On the Cauchy Problem Associated to the Brinkman Flow in R3+

Michel Molina del Sol | Eduardo Arbieto Alarcón | Rafael José Iorio Junior

**Keywords:**Partial Differencial Equations | Integral-differential nonlinear evolution equations | Cauchy problem | Local and global well-posedness.

In this work we continue our study of the Cauchy problem associated to the Brinkman equations (see (2)-(3) below) which model fluid flow in certain types of porous media. Here we will consider the flow in the upper half-space under the assumption that the plane z = 0 is impenetrable to the fluid. This means that we will have to introduce boundary conditions that must be attached to the Brinkman equations. We study local and global well- posedness in appropriate Sobolev spaces introduced below, using Kato’s theory for quasilinear equations, parabolic regularization and a comparison principle for the solutions of the problem.