Skip to content
DraconiticResearch
Search Sources API
197,768 studies31 sourcesUpdated 56 min ago
PreprintarXiv licencearXiv

Sharp Hxs Ill-posedness of the Hard-sphere Boltzmann Equation

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

Abstract

We investigate the ill-posedness mechanism of the hard-sphere Boltzmann equation in Hxs Sobolev space. Via a direct construction, we prove a strong-weak type ill-posedness result in the low-regularity regime s<1, establishing a sharp threshold in connection to the local s>1 well-posedness result [11]. Instead of originating from the large-velocity growth of the collision kernel, this illposedness is generated by the loss term and dispersive effects. Consequently, we prove a dispersion-driven nonlinear instability mechanism for the hard-sphere Boltzmann equation, and provide a capstone of the ill-posedness series [18,20].

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 global stability for the hard-sphere Boltzmann equation near equilibrium
    Xuwen Chen, Shunlin Shen, Zhifei Zhang
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  3. Fluid dynamic limit of Boltzmann equation for granular hard-spheres in a nearly elastic regime
    Ricardo J. Alonso, Bertrand Lods, Isabelle Tristani
    Mémoires de la Société mathématique de FrancePeer-reviewedLicence not statedFull text16 citedGeneral mathematicsOpenAlex
  4. Bargmann-Fock Representation and Global Estimates for the Linearized Hard-Sphere Boltzmann Operator
    Ilya Karlin
    arXivPreprintCC BY 4.0Full textMathematical physicsarXiv
  5. Time-asymptotic expansion and fluid approximation of the 1D Boltzmann equation with hard potentials
    Hui-Tzu Chang, Haitao Wang, Kung-Chien Wu
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv

Cite

BibTeXRIS
BibTeX
@misc{chen2026sharp,
  title = {{Sharp $H_x^s$ Ill-posedness of the Hard-sphere Boltzmann Equation}},
  author = {Xuwen Chen and Yan Guo and Shunlin Shen and Zhifei Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2606.01331},
  eprint = {2606.01331},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.01331},
}
RIS
TY  - UNPB
TI  - Sharp H_x^s Ill-posedness of the Hard-sphere Boltzmann Equation
AU  - Xuwen Chen
AU  - Yan Guo
AU  - Shunlin Shen
AU  - Zhifei Zhang
PY  - 2026
DA  - 2026/09/13
PB  - arXiv
DO  - 10.48550/arxiv.2606.01331
UR  - https://arxiv.org/abs/2606.01331
AB  - We investigate the ill-posedness mechanism of the hard-sphere Boltzmann equation in H_x^s Sobolev space. Via a direct construction, we prove a strong-weak type ill-posedness result in the low-regularity regime s<1, establishing a sharp threshold in connection to the local s>1 well-posedness result [11]. Instead of originating from the large-velocity growth of the collision kernel, this illposedness is generated by the loss term and dispersive effects. Consequently, we prove a dispersion-driven nonlinear instability mechanism for the hard-sphere Boltzmann equation, and provide a capstone of the ill-posedness series [18,20].
ER  -