论文标题
在$ l^2 $的冲击波稳定性上，用于有限的汉堡解决方案
On the $L^2$ stability of shock waves for finite-entropy solutions of Burgers
论文作者
论文摘要
我们证明了$ l^2 $稳定性估计弱的且可能\ emph {non-entropic}之间的熵冲击，标量保护法的解决方案$ \ partial_t u+\ partial_x f（u）= 0 $，具有严格的convex prom function $ f $。这概括了Leger和Vasseur的先前结果，他们在熵解决方案中证明了$ l^2 $稳定性。我们的主要结果，估计\ begin {align*} \ int _ {\ mathbb r} | u（t，t，\ cdot）-u_0^{shock}（\ cdot -x（t））|^2 \，dx \ leq \ leq \ leq \ leq \ int _ { +cμ_+（[0，t] \ times \ r），\ end {align*}对于某些Lipschitz换算$ x（t）$，包括一个错误术语，该术语占熵生产量的积极部分$μ= \ partial_t（u^2/2）+^2/2）+\ partial_x q（u^partial_x q（u）$，$ q'（$ q'（u）$ q'（u）（u）= uf'（u）= uf'（u）=在这种一般的非全面环境中，稳定性估计值与非对称相互作用粒子系统的流体动力极限有关。 我们的证明调整了Leger和Vasseur设计的方案，其中一个人构造了一个偏移$ x（t）$，该计划允许从左侧的时间衍生上的上方绑定。主要区别在于一个事实，即我们的解决方案$ u（t，\ cdot）$可能在$ x = x（t）$的情况下会出现非电气冲击，并且在这种情况下需要新的界限。我们还将此稳定性估算到具有有限变化的初始数据。
We prove $L^2$ stability estimates for entropic shocks among weak, possibly \emph{non-entropic}, solutions of scalar conservation laws $\partial_t u+\partial_x f(u)=0$ with strictly convex flux function $f$. This generalizes previous results by Leger and Vasseur, who proved $L^2$ stability among entropy solutions. Our main result, the estimate \begin{align*} \int_{\mathbb R} |u(t,\cdot)-u_0^{shock}(\cdot -x(t))|^2\,dx\leq \int_{\mathbb R}|u_0-u_0^{shock}|^2 +Cμ_+([0,t]\times\R), \end{align*} for some Lipschitz shift $x(t)$, includes an error term accounting for the positive part of the entropy production measure $μ=\partial_t(u^2/2)+\partial_x q(u)$, where $q'(u)=uf'(u)$. Stability estimates in this general non-entropic setting are of interest in connection with large deviation principles for the hydrodynamic limit of asymmetric interacting particle systems. Our proof adapts the scheme devised by Leger and Vasseur, where one constructs a shift $x(t)$ which allows to bound from above the time-derivative of the left-hand side. The main difference lies in the fact that our solution $u(t,\cdot)$ may present a non-entropic shock at $x=x(t)$ and new bounds are needed in that situation. We also generalize this stability estimate to initial data with bounded variation.
