arXiv 2603.266502026-03math.AP

非线性动能福克-普朗克方程的基本解

The fundamental solution of a nonlinear kinetic Fokker-Planck equation

IRIMAS, Université de Haute-Alsace（阿尔萨斯高地大学）；CEREMADE (CNRS UMR n ∘ 7534), PSL university, Université Paris-Dauphine（巴黎第九大学）；Inria-Paris, MOKAPLAN（法国国家信息与自动化研究所）

Giovanni Brigati, Guillaume Carlier, Jean Dolbeault

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AI总结：

本文研究非线性动能方程的基本解，涉及孔隙介质或快速扩散算子，通过自相似变换将问题转化为非线性动能福克-普朗克方程，揭示了自相似行为和稳定性性质。

摘要：

本文致力于研究涉及孔隙介质或快速扩散算子的非线性动能方程的基本解。此类非线性具有有趣的缩放性质，导致基本解呈现自相似行为。基本解指初始数据为狄拉克分布，并且控制大时间渐进行为的解。通过自相似变换，方程转化为具有调和约束的非线性动能福克-普朗克方程，中间渐进行为转化为特殊稳态解的稳定性性质，该解吸引大时间的解。在均匀情况下（纯非线性扩散），问题简化为具有巴伦布-帕特尔自相似轮廓的经典非线性扩散方程。意外的是，这种美丽结构在动能层面得以保持，对相对熵估计、详细中间渐进行为和非线性扩散极限在适应的函数空间中产生显著影响。

英文摘要：

This paper is devoted to a fundamental solution of a nonlinear kinetic equation involving a porous medium or fast diffusion operator acting on velocities. Such a nonlinearity has interesting scaling properties, which result in a self-similar behaviour of the fundamental solution. Here fundamental solution means a Dirac distribution initial datum which moreover governs the large time asymptotics of a large class of solutions. Using a self-similar change of variables, the equation becomes a nonlinear kinetic Fokker-Planck equation with harmonic confinement and the intermediate asymptotics regime is transformed into a stability property of a special stationary solution, which attracts the solutions for large times. In the homogeneous case (pure nonlinear diffusion), the problem is reduced to a classical nonlinear diffusion equation with Barenblatt-Pattle self-similar profiles. Unexpectedly, this beautiful structure is preserved at kinetic level, with remarkable consequences for relative entropy estimates, detailed intermediate asymptotics and nonlinear diffusion limits in adapted functional spaces.

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