Existence and uniqueness of the entropy solution of a stochastic conservation law with a Q‐Brownian motion
Existence and uniqueness of the entropy solution of a stochastic conservation law with a Q‐Brownian motion
复制标题
Q-布朗运动随机守恒定律熵解的存在唯一性
DOI：
10.1002/mma.6329
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发表时间：
2020
影响因子：
2.9
通讯作者：
Hilhorst Danielle
中科院分区：
文献类型：
--
作者：
Funaki Tadahisa;Gao Yueyuan;Hilhorst Danielle
本文证明了一类具有乘性源项的一阶随机守恒律的熵解的存在唯一性。在定义了随机守恒律的测值弱熵解之后，我们给出了Kato不等式，作为推论，我们得到了测值弱熵解的唯一性，它与问题的唯一弱熵解重合。利用变量加倍的方法证明了Kato不等式，并利用隐式时间离散化方法证明了伴随的随机非线性抛物问题强解的存在唯一性，证明了其收敛到随机守恒律的测值熵解，从而证明了测值熵解的存在性.
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‐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.