发表机构
Korea Advanced Institute of Science and Technology; Ulsan National Institute of Science and Technology（韩国科学技术院; 蔚山科学技术院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对标量粘性守恒律中退化Oleinik激波与稀疏波复合波，通过建立新的退化Hardy-Poincaré不等式，在无小性假设下证明了大$L^2$扰动的均匀稳定性，并推出无粘极限下的稳定性。
AI 中文摘要
我们研究标量粘性守恒律中由退化Oleinik激波和稀疏波组成的复合波的均匀稳定性。我们考虑三次型通量以及复合波在$H^1$强解类中任意大的$L^2$初始扰动。我们获得了此类大扰动的均匀$L^2$估计。特别地，该估计蕴含了退化Oleinik激波在动态平移下的收缩性。作为进一步推论，关于粘性的均匀性意味着由退化激波和稀疏波组成的相关自相似解在无粘极限类中是稳定的。主要的分析困难在于，支撑$a$-收缩方法的标准加权Poincaré不等式在线性层面无法产生所需的强制性。我们通过建立一个新的适应激波剖面退化结构的退化Hardy--Poincaré不等式来克服这一障碍。结合精确的非线性分解，该线性不等式产生了一个严格的强制性估计，该估计控制二次能量以及高阶非线性项，且不需要对扰动做任何小性假设。
英文摘要
We study the uniform stability of composite waves consisting of a degenerate Oleinik shock and a rarefaction wave for scalar viscous conservation laws. We consider cubic-type fluxes and arbitrarily large $L^2$ initial perturbations of the composite wave within the $H^1$ strong-solution class. We obtain a uniform $L^2$ estimate for such large perturbations. In particular, the estimate implies the contraction of a degenerate Oleinik shock, up to a dynamical shift. As a further consequence, the uniformity with respect to viscosity implies that the associated self-similar solution, composed of the degenerate shock and rarefaction, is stable in the class of inviscid limits. The main analytical difficulty is that the standard weighted Poincaré inequality underlying the $a$-contraction method fails to yield the required coercivity at the linear level. We overcome this obstruction by establishing a new degenerate Hardy--Poincaré inequality adapted to the degenerate structure of the shock profile. Combined with an exact nonlinear decomposition, this linear inequality yields a strict coercivity estimate that controls the quadratic energy together with the higher-order nonlinear terms, without any smallness assumption on the perturbation.
Comments46 pages
