An extrapolation result in the variational setting: improved regularity, compactness, and applications to quasilinear systems

In this paper we consider the variational setting for SPDE on a Gelfand triple (V, H, V∗) . Under the standard conditions on a linear coercive pair (A, B), and a symmetry condition on A we manage to extrapolate the classical L2 -estimates in time to Lp -estimates for some p > 2 without any further conditions on (A, B). As a consequence we obtain several other a priori regularity results of the paths of the solution. Under the assumption that V embeds compactly into H, we derive a universal compactness result quantifying over all (A, B). As an application of the compactness result we prove global existence of weak solutions to a system of second order quasi-linear equations.