具有Lévy噪声初始数据的Burgers湍流中的激波结构
Structure of shocks in Burgers turbulence with Lévy noise initial data
摘要：
本文研究了一维无粘Burgers方程在初始速度为Lévy噪声时的激波结构，以及当初始势为双边Lévy过程$ψ_0$时的激波结构.当$ψ_0$在Vigon意义下是突变的或与$\limsup_{|h| \downarrow 0} h^{-2} ψ_0(h) = \infty$有界变化时，我们证明了零速度点集是再生的，并且在后一种情况下，这一集等于拉格朗日正则点集，这是非空的。当$ψ_0$是突变的，我们表明，激波结构是离散的。当$ψ_0$被侵蚀，我们表明，有没有稀疏间隔。
英文摘要：
We study the structure of the shocks for the inviscid Burgers equation in dimension 1 when the initial velocity is given by Lévy noise, or equivalently when the initial potential is a two-sided Lévy process $ψ_0$. When $ψ_0$ is abrupt in the sense of Vigon or has bounded variation with $\limsup_{|h| \downarrow 0} h^{-2} ψ_0(h) = \infty$, we prove that the set of points with zero velocity is regenerative, and that in the latter case this set is equal to the set of Lagrangian regular points, which is non-empty. When $ψ_0$ is abrupt we show that the shock structure is discrete. When $ψ_0$ is eroded we show that there are no rarefaction intervals.
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