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Local well-posedness for the Boltzmann equation with hard potentials

Hao-Guang Li, Wei-Xi Li, Chao-Jiang Xu

TL;DR

This work proves a local-in-time well-posedness result for the spatially inhomogeneous Boltzmann equation with full-range hard potentials and non-cutoff kernel in a non-perturbative setting, under macroscopic bounds. The authors introduce a framework that tolerates polynomial velocity decay and exploit a novel combination of hypoelliptic regularization with interpolation inequalities to control weight-loss in velocity moments. Central to the approach are precise coercivity, trilinear, moment, and commutator estimates, plus a hypoelliptic mechanism that translates velocity diffusion into spatial regularity. The resulting local weak solution is nonnegative, satisfies the macroscopic bounds, and is unique within the prescribed class, advancing the conditional-regularity program for rough data in non-perturbative hard-potential regimes.

Abstract

We consider the spatially inhomogeneous non-cutoff Boltzmann equation with hard potentials in the non-perturbative setting. For initial data with polynomial decay in the velocity variable, we establish the local-in-time existence and uniqueness of weak solutions, conditional to pointwise bounds on the hydrodynamic quantities (mass, energy, and entropy). Compared to the soft potential case, the key challenge for full-range hard potentials lies in the more severe loss of velocity moments. The proof combines a hypoelliptic estimate with interpolation inequalities to handle the moment-loss terms.

Local well-posedness for the Boltzmann equation with hard potentials

TL;DR

This work proves a local-in-time well-posedness result for the spatially inhomogeneous Boltzmann equation with full-range hard potentials and non-cutoff kernel in a non-perturbative setting, under macroscopic bounds. The authors introduce a framework that tolerates polynomial velocity decay and exploit a novel combination of hypoelliptic regularization with interpolation inequalities to control weight-loss in velocity moments. Central to the approach are precise coercivity, trilinear, moment, and commutator estimates, plus a hypoelliptic mechanism that translates velocity diffusion into spatial regularity. The resulting local weak solution is nonnegative, satisfies the macroscopic bounds, and is unique within the prescribed class, advancing the conditional-regularity program for rough data in non-perturbative hard-potential regimes.

Abstract

We consider the spatially inhomogeneous non-cutoff Boltzmann equation with hard potentials in the non-perturbative setting. For initial data with polynomial decay in the velocity variable, we establish the local-in-time existence and uniqueness of weak solutions, conditional to pointwise bounds on the hydrodynamic quantities (mass, energy, and entropy). Compared to the soft potential case, the key challenge for full-range hard potentials lies in the more severe loss of velocity moments. The proof combines a hypoelliptic estimate with interpolation inequalities to handle the moment-loss terms.
Paper Structure (22 sections, 22 theorems, 342 equations)

This paper contains 22 sections, 22 theorems, 342 equations.

Table of Contents

  1. Introduction
  2. Notations and Main Result
  3. Related works
  4. Difficulty and methodology for non-perturbative setting
  5. Proof Scheme for Theorem \ref{['local-exsitence']}
  6. Choice of parameters in weight functions
  7. Organization of the paper and conventions for constants
  8. Preliminary Estimates
  9. Trilinear bound and coercivity estimate
  10. Interpolation inequalities
  11. Moment estimate for the collision operator
  12. $L^2$-estimates for the collision operator
  13. Hypoelliptic estimates
  14. Upper bound of trilinear terms
  15. Coercivity estimate in the velocity variable
  16. ...and 7 more sections

Key Result

Theorem 1.3

Suppose that the collision kernel $B$ takes the form kern with parameters $0 < s < 1$ and $\gamma\geq 0$. Then there exist two constants $\ell_0>\ell$, depending only on $s$ and $\gamma$, such that the following existence and uniqueness results hold.

Theorems & Definitions (55)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Lemma 2.1
  • proof
  • ...and 45 more