Generalızed Local Mıxed Morrey Estımates for Lınear Ellıptıc Systems with Dıscontınuous Coeffıcıents

Abstract

We consider the 2b-order linear elliptic systems (Equation presented) in the generalized local mixed Morrey spaces M{x0}p,ϕ(Rn) and generalized mixed Morrey spaces Mp,ϕ (Rn ), where the principal coefficients Aα are functions with vanishing mean oscillation. We obtain local regularity results for the strong solutions to L(x, D) in the spaces M{x0}p,ϕ(Rn) and Mp,ϕ (Rn ) . Solutions to the linear elliptic systems with discontinuous coefficients are used in many practical applications. However, theoretical understanding of the solutions to these equations is incomplete. We can apply this local regularity results in generalized local mixed Morrey spaces to study the regularity in generalized local mixed Morrey spaces of of the Navier-Stokes equations. Solutions to the Navier-Stokes equations are used in many practical applications. However, theoretical understanding of the solutions to these equations is incomplete. In particular, solutions of the Navier-Stokes equations often include turbulence, which remains one of the greatest unsolved problems in physics, despite its immense importance in science and engineering. The possibilities nowadays to exploit supercomputers and highly developed numerical methods for nonlinear partial differential equations allow us to determine even the general solutions to the Navier-Stokes equations. However the difficulties become greater with increasing Reynolds number. This has to do with the particular structure of the solutions at high Reynolds numbers. Note that in the limiting case of high Reynolds numbers, most of these exact solutions have a boundary-layer character.