DEGENERATE PARABOLIC STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS: QUASILINEAR CASE

• 成果类型: Article
• 署名作者: Debussche, Arnaud; Hofmanova, Martina; Vovelle, Julien
• 署名单位: Ecole Normale Superieure de Rennes (ENS Rennes); Max Planck Society; Centre National de la Recherche Scientifique (CNRS); Ecole Centrale de Lyon; Institut National des Sciences Appliquees de Lyon - INSA Lyon; Universite Claude Bernard Lyon 1; Universite Jean Monnet
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/15-AOP1013
• 发表日期: 2016
• 页码: 1916-1955
• 关键词: scalar conservation-laws; kinetic formulation; wave-equations

摘要

In this paper, we study the Cauchy problem for a quasilinear degenerate parabolic stochastic partial differential equation driven by a cylindrical Wiener process. In particular, we adapt the notion of kinetic formulation and kinetic solution and develop a well-posedness theory that includes also an L-1-contraction property. In comparison to the previous works of the authors concerning stochastic hyperbolic conservation laws [J. Funct. Anal. 259 (2010) 1014-1042] and semilinear degenerate parabolic SPDEs [Stochastic Process. Appl. 123 (2013) 4294-4336], the present result contains two new ingredients that provide simpler and more effective method of the proof: a generalized Ito formula that permits a rigorous derivation of the kinetic formulation even in the case of weak solutions of certain nondegenerate approximations and a direct proof of strong convergence of these approximations to the desired kinetic solution of the degenerate problem.