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Non-cutoff Boltzmann equation with soft potentials in the whole space

Kleber Carrapatoso, Pierre Gervais

Abstract

We prove the existence and uniqueness of global solutions to the Boltzmann equation with non-cutoff soft potentials in the whole space when the initial data is a small perturbation of a Maxwellian with polynomial decay in velocity. Our method is based in the decomposition of the desired solution into two parts: one with polynomial decay in velocity satisfying the Boltzmann equation with only a dissipative part of the linearized operator ; the other with Gaussian decay in velocity verifying the Boltzmann equation with a coupling term.

Non-cutoff Boltzmann equation with soft potentials in the whole space

Abstract

We prove the existence and uniqueness of global solutions to the Boltzmann equation with non-cutoff soft potentials in the whole space when the initial data is a small perturbation of a Maxwellian with polynomial decay in velocity. Our method is based in the decomposition of the desired solution into two parts: one with polynomial decay in velocity satisfying the Boltzmann equation with only a dissipative part of the linearized operator ; the other with Gaussian decay in velocity verifying the Boltzmann equation with a coupling term.
Paper Structure (25 sections, 24 theorems, 371 equations, 1 figure)

This paper contains 25 sections, 24 theorems, 371 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main result
  3. Notations
  4. Estimates on the collision operator
  5. Auxiliary results
  6. Homogeneous estimates in polynomially weighted spaces
  7. Linear theory
  8. Estimates on $\mathscr L$
  9. Estimates on $\mathscr L - v \cdot \nabla_x$ in exponentially weighted spaces
  10. Estimates on $\mathcal{B}-v \cdot \nabla_x$ in polynomially weighted spaces
  11. Nonlinear estimates
  12. Nonlinear estimates in Sobolev-type spaces
  13. Nonlinear estimates in Fourier-based spaces
  14. Proof of the main result
  15. Stability of the scheme
  16. ...and 10 more sections

Key Result

Theorem 1.1

Assume eq:noncutoff1--eq:noncutoff2 hold. Consider $k > 13/2 + 7|\gamma| / 2 + 8s$ and define the weight function $m = \langle v \rangle^k$. There exists ${\varepsilon}_0 > 0$ small enough such that any initial data $f_0 \in \mathbf{X}(m)$ satisfying $\| f_0 \|_{\mathbf{X}(m)} \leqslant {\varepsilon If moreover the initial data $\widehat{f}_0 \in L^p_\xi L^2_v(\langle v \rangle^{-8s} m)$ satisfies

Figures (1)

  • Figure 1: The changes of variables $(v, v') \rightarrow (v, v_*)$ and $(v', v_*) \rightarrow (v, v_*)$.

Theorems & Definitions (41)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4: Cancellation lemma
  • Lemma 2.5
  • Lemma 2.6
  • ...and 31 more