Higher order moments for SPDE with monotone nonlinearities

Abstract

This paper introduces a new p-dependent coercivity condition through which L pmoments 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.

The critical variational setting for stochastic evolution equations

“…Since it does not give more flexibility in the examples we consider, we prefer to work with (3.8). A variant of condition (3.15) with η = p−2 2 with p > 2 is considered in [18], where it is used to establish bounds on higher order moments of the solution.…”

“…Further progress was made in the papers [10], where the Lévy case was considered, and in [28], where the local monotonicity conditions are weakened. Recently in the papers [18,31] a new type of coercivity condition has been found which gives sufficient conditions for L p (Ω)-estimates as well.…”