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Uniqueness and Zeroth-Order Analysis of Weak Solutions to the Non-cutoff Boltzmann equation

Dingqun Deng, Shota Sakamoto

Abstract

We establish the uniqueness of large solutions to the non-cutoff Boltzmann equation with moderate soft potentials. Specifically, the weak solution $F=μ+μ^{\frac{1}{2}}f$ is unique as long as it has finite energy, in the sense that the norm $\|f\|_{L^\infty_t L^{r}_{x,v}}+\|f\|_{L^\infty_t L^2_{x,v}}$ remains bounded (arbitrary large) for some sufficiently large $r>0$. Our approach applies dilated dyadic decompositions in phase space $(v,ξ,η)$ to capture hypoellipticity and to reduce the fractional derivative structure $(-Δ_v)^{s}$ of the Boltzmann collision operator to zeroth order. The difficulties posed by the large solution are overcome through the negative-order hypoelliptic estimate that gains integrability in $(t,x)$.

Uniqueness and Zeroth-Order Analysis of Weak Solutions to the Non-cutoff Boltzmann equation

Abstract

We establish the uniqueness of large solutions to the non-cutoff Boltzmann equation with moderate soft potentials. Specifically, the weak solution is unique as long as it has finite energy, in the sense that the norm remains bounded (arbitrary large) for some sufficiently large . Our approach applies dilated dyadic decompositions in phase space to capture hypoellipticity and to reduce the fractional derivative structure of the Boltzmann collision operator to zeroth order. The difficulties posed by the large solution are overcome through the negative-order hypoelliptic estimate that gains integrability in .
Paper Structure (60 sections, 17 theorems, 499 equations)

This paper contains 60 sections, 17 theorems, 499 equations.

Table of Contents

  1. Introduction
  2. Non-cutoff Boltzmann model and difficulties of uniqueness
  3. Reformulations
  4. Difficulties of uniqueness for nonlinear kinetic equations
  5. Main result
  6. Notations
  7. Miscellany
  8. $L^p$ spaces
  9. Bessel potential
  10. Dissipation norm
  11. The a priori assumption
  12. The weak solution
  13. Main novelty and ideas for the uniqueness of large solution
  14. Toolbox
  15. Tools for pseudo-differential calculus
  16. ...and 45 more sections

Key Result

Theorem 1.1

Fix any $M_0>0$. Assume the moderate soft potential case, where $(\gamma, s)$ satisfy AssGas. Then there exist large constants $r^*=r^*(d,\gamma,s)\in(2,\infty)$ and $C_{\gamma,s,d}>0$, and a small time $T=T(M_0,\gamma,s,d)>0$ such that, if $\phi_1$ and $\phi_2$ are two solutions of the non-cutoff B Then Consequently, since the uniqueness time depends only on $M_0,\gamma,s,d$, the solution is uni

Theorems & Definitions (29)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 2.1: Deng2020a
  • Theorem 2.2: Seeger2018
  • Lemma 2.3: Alexandre2011 and Morimoto2016
  • Lemma 2.4
  • proof
  • Lemma 2.5: Angular integrals for velocity and frequency
  • proof
  • Lemma 2.6: Regular change of variable
  • ...and 19 more