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PreprintCC BY 4.0arXiv

Well-posedness for a class of n× n hyperbolic systems and shock stability for the viscous approximation

  • Rahul Barthwal
  • Lilu Sahu
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Abstract

We establish the global well-posedness of weak solutions to the Cauchy problem for a broad class of multi-dimensional n× n Keyfitz-Kranzer type systems with homogeneous flux. Our proof relies on a novel viscous approximation that yields the necessary compactness estimates. By exploiting the specific structure of this proposed approximation, we derive a priori uniform L∞ bounds and prove the L1loc precompactness of the sequence of approximate solutions. This framework allows us to rigorously justify the vanishing viscosity limit. Furthermore, under suitable assumptions on the initial data, we employ the relative entropy method to analyze the L2-time decay of large perturbations of the viscous shock, up to a dynamical shift.Evan

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Cite

BibTeXRIS
BibTeX
@misc{barthwal2026wellposedness,
  title = {{Well-posedness for a class of $n\times n$ hyperbolic systems and shock stability for the viscous approximation}},
  author = {Rahul Barthwal and Lilu Sahu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.13989},
  eprint = {2609.13989},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.13989},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Well-posedness for a class of n× n hyperbolic systems and shock stability for the viscous approximation
AU  - Rahul Barthwal
AU  - Lilu Sahu
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2609.13989
UR  - https://arxiv.org/abs/2609.13989
AB  - We establish the global well-posedness of weak solutions to the Cauchy problem for a broad class of multi-dimensional n× n Keyfitz-Kranzer type systems with homogeneous flux. Our proof relies on a novel viscous approximation that yields the necessary compactness estimates. By exploiting the specific structure of this proposed approximation, we derive a priori uniform L^∞ bounds and prove the L^1_loc precompactness of the sequence of approximate solutions. This framework allows us to rigorously justify the vanishing viscosity limit. Furthermore, under suitable assumptions on the initial data, we employ the relative entropy method to analyze the L^2-time decay of large perturbations of the viscous shock, up to a dynamical shift.Evan
ER  -