论文标题
猎人 - 萨克斯顿方程的保守解决方案的渐近行为
Asymptotic behavior of conservative solutions to the Hunter-Saxton equation
论文作者
论文摘要
在本文中，我们研究了（能量）保守解决方案（能量）对猎人 - 萨克斯顿方程的近期渐近行为，该框架由溶液的发展及其能量度量组成。我们描述了保守解决方案的大渐近扩展，并严格验证$ l^{\ infty}（\ Mathbb {r}）$和$ {\ dot {h}}}^1（\ Mathbb {r}）$ Space的有效性。领先的订单项由仅由系统的总能量确定的扭结波给出。作为推论，我们还表明能量度量的奇异部分会收敛到零，因为时间是正数或负无穷大。在初始能量度量尾部的一些自然衰减率假设下，我们严格地提供了$ l^{\ infty}（\ Mathbb {r}）$和$ {\ dot {h}}}}^1（\ Mathbb {r}）$中的最佳误差估计。随着时间到达无穷大，在初始数据的相同假设下，也可以获得解决方案的点收敛性和侧重呈增长率。我们的结果的证明在很大程度上取决于对用于测量值的初始数据设计的广义特征的精心分析，以及用于保守解决方案的明确公式。
In this paper we study the large time asymptotic behavior of (energy) conservative solutions to the Hunter-Saxton equation in a generalized framework that consists of the evolutions of solution and its energy measure. We describe the large time asymptotic expansions of the conservative solutions, and rigorously verify the validity of the leading order term in $L^{\infty}(\mathbb{R})$ and ${\dot{H}}^1(\mathbb{R})$ spaces respectively. The leading order term is given by a kink-wave that is determined by the total energy of the system only. As a corollary, we also show that the singular part of the energy measure converges to zero, as the time goes to either positive or negative infinity. Under some natural decay rate assumptions on the tails of the initial energy measure, we rigorously provide the optimal error estimates in $L^{\infty}(\mathbb{R})$ and ${\dot{H}}^1(\mathbb{R})$. As the time goes to infinity, the pointwise convergence and pointwise growth rate for the solution are also obtained under the same assumptions on the initial data. The proofs of our results rely heavily on the elaborate analysis of the generalized characteristics designed for the measure-valued initial data, and explicit formulae for conservative solutions.
