A selfdual variational principle with minimal hypothesis and applications to stationary, dynamic and stochastic equations

Abstract

We revisit the self-dual variational approach to the resolution of non-linear stationary, parabolic and stochastic partial differential equations involving a monotone non-linearity, by relaxing the hypothesis under which it is applicable, in two directions: The prohibitive coercivity condition which was problematic for applications to evolution equations, be them deterministic or stochastic, and the restrictive “regularity” property that required the domain of the non-linear operator under study to be large. The new principle is applied to give more direct proofs for the existence of weak solutions for incompressible Navier-Stokes equations in their stationary, dynamic or stochastic versions.