Quasilinear parabolic stochastic evolution equations via maximal $ L^{p} $-regularity

TL;DR

This paper establishes local existence and uniqueness for quasilinear stochastic parabolic equations using maximal L^p-regularity, with applications to fluid dynamics and convection-diffusion problems, including conditions for global solutions.

Contribution

It extends maximal L^p-regularity theory to quasilinear stochastic equations, providing new well-posedness results and applications to complex fluid models.

Findings

Proved local existence and uniqueness of solutions.

Applied theory to convection-diffusion and Navier-Stokes equations.

Established conditions for global solutions in specific cases.

Abstract

We study the Cauchy problem for an abstract quasilinear stochastic parabolic evolution equation on a Banach space driven by a cylindrical Brownian motion. We prove existence and uniqueness of a local strong solution up to a maximal stopping time, that is characterised by a blow-up alternative. The key idea is an iterative application of the theory about maximal $ L^p $- regularity for semilinear stochastic evolution equations by Van Neerven, Veraar and Weis. We apply our local well-posedness result to a convection-diffusion equation on a bounded domain with Dirichlet, Neumann or mixed boudary conditions and to a generalized Navier-Stokes equation describing non-Newtonian fluids. In the first example, we can even show that the solution exists globally.