### Abstract

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.

Original language | English |
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Pages (from-to) | 5860-5886 |

Number of pages | 27 |

Journal | Mathematical Methods in the Applied Sciences |

Volume | 43 |

Issue number | 9 |

DOIs | |

Publication status | Published - 2020 Jun 1 |

### Keywords

- Kato inequality
- Q-Brownian motion
- associated parabolic problem
- existence and uniqueness of the entropy solution
- stochastic first-order conservation law

### ASJC Scopus subject areas

- Mathematics(all)
- Engineering(all)

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## Cite this

*Mathematical Methods in the Applied Sciences*,

*43*(9), 5860-5886. https://doi.org/10.1002/mma.6329