关于磁场中带噪声的随机磁流体动力学系统弱鞅解的存在性

中文摘要 AI 辅助

我们证明了随机磁流体动力学（MHD）系统的解的存在性，其中随机性通过随机初始数据和仅出现在感应方程中的随机积分引入，而流体方程保持确定性。我们关注绝热指数 $\gamma$ 满足 $\gamma > \frac{3}{2}$ 的情形。存在性证明采用惩罚方法。我们定义了鞅解的概念，并建立了其存在的充分条件。随后，证明通过随机紧性方法进行。利用能量不等式，我们推导出关于期望的先验估计。由于问题的随机性质，我们证明了惩罚解在分布意义上的收敛，并验证了极限对象是一个鞅解。

英文摘要

We prove the existence of solutions to the stochastic magnetohydrodynamics (MHD) system, where randomness is introduced through random initial data and a stochastic integral appearing solely in the induction equation, while the fluid equations remain deterministic. We focus on the case where the adiabatic exponent $γ$ satisfies $γ> \frac{3}{2}$. The existence proof is carried out using the penalization method. We define the notion of a martingale solution and establish sufficient conditions for its existence. The proof then proceeds by means of the stochastic compactness method. Using an energy inequality, we derive a priori estimates in terms of expectation. Due to the stochastic nature of the problem, we demonstrate convergence in law of the penalized solutions and verify that the limiting object is a martingale solution.

发表机构

• Charles University, Faculty of Mathematics and Physics（查理大学数学与物理学院）
• Institute of Mathematics of the Academy of Sciences of the Czech Republic（捷克科学院数学研究所）

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