Special Session 108: New Developments in porous media

Interior $W^{1,q}$ estimates for solutions of nonlinear degenerate parabolic systems

Truyen Nguyen
University of Akron
We consider nonlinear parabolic systems of the form $u_t = \mbox{div}\, \mathbf{A}(x,t,u,\nabla u) + \mathbf{B}(x,t, u,\nabla u )$ which include those of $p$-Laplacian type. In this talk, we will discuss some results concerning local integrability of gradients of weak solutions to the system. In particular, we derive interior $L^q$ estimates for the gradient when $\mathbf{A}$ is possibly dicontinuous in the $x$ variable. The dependence of the principal part on the $u$ variable made it difficult to perform any scaling analysis and we handle it by using the intrinsic geometry method of DiBenedetto together with our two-parameter scaling technique.