<p>This paper studies the large-time asymptotic behavior of the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\lambda$</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>-dissipative solutions (<InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\lambda \in [0,1]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>) to the Hunter–Saxton equation by using the explicit characteristics. The leading order term in the large-time asymptotic expansions in both spaces <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$L^{\infty }(\mathbb {R})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>∞</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">${\dot{H}}^1(\mathbb {R})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mover accent="true"> <mi>H</mi> <mo>˙</mo> </mover> </mrow> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is shown to be given by a special self-similar solution usually referred to as the kink wave, which is determined by the total remaining energy after accounting for all possible sudden energy releases (blow-ups) in the system. This remaining energy can be calculated based on the initial total energy, the singular measures resulting from the initial data, and the dissipation parameter <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\lambda$</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>. Importantly, our results encompass both the energy conservative (<InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\lambda =0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>) and energy dissipative (<InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\lambda =1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>) solutions.</p>

On the large-time asymptotic behaviors of $\lambda$-dissipative solutions to the Hunter–Saxton equation

期刊论文
发表日期：

2025年12月4日

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Yu Gao,
Hao Liu

摘要

This paper studies the large-time asymptotic behavior of the $\lambda$ $λ$ -dissipative solutions ( $\lambda \in [0,1]$ $λ \in [0, 1]$ ) to the Hunter–Saxton equation by using the explicit characteristics. The leading order term in the large-time asymptotic expansions in both spaces $L^{\infty }(\mathbb {R})$ $L^{\infty} (R)$ and ${\dot{H}}^1(\mathbb {R})$ ${\dot{H}}^{1} (R)$ is shown to be given by a special self-similar solution usually referred to as the kink wave, which is determined by the total remaining energy after accounting for all possible sudden energy releases (blow-ups) in the system. This remaining energy can be calculated based on the initial total energy, the singular measures resulting from the initial data, and the dissipation parameter $\lambda$ $λ$ . Importantly, our results encompass both the energy conservative ( $\lambda =0$ $λ = 0$ ) and energy dissipative ( $\lambda =1$ $λ = 1$ ) solutions.

On the large-time asymptotic behaviors of \(\lambda\)-dissipative solutions to the Hunter–Saxton equation

摘要