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Entropy-dissipation methods have been developed recently to investigate the well-posedness and the qualitative behavior of solutions to nonlinear parabolic equations. The strength of the method lies in its flexibility and applicability to a large class of nonlinear equations. In this talk, two aspects of entropy methods will be detailed. First, a priori estimates for higher-order parabolic equations will be derived by means of (Lyapunov) functionals which are called entropies. The estimations are usually based on skillful integration by parts. These integrations can be made systematic by formulating the task as a decision problem in real algebraic geometry, which can be solved in an algorithmic way. The method is applied to a fourth- and sixth-order quantum diffusion problem for semiconductors.
Second, we present a technique to derive a priori estimates for cross-diffusion systems whose diffusion matrix may be non-symmetric and not positive definite. The key idea is to exploit a formal gradient-flow structure. The corresponding entropy (or free energy) functional yields new variables which make the diffusion matrix positive definite. In certain cases, the new variables also allow for the proof of bounded solutions, although no classical maximum principle can be used. The method is applied to a diffusion system arising in tumor-growth modeling.
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