AI 中文总结
研究具有正则化热噪声的齐次朗道方程，基于确定性朗道方程结构等驱动因素构建模型。通过三级近似方案证明适度软势下概率弱解存在性，解满足相关守恒和估计，对特定噪声基还得到精细熵不等式。
AI 中文摘要
我们引入并分析了一个具有正则化热噪声的波动齐次朗道方程。该模型由确定性朗道方程的非局部梯度流结构、涨落耗散原理以及类卡茨保守朗道粒子系统的鞅涨落协方差所驱动。噪声以朗道散度形式书写，在速度对中反对称，并在引入速度相关性后按斯特拉托诺维奇意义解释。为处理平方根迁移率的真空奇异性和非局部斯特拉托诺维奇到伊藤修正，我们用正则系数代替迁移率。对于适度软势，我们证明了正则化波动齐次朗道方程概率弱解的存在性。证明基于结合伽辽金近似、系数正则化、人工扩散以及\(L^2\)和\(L^1\)框架下紧致性的三级近似方案。解满足质量守恒、能量不等式和熵耗散估计。最后，对于满足切向无散条件的一类特殊允许噪声基，我们得到了一个精细的熵不等式，其中期望熵相对于初始熵是非增的。
英文摘要
We introduce and analyze a fluctuating homogeneous Landau equation with regularised thermal noise. The model is motivated by the nonlocal gradient flow structure of the deterministic Landau equation, the fluctuation--dissipation principle, and the covariance of the martingale fluctuations of a Kac-like conservative Landau particle system. The noise is written in Landau-divergence form, is antisymmetric in the pair of velocities, and is interpreted in the Stratonovich sense after introducing a velocity correlation. To handle the vacuum singularity of the square-root mobility and the nonlocal Stratonovich-to-Itô correction, we replace the mobility by a regular coefficient. For moderately soft potentials, we prove the existence of probabilistic weak solutions to the regularised fluctuating homogeneous Landau equation. The proof is based on a three-level approximation scheme combining Galerkin approximations, coefficient regularisations, artificial diffusion, and compactness in both $L^2$ and $L^1$ frameworks. The solutions satisfy mass conservation, an energy inequality, and the entropy dissipation estimate. Finally, for a special class of admissible noise bases satisfying a tangential divergence-free condition, we obtain a refined entropy inequality in which the expected entropy is non-increasing relative to the initial entropy.
Comments102 pages
