L2-contraction and Asymptotic Stability of Monotone Shock Waves for Cubic and Quartic KdV-Burgers Equations
Abstract
We investigate the nonlinear stability of monotone traveling shock waves arising in the generalized Korteweg-de Vries-Burgers equation with a power-law flux, focusing on the cubic and quartic cases p=3 and p=4, respectively, in the presence of both dissipative and dispersive effects. Since shocks connecting distinct end states destroy the integrable structure of the equation, we introduce a slowly varying reference connection to establish global well-posedness for the Cauchy problem in an affine Sobolev space. A key ingredient in the stability analysis is to establish a new two-sided comparison estimate for the derivative of a monotone shock-wave profile. After a monotone coordinate transformation, this estimate allows the viscous dissipation to be compared with the natural vanishing weight arising in a shifted Poincare inequality. Together with an L∞ bootstrap controlling the higher-order relative-flux terms, this yields a global L2-contraction property modulo a time-dependent shift for sufficiently small H1 perturbations in the admissible parameter regimes. We obtain exponential decay of the shock profile and show asymptotic convergence to the shifted shock in Lq(R), 2<q≤ ∞, with vanishing shift velocity.
Related
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BibTeX
@misc{du2026l2contraction,
title = {{L2-contraction and Asymptotic Stability of Monotone Shock Waves for Cubic and Quartic KdV-Burgers Equations}},
author = {Yi Du and Tao Yan and Yushu Zeng and Shihui Zhu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00595},
eprint = {2611.00595},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00595},
}
RIS
TY - UNPB
TI - L2-contraction and Asymptotic Stability of Monotone Shock Waves for Cubic and Quartic KdV-Burgers Equations
AU - Yi Du
AU - Tao Yan
AU - Yushu Zeng
AU - Shihui Zhu
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00595
UR - https://arxiv.org/abs/2611.00595
AB - We investigate the nonlinear stability of monotone traveling shock waves arising in the generalized Korteweg-de Vries-Burgers equation with a power-law flux, focusing on the cubic and quartic cases p=3 and p=4, respectively, in the presence of both dissipative and dispersive effects. Since shocks connecting distinct end states destroy the integrable structure of the equation, we introduce a slowly varying reference connection to establish global well-posedness for the Cauchy problem in an affine Sobolev space. A key ingredient in the stability analysis is to establish a new two-sided comparison estimate for the derivative of a monotone shock-wave profile. After a monotone coordinate transformation, this estimate allows the viscous dissipation to be compared with the natural vanishing weight arising in a shifted Poincare inequality. Together with an L^∞ bootstrap controlling the higher-order relative-flux terms, this yields a global L^2-contraction property modulo a time-dependent shift for sufficiently small H^1 perturbations in the admissible parameter regimes. We obtain exponential decay of the shock profile and show asymptotic convergence to the shifted shock in L^q(R), 2<q≤ ∞, with vanishing shift velocity.
ER -