Nonlinear stability of composite waves of traveling wave and rarefaction wave for a parabolic-hyperbolic system arising from chemotaxis
Abstract
We study the nonlinear stability of a composite wave consisting of a traveling wave and a rarefaction wave for a parabolic-hyperbolic system arising from chemotaxis. We prove that if the initial value is a small H1-type perturbation of composite wave, then the system admits a global solution that converges toward the composite wave with an absolutely continuous shift. The proof combines the weighted relative-entropy mechanism for viscous shocks with the energy structure of rarefaction waves. A key ingredient is the inclusion of a rarefaction modulation factor in the weighted relative entropy; its derivatives combine with the terms generated by the rarefaction profile to produce a rarefaction dissipation. Moreover, the spatial separation of the two waves yields time-integrable interaction errors caused by the non-exact superposition. Since diffusion acts only on the density component, the full H1 estimate is closed by exploiting the coupling structure of the system to recover the missing dissipation of the hyperbolic component.
Related
- Asymptotic stability of strong rarefaction waves to a parabolic-hyperbolic system arising from chemotaxis
- Stability of composite waves consisting of a non-classical Liu shock and a rarefaction wave for a non-convex viscoelastic system
- Nonlinear Stability of Nonconstant Steady States in Chemotaxis--Consumption Systems with General Motility
- Large-Time Behavior towards Composite Waves of Degenerate Shock and Rarefaction Wave for Modified KdV-Burgers Equation
- Global stability of superposition of viscous contact wave and rarefaction waves for compressible Navier-Stokes system with temperature-dependent transport coefficients and large data
BibTeX
@misc{liu2026nonlinear,
title = {{Nonlinear stability of composite waves of traveling wave and rarefaction wave for a parabolic-hyperbolic system arising from chemotaxis}},
author = {Sitong Liu and Jingyu Li},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23448},
eprint = {2609.23448},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23448},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Nonlinear stability of composite waves of traveling wave and rarefaction wave for a parabolic-hyperbolic system arising from chemotaxis
AU - Sitong Liu
AU - Jingyu Li
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23448
UR - https://arxiv.org/abs/2609.23448
AB - We study the nonlinear stability of a composite wave consisting of a traveling wave and a rarefaction wave for a parabolic-hyperbolic system arising from chemotaxis. We prove that if the initial value is a small H^1-type perturbation of composite wave, then the system admits a global solution that converges toward the composite wave with an absolutely continuous shift. The proof combines the weighted relative-entropy mechanism for viscous shocks with the energy structure of rarefaction waves. A key ingredient is the inclusion of a rarefaction modulation factor in the weighted relative entropy; its derivatives combine with the terms generated by the rarefaction profile to produce a rarefaction dissipation. Moreover, the spatial separation of the two waves yields time-integrable interaction errors caused by the non-exact superposition. Since diffusion acts only on the density component, the full H^1 estimate is closed by exploiting the coupling structure of the system to recover the missing dissipation of the hyperbolic component.
ER -