Besov Regularität von parabolischen partiellen Differentialgleichungen auf Lipschitz Gebieten

This project is concerned with partial differential equations (PDEs) of parabolic type on Lipschitz domains. We want to study the regularity of the solutions of such equations and are particularly interested in smoothness estimates in specific scales of Besov spaces, which determine the approximation order of adaptive and other nonlinear numerical approximation schemes. We wish to show that the Besov regularity is high enough to justify the use of adaptive schemes compared to non-adaptive (uniform) schemes. We shall mainly be concerned with approximation schemes based on wavelets.

Our starting point is to improve existing results for the heat equation and extend these results to linear parabolic PDEs with variable coefficients and also to nonlinear parabolic PDEs. We also want to show that when restricting ourselves to polygonal and polyhedral domains (compared to general Lipschitz domains) we get even higher Besov regularity.

Since the approximation rates obtained this way still depend on the spatial dimension, we secondly wish to study regularity of parabolic PDEs in generalized dominating mixed smoothness spaces. By using tensor-wavelets we want to beat the so-called 'curse of dimension' and obtain convergence rates independent of the underlying dimension of the domain.