Large-Time Behavior towards Composite Waves of Degenerate Shock and Rarefaction Wave for Modified KdV-Burgers Equation
Abstract
In this paper, we study the modified Korteweg-de Vries-Burgers (mKdVB) equation ut + (u3)x = μuxx − κuxxx 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 H1 perturbations. The shock strength δS and the rarefaction strength δR need not be small and are restricted only by the explicit ratio δR≤5/18δS and the quantity κδS2/μ2 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].
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BibTeX
@misc{eun2026largetime,
title = {{Large-Time Behavior towards Composite Waves of Degenerate Shock and Rarefaction Wave for Modified KdV-Burgers Equation}},
author = {Namhyun Eun and Sungho Han and Jeongho Kim},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20591},
eprint = {2609.20591},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20591},
}
RIS
TY - UNPB
TI - Large-Time Behavior towards Composite Waves of Degenerate Shock and Rarefaction Wave for Modified KdV-Burgers Equation
AU - Namhyun Eun
AU - Sungho Han
AU - Jeongho Kim
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.20591
UR - https://arxiv.org/abs/2609.20591
AB - 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≤5/18δ_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].
ER -