随机 Landau--Lifshitz--Baryakhtar 方程在 $\mathbb{R}^d$ 中的适定性

中文摘要 AI 辅助

本文研究 $\mathbb{R}^d$（$d=1,2,3$）上受 Stratonovich 高斯扰动影响的随机 Landau--Lifshitz--Baryakhtar 方程的柯西问题。我们为 $\mathbb{L}^2$、$\mathbb{H}^1$ 和 $\mathbb{H}^2$ 中的初始数据建立了低正则性适定性理论。对于 $\mathbb{L}^2$ 初始数据，我们在一维和二维中证明了全局极弱解的存在性和路径唯一性，而在三维中，我们在每个有限时间区间上构造了鞅极弱解。对于任意 $\mathbb{H}^1$ 初始数据，我们证明了在所有维度 $d\leq3$ 中全局路径弱解的存在性和唯一性。此外，对于 $\mathbb{H}^2$ 初始数据，我们在相同维度范围内建立了全局路径强解的存在性和唯一性。分析基于频率截断近似方案与全空间环境中的随机紧性论证相结合。全局延拓的关键步骤是在 $\mathbb{H}^1$ 正则性下证明随机有效场平衡，其中漂移项仅存在于负 Sobolev 空间中。空间磨光化与极限过程得出局部弱解的能量恒等式。随后，一个强制性修正给出全局能量矩并排除有限时间爆破。

英文摘要

This paper studies the Cauchy problem for the stochastic Landau--Lifshitz--Baryakhtar equation on $\mathbb{R}^d$, $d=1,2,3$, subject to Stratonovich Gaussian perturbations. We establish a low-regularity well-posedness theory for initial data in $\mathbb{L}^2$, $\mathbb{H}^1$, and $\mathbb{H}^2$. For $\mathbb{L}^2$ initial data, we prove the existence and pathwise uniqueness of global very weak solutions in dimensions one and two, while in dimension three we construct martingale very weak solutions on every finite time interval. For arbitrary $\mathbb{H}^1$ initial data, we prove the existence and uniqueness of global pathwise weak solutions in all dimensions $d\leq3$. Furthermore, for $\mathbb{H}^2$ initial data, we establish the existence and uniqueness of global pathwise strong solutions in the same range of dimensions. The analysis is based on a frequency-truncation approximation scheme combined with stochastic compactness arguments in the whole-space setting. The key step in global continuation is to justify the stochastic effective-field balance at $\mathbb{H}^1$ regularity, where the drift is available only in a negative Sobolev space. Spatial mollification and passage to the limit yield the energy identity for local weak solutions. A coercive modification then gives global energy moments and excludes finite-time blow-up.

发表机构

• Huazhong University of Science and Technology（华中科技大学）

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