Abstract elliptic operators appearing in atmospheric dispersion

Veli B Shakhmurov,Aida Sahmurova

doi:10.1186/1687-2770-2014-43

Abstract

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.

Highlights

1 Introduction It is well known that many classes of PDEs, pseudo DEs and integro DEs can be expressed as a differential-operator equation (DOE)
Theorem A Let E be a Banach space, A be a φ-positive operator in E with bound M
Jλ f, where Ojλ = Oj + λ are local operators generated by boundary value problems (BVPs) with constant coefficients of type ( . )-( . ) and Kjλ and jλ are uniformly bounded operators defined in the proof of Theorem

Summary

Introduction

It is well known that many classes of PDEs, pseudo DEs and integro DEs can be expressed as a differential-operator equation (DOE). DαW m,p ; E(A), E ⊂ Lp ; E(A), E κ,p is continuous and there exists a positive constant Cμ such that for all u ∈ Wpl ( ; E(A), E) the uniform estimate holds: Dα u Lp( ;(E(A),E)κ,p) ≤ Cμ hμ u W m,p( ;E(A),E) + h–( –μ) u Lp( ;E). Theorem A Let E be a Banach space, A be a φ-positive operator in E with bound M, φ. Assume the following conditions are satisfied: ( ) a = , a ∈ S(φ ) ∩ C/R+, for φ + φ < π , p ∈ ( , ∞); ( ) η = (– )m α β – (– )m α β = , |αk| + |βk| > ; ( ) A is a R-positive operator in a UMD-space E, m is a nonnegative integer. By continuing this process n times we obtain the conclusion

Consider the BVP for DOE with variable coefficients n

Let all conditions of Theorem hold for m

Bki w