Degenerate parabolic stochastic partial differential equations: Quasilinear case

• Debussche, Arnaud ^[2] ; Hofmanová, Martina ^[1] ; Vovelle, Julien ^[3]
  1. [1] Max Planck Institute for Mathematics in the Sciences

     Max Planck Institute for Mathematics in the Sciences

     Kreisfreie Stadt Leipzig, Alemania

     
  2. [2] ENS Rennes
  3. [3] Université Lyon I

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• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 44, Nº. 3, 2016, págs. 1916-1955
• Idioma: inglés
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  • 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 L1-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 Itô 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.