Global existence, uniqueness and stability for nonlinear dissipative systems of bulk-interface interaction
- Disser, Karoline
2010 Mathematics Subject Classification
- 35K20 35K51 35R01 35R05 35B50 35B40
- Bulk-interface interaction, bulk-surface interaction, gradient structure, fast diffusion, porous media equation, nonlinear parabolic system, maximum principle, Poincaré inequality, exponential stability, maximal parabolic regularity, Schaefer's fixed point theorem
We consider a general class of nonlinear parabolic systems corresponding to thermodynamically consistent gradient structure models of bulk-interface interaction. The setting includes non-smooth geometries and e.g. slow, fast and "entropic'' diffusion processes under mass conservation. The main results are global well-posedness and exponential stability of equilibria. As a part of the proof, we show bulk-interface maximum principles and a bulk-interface Poincaré inequality. The method of proof for global existence is a simple but very versatile combination of maximal parabolic regularity of the linearization, a priori L∞-bounds and a Schaefer's fixed point argument. This allows us to extend the setting e.g. to Allen-Cahn dissipative dynamics and to include large classes of inhomogeneous boundary conditions and external forces.