Singular limits of the quasi-linear Kolmogorov-type equation with a source term

IF 0.5 4区数学 Q4 MATHEMATICS, APPLIED

Journal of Hyperbolic Differential Equations Pub Date : 2019-07-09 DOI:10.1142/s0219891621500247

I. Kuznetsov, S. Sazhenkov

引用次数: 2

Abstract

Existence, uniqueness and stability of kinetic and entropy solutions to the boundary value problem associated with the Kolmogorov-type, genuinely nonlinear, degenerate hyperbolic–parabolic (ultra-parabolic) equation with a smooth source term is established. In addition, we consider the case when the source term contains a small positive parameter and collapses to the Dirac delta-function, as this parameter tends to zero. In this case, the limiting passage from the original equation with the smooth source to the impulsive ultra-parabolic equation is investigated and the formal limit is rigorously justified. Our proofs rely on the use of kinetic equations and the compensated compactness method for genuinely nonlinear balance laws.

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具有源项的拟线性kolmogorov型方程的奇异极限

建立了具有光滑源项的Kolmogorov型真非线性退化双曲-抛物（超抛物）方程边值问题的动力学和熵解的存在性、唯一性和稳定性。此外，我们还考虑了源项包含一个小的正参数并折叠为狄拉克德尔塔函数的情况，因为该参数趋于零。在这种情况下，研究了从具有光滑源的原始方程到脉冲超抛物方程的极限通道，并严格证明了形式极限。我们的证明依赖于使用动力学方程和真正非线性平衡定律的补偿紧致性方法。

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.