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Global renormalized solutions for hard potential non-cutoff Boltzmann equation without defect measure

Yi-Long Luo, Jing-Xin Nie

Abstract

The existence of global renormalized solutions to the Boltzmann equation with long-range interactions without angular cutoff was first established by Alexandre and Villani [Comm. Pure Appl. Math., 55(1), 30-70, 2002]. Their result relies on a definition of renormalized solutions involving a non-negative defect measure. In this paper, we address this issue for the inverse power law model in the case of hard potentials ($0 \leq γ\leq 1$). By exploiting the stronger coercivity estimates provided by hard potentials, we prove that the defect measure actually vanishes. Consequently, we establish the global existence of renormalized solutions for the non-cutoff Boltzmann equation with hard potentials in the standard sense, without any defect measure. Finally, we construct a counterexample showing that the approach developed for the hard potential case fails for soft potential model ($-3 < γ< 0$).

Global renormalized solutions for hard potential non-cutoff Boltzmann equation without defect measure

Abstract

The existence of global renormalized solutions to the Boltzmann equation with long-range interactions without angular cutoff was first established by Alexandre and Villani [Comm. Pure Appl. Math., 55(1), 30-70, 2002]. Their result relies on a definition of renormalized solutions involving a non-negative defect measure. In this paper, we address this issue for the inverse power law model in the case of hard potentials (). By exploiting the stronger coercivity estimates provided by hard potentials, we prove that the defect measure actually vanishes. Consequently, we establish the global existence of renormalized solutions for the non-cutoff Boltzmann equation with hard potentials in the standard sense, without any defect measure. Finally, we construct a counterexample showing that the approach developed for the hard potential case fails for soft potential model ().
Paper Structure (28 sections, 40 theorems, 393 equations)

This paper contains 28 sections, 40 theorems, 393 equations.

Table of Contents

  1. Introduction and main results
  2. Motivations
  3. Properties of momentum transfer
  4. Notations and main results
  5. Notations
  6. Initial data
  7. Definition of renormalized solutions
  8. Comparison with the work Alex-Vill-2002-CPAM of Alexandre and Villani
  9. Historical remarks.
  10. Sketch of proofs
  11. Organizations of current paper
  12. Toolbox
  13. global existence of approximation equation
  14. integrability of the renormalized collision operator
  15. Cancellation lemma
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $0 \leq \gamma \leq 1$ and $\nu < \alpha < 1$ with $\nu \in [0, 1/2]$ defined in nu. Suppose that the initial data $f_0$ satisfies f0 and f0-p. Then the Cauchy problem Boltz-f0 admits a renormalized solution in the sense of Definition def-re, satisfying the following statements:

Theorems & Definitions (80)

  • Definition 1.1: Renormalized solution
  • Remark 1.1
  • Theorem 1.1
  • Remark 1.2: Soft potential model $-3 < \gamma < 0$
  • Definition 2.1
  • Lemma 2.1: Dunford-Pettis, Dunford-Schwartz-1958
  • Lemma 2.2: Vitali Convergence Theorem, Dunford-Schwartz-1958Folland-Real
  • Lemma 2.3: Monotone Convergence Theorem, Brezis-2010
  • Lemma 2.4: Lebesgue's Dominated Convergence Theorem, Brezis-2010
  • Lemma 2.5: Fatou's lemma, Brezis-2010
  • ...and 70 more