The Annals of Probability

Well-posedness and regularity for quasilinear degenerate parabolic-hyperbolic SPDE

Benjamin Gess and Martina Hofmanová

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Abstract

We study quasilinear degenerate parabolic-hyperbolic stochastic partial differential equations with general multiplicative noise within the framework of kinetic solutions. Our results are twofold: First, we establish new regularity results based on averaging techniques. Second, we prove the existence and uniqueness of solutions in a full $L^{1}$ setting requiring no growth assumptions on the nonlinearities. In addition, we prove a comparison result and an $L^{1}$-contraction property for the solutions, generalizing the results obtained in [Ann. Probab. 44 (2016) 1916–1955].

Article information

Source
Ann. Probab., Volume 46, Number 5 (2018), 2495-2544.

Dates
Received: June 2017
First available in Project Euclid: 24 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.aop/1535097633

Digital Object Identifier
doi:10.1214/17-AOP1231

Mathematical Reviews number (MathSciNet)
MR3846832

Zentralblatt MATH identifier
06964342

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60] 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15]

Keywords
Quasilinear degenerate parabolic stochastic partial differential equation kinetic formulation kinetic solution velocity averaging lemmas renormalized solutions

Citation

Gess, Benjamin; Hofmanová, Martina. Well-posedness and regularity for quasilinear degenerate parabolic-hyperbolic SPDE. Ann. Probab. 46 (2018), no. 5, 2495--2544. doi:10.1214/17-AOP1231. https://projecteuclid.org/euclid.aop/1535097633


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