Kalisch Henrik, Mitrovic Darko
Department of Mathematics, University of Bergen, Faculty of Mathematics, Allegaten 41, 5007 Bergen, Norway.
Faculty of Mathematics, University of Vienna, Oskar Morgenstern-Platz 1, 1090 Vienna, Austria.
Int J Appl Comput Math. 2022;8(4):175. doi: 10.1007/s40819-022-01368-4. Epub 2022 Jun 28.
A weak notion of solution for systems of conservation laws in one dimension is put forward. In the framework introduced here, it can be shown that the Cauchy problem for any system of conservation laws has a solution. The solution concept is an extension of the notion of singular -shocks which have been used to provide solutions for Riemann problems in various systems, for example in cases where strict hyperbolicity or the genuine-nonlinearity condition are not satisfied, or in cases where initial conditions have large variation. We also introduce admissibility conditions which eliminate a wide range of unreasonable solutions. Finally, we provide an example from the shallow water system which justifies introduction of -distributions as a part of solutions to systems of conservation laws.
提出了一维守恒律系统的一种弱解概念。在此引入的框架下，可以证明任何守恒律系统的柯西问题都有解。该解的概念是奇异(\delta)-激波概念的扩展，奇异(\delta)-激波已被用于为各种系统中的黎曼问题提供解，例如在不满足严格双曲性或真正非线性条件的情况下，或者在初始条件有大的变化的情况下。我们还引入了可容许性条件，这些条件消除了大量不合理的解。最后，我们给出一个来自浅水系统的例子，它证明了引入(\delta)-分布作为守恒律系统解的一部分是合理的。
