Skip to content
DraconiticResearch
Search Sources API
201,683 studies31 sourcesUpdated 38 min ago
PreprintarXiv licencearXiv

Sharp global stability for the hard-sphere Boltzmann equation near equilibrium

  • Xuwen Chen
  • Shunlin Shen
  • Zhifei Zhang
Open at sourcePDFDOICite

Abstract

We study the hard-sphere Boltzmann equation near a global Maxwellian and have found the bifurcation between the stability and instability in Hxs Sobolev space. We prove a strong-weak type instability result in the s<1 regularity regime, and establish the global stability under the regularity s>1 via the interplay between the dissipation and dispersive effects. In this intricate competition of dispersion, dissipation, large-velocity growth, and the asymmetric nature of the gain and loss terms, we prove that the instability is jointly generated by the dispersive effect and the loss term structure. To reach the sharp regularity for the global stability, we incorporate the dispersive effect into the dissipative energy framework for the analysis near equilibrium. This hence concludes sharp threshold and provides a complete characterization of global well-posedness near equilibrium.

Related

  1. Boltzmann Equation and Hard Sphere Gas with Singular Initial Data: Well-posedness, Kinetic Limit, and Weak Stability
    Chenjiayue Qi
    arXivPreprintarXiv licenceFull textAnalysis & PDEsMathematical physicsarXiv
  2. Sharp Hxs Ill-posedness of the Hard-sphere Boltzmann Equation
    Xuwen Chen, Yan Guo, Shunlin Shen +1
    arXivPreprintarXiv licenceFull textAnalysis & PDEsMathematical physicsarXiv
  3. The Binary-ternary Boltzmann Equation Near Equilibrium
    Xianshen Hu, Linjie Xiong, Zhihong Xu
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  4. Bargmann-Fock Representation and Global Estimates for the Linearized Hard-Sphere Boltzmann Operator
    Ilya Karlin
    arXivPreprintCC BY 4.0Full textMathematical physicsarXiv
  5. Local stability and rates of convergence to equilibrium for the Nonlinear Renewal Equation; applications to Hawkes processes
    Céline Duval, Eric Luçon
    NonlinearityPeer-reviewedCC BY 4.0Full textDynamical systemsProbabilityHAL

Cite

BibTeXRIS
BibTeX
@misc{chen2026sharp,
  title = {{Sharp global stability for the hard-sphere Boltzmann equation near equilibrium}},
  author = {Xuwen Chen and Shunlin Shen and Zhifei Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.38759},
  eprint = {2610.38759},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.38759},
}
RIS
TY  - UNPB
TI  - Sharp global stability for the hard-sphere Boltzmann equation near equilibrium
AU  - Xuwen Chen
AU  - Shunlin Shen
AU  - Zhifei Zhang
PY  - 2026
DA  - 2026/10/28
PB  - arXiv
DO  - 10.48550/arxiv.2610.38759
UR  - https://arxiv.org/abs/2610.38759
AB  - We study the hard-sphere Boltzmann equation near a global Maxwellian and have found the bifurcation between the stability and instability in H_x^s Sobolev space. We prove a strong-weak type instability result in the s<1 regularity regime, and establish the global stability under the regularity s>1 via the interplay between the dissipation and dispersive effects. In this intricate competition of dispersion, dissipation, large-velocity growth, and the asymmetric nature of the gain and loss terms, we prove that the instability is jointly generated by the dispersive effect and the loss term structure. To reach the sharp regularity for the global stability, we incorporate the dispersive effect into the dissipative energy framework for the analysis near equilibrium. This hence concludes sharp threshold and provides a complete characterization of global well-posedness near equilibrium.
ER  -