Lars Diening

Sobolev Spaces with variable Exponents and Applications to Fluid Dynamics

We consider the model of electrorheological fluids by Rajagopal and Ruzicka. The equation basically behaves like the Navier-Stokes equation with the Laplacian replaced by the p(x)-Laplacian. The natural spaces to solve this equation are the Sobolev spaces with variable Exponents.

After a brief introduction to the theory of these spaces, we show how the theory of Lipschitz-approximations can be generalized to this case of variable exponents. We show existence of solutions for rather small values of p.