On the failure of the chain rule for the divergence of Sobolev vector fields

IF 0.5 4区数学 Q4 MATHEMATICS, APPLIED

Journal of Hyperbolic Differential Equations Pub Date : 2022-04-04 DOI:10.1142/s0219891623500108

Miriam Buck, S. Modena

引用次数: 2

Abstract

We construct a large class of incompressible vector fields with Sobolev regularity, in dimension [Formula: see text], for which the chain rule problem has a negative answer. In particular, for any renormalization map [Formula: see text] (satisfying suitable assumptions) and any (distributional) renormalization defect [Formula: see text] of the form [Formula: see text], where [Formula: see text] is an [Formula: see text] vector field, we can construct an incompressible Sobolev vector field [Formula: see text] and a density [Formula: see text] for which [Formula: see text] but [Formula: see text], provided [Formula: see text].

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关于Sobolev向量场发散的链式规则的失效

我们构造了一大类具有Sobolev正则性的不可压缩向量场，其维数为[公式:见文本]，其中链式法则问题有一个否定的答案。特别地，对于任何重整化映射[公式:见文](满足适当的假设)和任何(分布的)重整化缺陷[公式:见文]的形式[公式:见文]，其中[公式:见文]是一个[公式:见文]向量场，我们可以构造一个不可压缩的Sobolev向量场[公式:见文]和一个密度[公式:见文]，其中[公式:见文]但[公式:见文]，提供[公式:见文]。

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.