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 $(V,H,V^{\ast })$ . Under the standard conditions on a linear coercive pair (A, B), and a symmetry condition on A we manage to extrapolate the classical ${\text{L}}^2$ -estimates in time to ${\text{L}}^p$ -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.