Luca Hornung | Searcharxiv

arXiv2017

Strong solutions to a nonlinear stochastic Maxwell equation with a retarded material law

We study the Cauchy problem for a semilinear stochastic Maxwell equation with Kerr-type nonlinearity and a retarded material law. We show existence and uniqueness of strong solutions using a refined Faedo-Galerkin method and spectral multiplier theorems for the Hodge-Laplacian. We also make use of a rescaling transformation that reduces the problem to an equation with additive noise to get an appropriate a priori estimate for the solution.

math.PR↗

arXiv2017

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

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.

math.FA↗