Higher order moments for SPDE with monotone nonlinearities

TL;DR

This paper develops a new $p$-dependent coercivity condition that enables the derivation of $L^p$-moments for solutions to a broad class of SPDEs, extending classical results and unifying various cases.

Contribution

Introduces an optimal $p$-dependent coercivity condition for SPDEs, allowing for $L^p$-moment estimates across multiple classical and higher order SPDEs.

Findings

Derived $L^p$-moments for stochastic heat equations with boundary conditions

Established $L^p$-moments for Burgers' and Navier-Stokes equations in 2D

Unified treatment of systems and higher order SPDEs using the new coercivity condition

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.