Asymptotic stability of strong rarefaction waves to a parabolic-hyperbolic system arising from chemotaxis
Abstract
We are interested in the asymptotic behavior of solutions toward strong rarefaction waves for a parabolic-hyperbolic system arising from chemotaxis. Suppose that the Riemann problem to the corresponding inviscid system admits rarefaction waves. We show that if the initial data is a small perturbation of an approximate rarefaction wave, then the Cauchy problem has a unique global solution that converges to the rarefaction wave asymptotically in time. The stability results hold regardless of the wave strengths. The waves can be either a single rarefaction wave or a composite of two rarefaction waves. In the composite case, we separate the initial centers of the two approximate profiles to make their interaction error term small while retaining convergence to the prescribed Riemann wave. The proofs are based on the energy method, where the key observations are the monotonicity of the approximate rarefaction waves with respect to both space and time, and the appropriate replacement of spatial derivative of the hyperbolic component with the parabolic component in proper forms.
Related
- Nonlinear stability of composite waves of traveling wave and rarefaction wave for a parabolic-hyperbolic system arising from chemotaxis
- Asymptotic Stability and Lp Time-Decay Rates of Rarefaction Waves for Systems of Viscous Conservation Laws
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- 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{liu2026asymptotic,
title = {{Asymptotic stability of strong rarefaction waves to a parabolic-hyperbolic system arising from chemotaxis}},
author = {Sitong Liu and Jingyu Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.01559},
eprint = {2606.01559},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.01559},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Asymptotic stability of strong rarefaction waves to a parabolic-hyperbolic system arising from chemotaxis
AU - Sitong Liu
AU - Jingyu Li
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2606.01559
UR - https://arxiv.org/abs/2606.01559
AB - We are interested in the asymptotic behavior of solutions toward strong rarefaction waves for a parabolic-hyperbolic system arising from chemotaxis. Suppose that the Riemann problem to the corresponding inviscid system admits rarefaction waves. We show that if the initial data is a small perturbation of an approximate rarefaction wave, then the Cauchy problem has a unique global solution that converges to the rarefaction wave asymptotically in time. The stability results hold regardless of the wave strengths. The waves can be either a single rarefaction wave or a composite of two rarefaction waves. In the composite case, we separate the initial centers of the two approximate profiles to make their interaction error term small while retaining convergence to the prescribed Riemann wave. The proofs are based on the energy method, where the key observations are the monotonicity of the approximate rarefaction waves with respect to both space and time, and the appropriate replacement of spatial derivative of the hyperbolic component with the parabolic component in proper forms.
ER -