Higher order moments for SPDE with monotone nonlinearities

Abstract

This paper introduces a new $p$-dependent coercivity condition through which $L^p$-moments for solutions can be obtained for a large class of SPDEs in the variational framework. If $p=2$, our condition reduces to the classically coercivity condition, which only yields second moments for the solution. The abstract result is shown to be optimal. Moreover, the results are applied to obtain $L^p$-moments of solutions for several classical SPDEs such as stochastic heat equations with Dirichlet and Neumann boundary conditions, Burgers' equation and the Navier-Stokes equations in two spatial dimensions. Furthermore, we can recover recent results for systems of SPDEs and higher order SPDEs using our unifying coercivity condition.