Existence and uniqueness of generalized solutions to hyperbolic systems with linear fluxes and stiff sources

IF 0.5 4区数学 Q4 MATHEMATICS, APPLIED

Journal of Hyperbolic Differential Equations Pub Date : 2021-09-01 DOI:10.1142/s021989162150020x

T. Pichard, N. Aguillon, B. Després, E. Godlewski, M. Ndjinga

引用次数: 1

Abstract

Motivated by the modeling of boiling two-phase flows, we study systems of balance laws with a source term defined as a discontinuous function of the unknown. Due to this discontinuous source term, the classical theory of partial differential equations (PDEs) is not sufficient here. Restricting to a simpler system with linear fluxes, a notion of generalized solution is developed. An important point in the construction of a solution is that the curve along which the source jumps, which we call the boiling curve, must never be tangent to the characteristics. This leads to exhibit sufficient conditions which ensure the existence and uniqueness of a solution in two different situations: first when the initial data is smooth and such that the boiling curve is either overcharacteristic or subcharacteristic; then with discontinuous initial data in the case of Riemann problems. A numerical illustration is given in this last case.

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具有线性通量和刚性源的双曲型系统广义解的存在唯一性

受沸腾两相流建模的启发，我们研究了源项定义为未知不连续函数的平衡律系统。由于这个不连续的源项，经典的偏微分方程理论在这里是不够的。将广义解的概念局限于具有线性通量的简单系统，提出了广义解的一个概念。构造解的一个重要点是，源跳跃的曲线，我们称之为沸腾曲线，决不能与特性相切。这导致在两种不同的情况下表现出确保解的存在性和唯一性的充分条件：首先，当初始数据是平滑的，并且沸腾曲线是过特征的或亚特征的；然后在黎曼问题的情况下使用不连续的初始数据。在最后一种情况下给出了数值说明。

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.