论文标题
全球存在于Camassa-Holm方程和传输噪声的耗散解决方案
Global existence of dissipative solutions to the Camassa--Holm equation with transport noise
论文作者
论文摘要
我们考虑了一个非线性随机部分微分方程（SPDE），该方程采用了由对流，依赖于位置的噪声项扰动的camassa-holm方程的形式。我们建立了耗时弱的弱星属解决方案的第一个全球及时存在结果，并具有一般有限能量的初始数据。该溶液是作为抛物线SPDE的经典溶液的极限。该证明将特定于模型的统计估计与紧凑技术的随机传播结合在一起，并系统地使用紧密度和A.S.在特定的准派空间上随机变量的表示。噪声函数的空间依赖性使得对先验估计和各种重态化的分析更加困难，从而产生了方程中的Martingale部分和二阶Stratonovich-Itô校正项引起的非线性项。
We consider a nonlinear stochastic partial differential equation (SPDE) that takes the form of the Camassa--Holm equation perturbed by a convective, position-dependent, noise term. We establish the first global-in-time existence result for dissipative weak martingale solutions to this SPDE, with general finite-energy initial data. The solution is obtained as the limit of classical solutions to parabolic SPDEs. The proof combines model-specific statistical estimates with stochastic propagation of compactness techniques, along with the systematic use of tightness and a.s. representations of random variables on specific quasi-Polish spaces. The spatial dependence of the noise function makes more difficult the analysis of a priori estimates and various renormalisations, giving rise to nonlinear terms induced by the martingale part of the equation and the second-order Stratonovich--Itô correction term.
