Quasilinear parabolic stochastic evolution equations via maximal L^p\n -regularity

2016/01/01 by Luca Hornung, Hornung, Luca
Computer Science · Economics, Econometrics and Finance · Mathematics · #35K57 #35K59 #35Q35 #58D25 #60H15 #60H30 #65J08 #76A05 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Partial Differential Equations #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1611.09241

openalex publication_date 2016/11/28 · openalex created_date 2022/08/16 · openalex updated_date 2026/07/28

Abstract

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

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