论文标题
二维随机非线性波方程的全局动力学
Global dynamics for the two-dimensional stochastic nonlinear wave equations
论文作者
论文摘要
我们研究了随机非线性波方程（SNLW）的全球时间动力学，并具有添加剂时空白噪声强迫，并在二维圆环上构成。我们本文的目标是两个方面。 （i）通过引入一个混合参数，将随机环境中的$ i $ - 方法与gronwall型参数相结合，我们首先证明了（重新归一化的）立方snlw在偏腹情况下的全球范围良好。我们的论点产生了解决方案Sobolev规范的双重指数增长。 （ii）然后，我们研究了散落的情况下的随机阻尼非线性波方程（SDNLW）。特别是，通过应用波尔加因的不变量参数，我们证明了（重新归一化的）depocations sdnlw的全球良好性相对于吉布斯的测量和吉布斯度量的不变性。
We study global-in-time dynamics of the stochastic nonlinear wave equations (SNLW) with an additive space-time white noise forcing, posed on the two-dimensional torus. Our goal in this paper is two-fold. (i) By introducing a hybrid argument, combining the $I$-method in the stochastic setting with a Gronwall-type argument, we first prove global well-posedness of the (renormalized) cubic SNLW in the defocusing case. Our argument yields a double exponential growth bound on the Sobolev norm of a solution. (ii) We then study the stochastic damped nonlinear wave equations (SdNLW) in the defocusing case. In particular, by applying Bourgain's invariant measure argument, we prove almost sure global well-posedness of the (renormalized) defocusing SdNLW with respect to the Gibbs measure and invariance of the Gibbs measure.
