Dérivation systématique d'estimations a priori pour des E.D.P. non-linéaires d'ordre supérieur et élimination des quantificateurs.

The proof of analytical and numerical properties of solutions to
nonlinear evolution equations is usually based on appropriate
a priori estimates and monotonicity properties of Lyapunov
functionals, which are called here entropies. These estimates
can be shown by subtle integration by parts. However,
such proofs are usually skillful and not systematic. In this
talk a systematic method for the derivation of a priori
estimates for a large class of nonlinear evolution equations
of higher order in one and several variables with periodic
boundary conditions is presented. This class of equations
contains, for instance, the porous medium equation, the thin-film
equation, and a semiconductor quantum fluid model.

The main idea is the identification of the integrations by parts
with polynomial manipulations. The proof of a priori estimates
is then formally equivalent to the solution of a decision
problem known in real algebraic geometry, which can be solved
algorithmically by quantifier elimination. The method also allows to
prove decay rates of the solutions to their equilibrium and to derive
new logarithmic Sobolev inequalities.