An extrapolation result in the variational setting: improved regularity, compactness, and applications to quasilinear systems
Sebastian Bechtel, Mark Veraar · Stochastic Partial Differential Equations Analysis and Computations · 2025
Abstract In this paper we consider the variational setting for SPDE on a Gelfand triple $$(V, H, V^*)$$ ( 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 $$\textrm{L}^2$$ L 2 -estimates in time to $$\textrm{L}^p$$ L p -estimates for some $$p>2$$ 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.