Sharp ill-posedness for the Hunter–Saxton equation on the line

IF 1.1 3区数学 Q1 MATHEMATICS

Journal of Evolution Equations Pub Date : 2024-04-01 DOI:10.1007/s00028-024-00962-x

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引用次数: 0

Abstract

The purpose of this paper is to give an exact division for the well-posedness and ill-posedness (non-existence) of the Hunter–Saxton equation on the line. Since the force term is not bounded in the classical Besov spaces (even in the classical Sobolev spaces), a new mixed space \(\mathcal {B}\) is constructed to overcome this difficulty. More precisely, if the initial data \(u_0\in \mathcal {B}\cap \dot{H}^{1}(\mathbb {R}),\) the local well-posedness of the Cauchy problem for the Hunter–Saxton equation is established in this space. Contrariwise, if \(u_0\in \mathcal {B}\) but \(u_0\notin \dot{H}^{1}(\mathbb {R}),\) the norm inflation and hence the ill-posedness is presented. It’s worth noting that this norm inflation occurs in the low frequency part, which exactly leads to a non-existence result. Moreover, the above result clarifies a corollary with physical significance such that all the smooth solutions in \(L^{\infty }(0,T;L^{\infty }(\mathbb {R}))\) must have the \(\dot{H}^1\) norm.

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线段上亨特-萨克斯顿方程的尖锐非确定性

摘要本文的目的是给出线性上亨特-萨克斯顿方程的好摆（well-posedness）和坏摆（non-posedness）的精确划分。由于力项在经典 Besov 空间（甚至在经典 Sobolev 空间）中都不受约束，因此本文构建了一个新的混合空间 \(\mathcal {B}\) 来克服这一困难。更确切地说，如果初始数据 \(u_0\in \mathcal {B}\cap \dot{H}^{1}(\mathbb {R}),\) 亨特-萨克斯顿方程的考奇问题的局部好求性在这个空间中成立。相反，如果 \(u_0\in \mathcal {B}\) 但是 \(u_0not\in \dot{H}^{1}(\mathbb {R}),\) 则会出现规范膨胀，进而出现问题。值得注意的是，这种规范膨胀发生在低频部分，这恰恰导致了不存在结果。此外，上述结果澄清了一个具有物理意义的推论，即所有在 \(L^{\infty }(0,T;L^{\infty }(\mathbb {R}))\) 中的平稳解必须具有 \(\dot{H}^1\) 规范。

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 Evolution Equations 数学-数学

CiteScore

2.30

自引率

7.10%

发文量

审稿时长

>12 weeks

期刊介绍： The Journal of Evolution Equations (JEE) publishes high-quality, peer-reviewed papers on equations dealing with time dependent systems and ranging from abstract theory to concrete applications. Research articles should contain new and important results. Survey articles on recent developments are also considered as important contributions to the field. Particular topics covered by the journal are: Linear and Nonlinear Semigroups Parabolic and Hyperbolic Partial Differential Equations Reaction Diffusion Equations Deterministic and Stochastic Control Systems Transport and Population Equations Volterra Equations Delay Equations Stochastic Processes and Dirichlet Forms Maximal Regularity and Functional Calculi Asymptotics and Qualitative Theory of Linear and Nonlinear Evolution Equations Evolution Equations in Mathematical Physics Elliptic Operators