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The particle paths of hyperbolic conservation laws
Mathematical Models and Methods in Applied Sciences ( IF 3.6 ) Pub Date : 2024-04-03 , DOI: 10.1142/s0218202524500209
Ulrik S. Fjordholm 1 , Ola H. Mæhlen 1 , Magnus C. Ørke 1
Affiliation  

Nonlinear scalar conservation laws are traditionally viewed as transport equations. We take instead the viewpoint of these PDEs as continuity equations with an implicitly defined velocity field. We show that a weak solution is the entropy solution if and only if the ODE corresponding to its velocity field is well-posed. We also show that the flow of the ODE is 1/2-Hölder regular. Finally, we give several examples showing that our results are sharp, and we provide explicit computations in the case of a Riemann problem.



中文翻译:

双曲守恒定律的粒子路径

非线性标量守恒定律传统上被视为输运方程。相反,我们将这些偏微分方程视为具有隐式定义的速度场的连续性方程。我们证明,当且仅当与其速度场对应的 ODE 适定时,弱解才是熵解。我们还证明 ODE 的流是 1/2-Hölder 正则的。最后,我们给出了几个例子,表明我们的结果很清晰,并且我们在黎曼问题的情况下提供了显式计算。

更新日期:2024-04-03
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