On Φ-variation for 1-d scalar conservation laws

IF 0.5 4区数学 Q4 MATHEMATICS, APPLIED

Journal of Hyperbolic Differential Equations Pub Date : 2020-12-01 DOI:10.1142/S0219891620500277

H. Jenssen, J. Ridder

引用次数: 2

Abstract

Let [Formula: see text] be a convex function satisfying [Formula: see text], [Formula: see text] for [Formula: see text], and [Formula: see text]. Consider the unique entropy admissible (i.e. Kružkov) solution [Formula: see text] of the scalar, 1-d Cauchy problem [Formula: see text], [Formula: see text]. For compactly supported data [Formula: see text] with bounded [Formula: see text]-variation, we realize the solution [Formula: see text] as a limit of front-tracking approximations and show that the [Formula: see text]-variation of (the right continuous version of) [Formula: see text] is non-increasing in time. We also establish the natural time-continuity estimate [Formula: see text] for [Formula: see text], where [Formula: see text] depends on [Formula: see text]. Finally, according to a theorem of Goffman–Moran–Waterman, any regulated function of compact support has bounded [Formula: see text]-variation for some [Formula: see text]. As a corollary we thus have: if [Formula: see text] is a regulated function, so is [Formula: see text] for all [Formula: see text].

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关于Φ-variation的一维标量守恒定律

设[公式：见文本]是满足[公式：参见文本]、[公式：详见文本]的[公式：请见文本]和[公式：参看文本]的凸函数。考虑标量一维柯西问题的唯一熵容许（即Kružkov）解[公式：见正文]，[公式：看正文]。对于具有有界[公式：见文本]-变差的紧凑支持数据[公式：看文本]，我们将解[公式：见图文本]实现为前跟踪近似的极限，并表明[公式：参见文本]的[公式：详见文本]变差在时间上是不增加的。我们还为[Former:see-text]建立了自然时间连续性估计[Former:see-text]，其中[Former:see-text]取决于[Former:see-text]。最后，根据Goffman–Moran–Waterman的一个定理，紧支撑的任何调节函数都有界[公式：见正文]-某些[公式：参见正文]的变差。因此，作为一个推论，我们有：如果[Former:see-text]是一个受调节的函数，那么对于所有[Former:see-text]来说，[Former:see-text]也是如此。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Hyperbolic Differential Equations 数学-物理：数学物理

CiteScore

1.10

自引率

0.00%

发文量

审稿时长

24 months

期刊介绍： This journal publishes original research papers on nonlinear hyperbolic problems and related topics, of mathematical and/or physical interest. Specifically, it invites papers on the theory and numerical analysis of hyperbolic conservation laws and of hyperbolic partial differential equations arising in mathematical physics. The Journal welcomes contributions in: Theory of nonlinear hyperbolic systems of conservation laws, addressing the issues of well-posedness and qualitative behavior of solutions, in one or several space dimensions. Hyperbolic differential equations of mathematical physics, such as the Einstein equations of general relativity, Dirac equations, Maxwell equations, relativistic fluid models, etc. Lorentzian geometry, particularly global geometric and causal theoretic aspects of spacetimes satisfying the Einstein equations. Nonlinear hyperbolic systems arising in continuum physics such as: hyperbolic models of fluid dynamics, mixed models of transonic flows, etc. General problems that are dominated (but not exclusively driven) by finite speed phenomena, such as dissipative and dispersive perturbations of hyperbolic systems, and models from statistical mechanics and other probabilistic models relevant to the derivation of fluid dynamical equations. Convergence analysis of numerical methods for hyperbolic equations: finite difference schemes, finite volumes schemes, etc.