Uniqueness and Weak-BV Stability for $$2\times 2$$ Conservation Laws
Uniqueness and Weak-BV Stability for $$2\times 2$$ Conservation Laws
复制标题
$$2 imes 2$$ 守恒定律的唯一性和弱 BV 稳定性
DOI：
10.1007/s00205-022-01813-0
复制
发表时间：
2022
影响因子：
2.5
通讯作者：
Vasseur, Alexis F.
中科院分区：
文献类型：
--
作者：
Chen, Geng;Krupa, Sam G.;Vasseur, Alexis F.
设一维双曲守恒律方程组具有两个未知量，且具有凸熵。我们考虑一族小BV函数，它们是该方程的整体解。对于任何小的BV初始数据，这样的整体解决方案是已知的存在。此外，他们被认为是唯一的证明所谓的驯服振荡条件，或有界变差条件的类空曲线。在本文中，我们表明，这些解决方案是稳定的，在一个更大的类弱（甚至可能不是BV）的解决方案的系统。这一结果扩展了经典的弱-强唯一性结果，允许比较光滑的解决方案。事实上，我们的结果将这些结果扩展到弱BV唯一性结果，其中只有一个解决方案被认为是smallBV，另一个解决方案可以来自一个大的类。作为结果的一个结果，在有2个未知量的系统中，类空曲线上的驯服振荡条件和有界变差条件对于BV理论中的解的唯一性是不必要的。该方法基于位移a-收缩理论，通过前沿跟踪方法生成合适的权函数。
Let a 1-d system of hyperbolic conservation laws, with two unknowns, be endowed with a convex entropy. We consider the family of smallBVfunctions which are global solutions of this equation. For any smallBVinitial data, such global solutions are known to exist. Moreover, they are known to be unique amongBVsolutions 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 evenBV) 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-BVuniqueness result, where only one of the solutions is supposed to be smallBV, 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 theBVtheory, in the case of systems with 2 unknowns. The method isbased, and builds up from the theory of a-contraction with shifts, where suitable weight functionsaare generated via the front tracking method.