Abstract elliptic operators appearing in atmospheric dispersion

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 L-p-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 L-p-spaces.