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arXiv · 1611.09241

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

Luca Hornung
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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.

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Luca Hornung. 2016-11-28. Quasilinear parabolic stochastic evolution equations via maximal $ L^{p} $-regularity. https://arxiv.org/abs/1611.09241

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