Existence of viscosity solutions with the optimal regularity of a two-peakon Hamilton–Jacobi equation

IF 0.5 4区数学 Q4 MATHEMATICS, APPLIED

Journal of Hyperbolic Differential Equations Pub Date : 2020-08-05 DOI:10.1142/s0219891621500156

Tomasz Cieślak, Jakub Siemianowski

引用次数: 0

Abstract

We study here a Hamilton–Jacobi equation with a quadratic and degenerate Hamiltonian, which comes from the dynamics of a multipeakon in the Camassa–Holm equation. It is given by a quadratic form with a singular positive semi-definite matrix. We increase the regularity of the value function considered in earlier works, which is known to be the viscosity solution. We prove that for a two-peakon Hamiltonian such solutions are actually [Formula: see text]-Hölder continuous in space and time-Lipschitz continuous. The time-Lipschitz regularity is proven in any dimension [Formula: see text]. Such a regularity is already known in the one-dimensional case and, moreover it is the best possible, as shown earlier.

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双峰Hamilton-Jacobi方程最优正则性黏度解的存在性

本文研究了一个二次简并Hamilton-Jacobi方程，它来源于Camassa-Holm方程中的多峰子动力学。它由一个奇异正半定矩阵的二次型给出。我们增加了早期作品中所考虑的值函数的规律性，即已知的粘度解。我们证明了对于一个双峰哈密顿量，这样的解实际上是[公式:见文本]-Hölder在空间和时间上连续的利普希茨连续的。时间- lipschitz正则性在任何维度上都得到了证明[公式:见原文]。这种规律性在一维情况下是已知的，而且如前面所示，它是最好的可能。

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.