奇异动力学方程及其应用
Singular kinetic equations and applications
摘要：
本文利用\cite{GIP15}中引入的仿控制分布方法研究了$\mathbb{R}^{2d}$上的奇异动力学方程。我们首先在动力学背景下发展了次控制演算，并在奇异项乘积定义良好的假设下，利用它建立了线性奇异动力学方程的整体适定性.我们还证明了如何所需的产品可以定义的情况下，奇异项是一个高斯随机场的概率计算。有趣的是，虽然正则化近似的第零Wiener混沌中的项不为零，但它们收敛于适当的加权Besov空间，并且不需要重整化。作为应用，利用熵方法得到了一类奇异系数非线性动力学方程的整体适定性。此外，我们还解决了具有奇异漂移的非线性动力学分布依赖的随机微分方程的鞅问题。
英文摘要：
In this paper we study singular kinetic equations on $\mathbb{R}^{2d}$ by the paracontrolled distribution method introduced in \cite{GIP15}. We first develop paracontrolled calculus in the kinetic setting, and use it to establish the global well-posedness for the linear singular kinetic equations under the assumptions that the products of singular terms are well-defined. We also demonstrate how the required products can be defined in the case that singular term is a Gaussian random field by probabilistic calculation. Interestingly, although the terms in the zeroth Wiener chaos of regularization approximation are not zero, they converge in suitable weighted Besov spaces and no renormalization is required. As applications the global well-posedness for a nonlinear kinetic equation with singular coefficients is obtained by the entropy method. Moreover, we also solve the martingale problem for nonlinear kinetic distribution dependent stochastic differential equations with singular drifts.
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