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Existence and uniqueness of the entropy solution of a stochastic conservation law with a Q ‐Brownian motion
Author(s) -
Funaki Tadahisa,
Gao Yueyuan,
Hilhorst Danielle
Publication year - 2020
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.6329
Subject(s) - mathematics , uniqueness , conservation law , entropy (arrow of time) , weak solution , mathematical analysis , discretization , brownian motion , entropy rate , joint quantum entropy , principle of maximum entropy , statistics , physics , quantum mechanics
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 Q ‐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.