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.

中文翻译：

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线上 Hunter-Saxton 方程的严重不适定性

摘要

本文的目的是对直线上的 Hunter-Saxton 方程的适定性和不适定性（不存在性）给出精确划分。由于力项在经典Besov空间中（甚至在经典Sobolev空间中）没有界限，因此构造一个新的混合空间\(\mathcal {B}\)来克服这个困难。更准确地说，如果初始数据\(u_0\in \mathcal {B}\cap \dot{H}^{1}(\mathbb {R}),\)则猎人的柯西问题的局部适定性–Saxton方程在此空间成立。相反，如果\(u_0\in \mathcal {B}\)但\(u_0\notin \dot{H}^{1}(\mathbb {R}),\)范数膨胀和不适定性为呈现。值得注意的是，这种范数膨胀发生在低频部分，这恰恰导致了不存在的结果。此外，上述结果阐明了一个具有物理意义的推论，即\(L^{\infty }(0,T;L^{\infty }(\mathbb {R}))\)中的所有光滑解必须具有\(\dot{H}^1\)范数。