论文标题
$ 2 \ times 2 $保护法律的独特性和弱BV稳定性
Uniqueness and weak-BV stability for $2\times 2$ conservation laws
论文作者
论文摘要
让1-D的双曲线保护定律系统具有两个未知数，并具有凸熵。我们认为小型$ bv $函数的家族是该方程式的全球解决方案。对于任何小$ bv $初始数据，已知这种全球解决方案存在。此外，众所周知，它们在$ bv $ solutions中是唯一的，可以验证所谓的温和振荡条件或类似太空曲线的有界变化条件。在本文中，我们表明这些解决方案在系统的较大类别（甚至可能不是$ bv $）的解决方案中是稳定的。该结果扩展了经典的弱唯一性结果，从而可以将其与平滑的解决方案进行比较。的确，我们的结果将这些结果扩展到弱$ bv $唯一性结果，其中只有一种解决方案应该是小$ bv $，而另一种解决方案可以来自大型类。由于我们的结果，对于$ BV $理论中解决方案的唯一性，对于具有2个未知数的系统而言，驯服振荡条件和符合空间曲线的有界变化条件并不是必需的。该方法基于$ l^2 $。它是由A Shifts A-Contaction的理论组成的，其中合适的权重函数$ A $是通过前跟踪方法生成的。
Let a 1-d system of hyperbolic conservation laws, with two unknowns, be endowed with a convex entropy. We consider the family of small $BV$ functions which are global solutions of this equation. For any small $BV$ initial data, such global solutions are known to exist. Moreover, they are known to be unique among $BV$ solutions verifying either the so-called Tame Oscillation Condition, or the Bounded Variation Condition on space-like curves. In this paper, we show that these solutions are stable in a larger class of weak (and possibly not even $BV$) solutions of the system. This result extends the classical weak-strong uniqueness results which allow comparison to a smooth solution. Indeed our result extends these results to a weak-$BV$ uniqueness result, where only one of the solutions is supposed to be small $BV$, and the other solution can come from a large class. As a consequence of our result, the Tame Oscillation Condition, and the Bounded Variation Condition on space-like curves are not necessary for the uniqueness of solutions in the $BV$ theory, in the case of systems with 2 unknowns. The method is $L^2$ based. It builds up from the theory of a-contraction with shifts, where suitable weight functions $a$ are generated via the front tracking method.
