Non-uniqueness of integral curves for autonomous Hamiltonian vector fields

IF 1.8 4区数学 Q1 MATHEMATICS

Differential and Integral Equations Pub Date : 2021-08-11 DOI:10.57262/die035-0708-411

V. Giri, M. Sorella

引用次数: 6

Abstract

In this work we prove the existence of an autonomous Hamiltonian vector field in W^{1,r}(T^d;R^d) with r=4 for which the associated transport equation has non-unique positive solutions. As a consequence of Ambrosio superposition principle, we show that this vector field has non-unique integral curves with a positive Lebesgue measure set of initial data and moreover we show that the Hamiltonian is not constant along these integral curves.

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自治哈密顿向量场积分曲线的非唯一性

在这项工作中，我们证明了在W^{1，r}（T^d；r^d）中，当r＝4时，一个自治的哈密顿向量场的存在性，其相关的输运方程具有非唯一的正解。作为Ambrosio叠加原理的结果，我们证明了这个向量场具有非唯一的积分曲线，具有初始数据的正Lebesgue测度集，并且我们还证明了哈密顿量沿着这些积分曲线是不恒定的。

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

Differential and Integral Equations MATHEMATICS, APPLIED-MATHEMATICS

CiteScore

2.40

自引率

0.00%

发文量

审稿时长

6-12 weeks

期刊介绍： Differential and Integral Equations will publish carefully selected research papers on mathematical aspects of differential and integral equations and on applications of the mathematical theory to issues arising in the sciences and in engineering. Papers submitted to this journal should be correct, new, and of interest to a substantial number of mathematicians working in these areas.