TY - JOUR

T1 - Existence and uniqueness of the entropy solution of a stochastic conservation law with a Q-Brownian motion

AU - Funaki, Tadahisa

AU - Gao, Yueyuan

AU - Hilhorst, Danielle

N1 - Funding Information:
The authors would like to thank Dr Perla El Kettani for the helpful discussions and the GDRI ReaDiNet as well as the research funding of MathAM‐OIL, AIST c/o AIMR, Tohoku University, for the financial support.
Publisher Copyright:
© 2020 John Wiley & Sons, Ltd.

PY - 2020/6/1

Y1 - 2020/6/1

N2 - In this paper, we prove the existence and uniqueness of the entropy solution for a first-order stochastic conservation law with a multiplicative source term involving a (Formula presented.) -Brownian motion. After having defined a measure-valued weak entropy solution of the stochastic conservation law, we present the Kato inequality, and as a corollary, we deduce the uniqueness of the measure-valued weak entropy solution, which coincides with the unique weak entropy solution of the problem. The Kato inequality is proved by a doubling of variables method; to that purpose, we prove the existence and the uniqueness of the strong solution of an associated stochastic nonlinear parabolic problem by means of an implicit time discretization scheme; we also prove its convergence to a measure-valued entropy solution of the stochastic conservation law, which proves the existence of the measure-valued entropy solution.

AB - In this paper, we prove the existence and uniqueness of the entropy solution for a first-order stochastic conservation law with a multiplicative source term involving a (Formula presented.) -Brownian motion. After having defined a measure-valued weak entropy solution of the stochastic conservation law, we present the Kato inequality, and as a corollary, we deduce the uniqueness of the measure-valued weak entropy solution, which coincides with the unique weak entropy solution of the problem. The Kato inequality is proved by a doubling of variables method; to that purpose, we prove the existence and the uniqueness of the strong solution of an associated stochastic nonlinear parabolic problem by means of an implicit time discretization scheme; we also prove its convergence to a measure-valued entropy solution of the stochastic conservation law, which proves the existence of the measure-valued entropy solution.

KW - Kato inequality

KW - Q-Brownian motion

KW - associated parabolic problem

KW - existence and uniqueness of the entropy solution

KW - stochastic first-order conservation law

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U2 - 10.1002/mma.6329

DO - 10.1002/mma.6329

M3 - Article

AN - SCOPUS:85081266670

VL - 43

SP - 5860

EP - 5886

JO - Mathematical Methods in the Applied Sciences

JF - Mathematical Methods in the Applied Sciences

SN - 0170-4214

IS - 9

ER -