction and appears in the nonlinear functions f i (u), with the terms g i , rep- resenting elevated point sources, a ki (x), b ki (x) are real-valued functions.The advection terms ω = ω(x) = (ω 1 (x), ω 2 (x), ω 3 (x)), describe transport from the velocity vector field of at- mospheric currents or wind.In this direction the work [11] and references there can be mentioned.The existence and uniqueness of solution of the problem (1.4) is established by the theoretic-operator method, i.e., this problem reduced to degenerate differential-operator equation.
Modern analysis methods, particularly abstract harmonic analysis, the operator theory, interpolation of Banach spaces, semigroups of linear operators, microlocal analysis, embedding and trace theorems in vector-valued Sobolev-Lions spaces are the main tools implemented to carry out the analysis.

Notations, definitions and background
Let γ = γ(x) be a positive measurable function on Ω ⊂ R n and E be a Banach space.Let L p,γ (Ω; E) denote the space of strongly measurable E-valued functions defined on Ω with the norm Let p =(p 1 , p 2 , . . ., p n ).L p,γ (G; E), G = ∏ n k=1 (0, b k ) will denote the space of all E-valued p-summable functions with mixed norm, i.e., the space of all measurable functions f defined on G equipped with norm For γ(x) ≡ 1 we will denote these spaces by L p (Ω; E) and L p (G; E), respectively (see e.g.[5] for E = C).
Let E 0 and E be two Banach spaces and E 0 is continuously and densely embeds into E. Let us consider the Sobolev-Lions-type space W m p,γ (a, b; E 0 , E), consisting of all functions u ∈ L p,γ (a, b; E 0 ) that have generalized derivatives u (m) ∈ L p,γ (a, b; E) with the norm Now, let we define E-valued Sobolev-Lions-type spaces with mixed L p and L p,γ norms.Let Consider E-valued weighted space defined by For definition of R-sectorial operator see e.g.[7, p. 39] In a similar way as in [21, Theorems 2.3, 2.4] we have the following result.
Theorem 2.1.Assume the following conditions be satisfied: (3) there exists a bounded linear extension operator from W m p,γ (Ω; Then, the embedding Consider the BVP for the degenerate ordinary DOE with parameter where is a linear operator in a Banach space E for x ∈ (0, 1), ε is a small positive and λ is a complex parameter.
Theorem 2.4.Suppose the Condition 2.2 is satisfied.Then, the operator B ε is uniformly R-positive in L p (0, 1; E).

Theorem 3.4. Let the Condition 3.2 hold and let
Then, problem (2.1) has a unique solution u ∈ W [2] p,α (G; E(A), E) for f ∈ L p (G; E), |arg λ| ≤ ϕ with sufficiently large |λ| and the coercive uniform estimate holds Proof.By assumption and by Theorem 2.1, for all h > 0 we have the following Ehrling-Nirenberg-Gagliardo-type estimate Let O ε denote the operator generated by the problem (3.2) and ∂ [1] u ∂x k .
By using the estimate (3.7) we obtain that there is a δ ∈ (0, 1) such that Hence, from perturbation theory of linear operators we obtain the assertion.

Abstract Cauchy problem for degenerate parabolic equation with parameter
Consider the initial and BVP for degenerate parabolic equation with parameter: where u = u(x, t) is a solution, δ ki , β ki are complex numbers, ε k are positive parameters, a k are complex-valued functions on G, A(x) is a linear operator in a Banach space E, domains G, G k, G k0 , G kb , σ ik and x (k) are defined in Section 2 and For p =(p 0 , p), p =(p 1 , p 2 , . . ., p n ), G T = (0, T) × G, L p,fl (G T ; E) will denote the space of all E-valued weighted p-summable functions with mixed norm.p,α (G T ; E(A), E) and the following coercive estimate holds Proof.The problem (4.1) can be expressed as the following abstract Cauchy problem . By [18, §1.14], O ε is a generator of an analytic semigroup in F. Then by virtue of [28,Theorem 4.2], problem (4.3) has a unique solution u ∈ W 1 p 0 (0, T; D(O ε ), F) for f ∈ L p 0 (0, T; F) and sufficiently large d > 0.Moreover, the following uniform estimate holds Since L p 0 (G T ; F) = L p(G T ; E), by Theorem 3.3 we have Hence, the assertion follows from the above estimate.

Degenerate parabolic DOE on the moving domain
Consider the degenerate problem (4.1)-(4.2) on the moving domain G(s) = ∏ n k=1 (0, b k (s)): where the end points b k (s) depend of a parameter s, x k ∈ (0, b k (s)) and b k (s) are positive continues function, G k0 (s), G kb (s) are domains defined in Section 2, replacing (0, b k ) by (0, b k (s)) and p,α ((G(s)); E(A), E) for f ∈ L p (G T (s); E) and sufficiently d > 0.Moreover, the following coercive uniform estimate holds Proof.Under the substitution τ k = x k b k (s) the problem (5.1)-(5.2) reduced to the following BVP in fixed domain G: where ãk (τ) = a k (x(τ)), The problem (5.4)-(5.5), is a particular case of (4.1)-(4.2).So, by virtue of Theorem 4.1 we obtain the required assertion.
Proof.Consider the following linear problem By Theorem 4.1 and in view of Proposition 5.1 there exists a unique solution w ∈ W 1, [2] p,α (G T ; E(A), E) of the problem (6.3) for f ∈ L p (G T ; E) and sufficiently large d > 0 and it satisfies the following coercive estimate uniformly with respect to b ∈ (0 , b 0 ], i.e., the constant C 0 does not depends on f ∈ L p (G T ; E) and b ∈ (0 b 0 ] where We want to solve the problem (6.1)-(6.2) locally by means of maximal regularity of the linear problem (6.3) via the contraction mapping theorem.For this purpose, let w be a solution of the linear BVP (6.3).Consider a ball For given υ ∈ B r , consider the following linearized problem where V = {υ ki }, υ ki ∈ B ki .Define a map Q on B r by Qυ = u, where u is solution of (6.4).We want to show that Q(B r ) ⊂ B r and that Q is a contraction operator provided T and b k are sufficiently small, and r is chosen properly.In view of separability properties of the problem (6.3) we have Qυ − w Y = u − w Y ≤ C 0 { F(x, V) − F(x, 0) X + [B(0, W) − B(x, V)]υ X }.
By assumption (5) of condition 6.2 we get where R = C 1 r + w Y is a fixed number.In view of above estimates, by suitable choice of µ R , L R and for sufficiently small T ∈ (0, T 0 ) and b k ∈ (0 , b 0k ] we have Qυ − w Y ≤ r, i.e.Q(B r ) ⊂ B r .
Moreover, in a similar way we obtain By suitable choice of µ R , L R and for sufficiently small T ∈ (0, T 0 ) and b k ∈ (0, b 0k ) we obtain Qυ − Q ῡ Y < η υ − ῡ Y , η < 1, i.e.Q is a contraction operator.Eventually, the contraction mapping principle implies a unique fixed point of Q in B r which is the unique strong solution u ∈ W 1, [2] p,α (G T ; E(A), E).