发表机构
Korea Institute for Advanced Study (KIAS); Korea Advanced Institute of Science and Technology (KAIST); Kyung Hee University（韩国高等研究院; 韩国科学技术院; 庆熙大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究修正KdV-Burgers方程，证明其退化激波与稀疏波复合波在小扰动下全局存在且时间渐近稳定，推广了无粘性结果并放宽稀疏波强度限制。
AI 中文摘要
本文研究修正Korteweg-de Vries-Burgers (mKdVB)方程 \\[u_t + (u^3)_x = \mu u_{xx} - \kappa u_{xxx}\\]，其中$\mu>0$且$\kappa>0$。由于通量是非凸的，（无粘）Riemann问题可以通过一个退化Oleinik激波和一个稀疏波的复合波来求解。我们证明了在小$H^1$扰动下，解的全局存在性以及相应的粘性-色散复合波在时间依赖的位移下的时间渐近稳定性。激波强度$\delta_S$和稀疏波强度$\delta_R$不必很小，仅受显式比值$\delta_R\le\frac{5}{18}\delta_S$和控制激波单调性的量$\kappa\delta_S^2/\mu^2$的限制。稳定性估计关于色散强度$\kappa$是一致的。作为稳定性分析的关键要素，我们在单调区域内建立了退化粘性-色散激波剖面的结构性质，包括其导数的双侧逐点界以及它收敛到两个端状态的速率。这些衰减速率与相应的纯粘性剖面的衰减速率一致，且常数与色散系数无关。在没有色散的情况下，即当$\kappa=0$时，相应的粘性复合波的时间渐近稳定性由Huang, Wang和Zhang [26]建立。本文工作将他们的结果推广到粘性-色散情形，并放宽了对稀疏波强度的限制。证明依赖于Kang和Vasseur [32,33,35]发展的带位移的$a$-压缩方法（用于粘性守恒律）。
英文摘要
In this paper, we study the modified Korteweg-de Vries-Burgers (mKdVB) equation \[u_t + (u^3)_x = μu_{xx} - κu_{xxx}\] with $μ>0$ and $κ>0$. Since the flux is non-convex, the (inviscid) Riemann problem may be solved by a composite wave of a degenerate Oleinik shock and a rarefaction wave. We prove the global existence of solutions and the time-asymptotic stability of the corresponding viscous-dispersive composite wave, up to a time-dependent shift, under small $H^1$ perturbations. The shock strength $δ_S$ and the rarefaction strength $δ_R$ need not be small and are restricted only by the explicit ratio $δ_R\le\frac{5}{18}δ_S$ and the quantity $κδ_S^2/μ^2$ governing the monotonicity of the shock. The stability estimate is uniform with respect to the dispersion strength $κ$. As a key ingredient of the stability analysis, we establish structural properties of the degenerate viscous-dispersive shock profile in the monotone regime, including two-sided pointwise bounds on its derivative and the rates at which it converges to its two end states. These decay rates coincide with those of the corresponding purely viscous profile, with constants independent of the dispersion coefficient. In the absence of dispersion, namely, when $κ=0$, the time-asymptotic stability of the corresponding viscous composite wave was established by Huang, Wang and Zhang [26]. The present work extends their result to the viscous-dispersive setting and relaxes the restriction on the rarefaction strength. The proof relies on the method of $a$-contraction with shifts (for viscous conservation laws) developed by Kang and Vasseur [32,33,35].
