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On the Cauchy problem of the Boltzmann equation with a very soft potential

Dingqun Deng

Abstract

The Cauchy problem for the Boltzmann equation with soft potential, in the framework of small perturbation of an equilibrium state, has been studied in many spaces. The method of strongly continuous semigroup has been applied by Caflisch\cite{Caflisch1980a} and Ukai-Asano\cite{Ukai1982} for the case of soft potential, where they obtained the $L^\infty$ solution without requiring any velocity deviation. By generalizing the estimate on linearized collision operator $L$ to the case of very soft potential, we obtain a similar global existence result for $γ\in[0,d)$. For soft potential, the spectrum structure of the linearized Boltzmann operator couldn't give spectral gap, so we use the method of integration by parts and consider a weighted velocity space in order to obtain algebraic decay in time.

On the Cauchy problem of the Boltzmann equation with a very soft potential

Abstract

The Cauchy problem for the Boltzmann equation with soft potential, in the framework of small perturbation of an equilibrium state, has been studied in many spaces. The method of strongly continuous semigroup has been applied by Caflisch\cite{Caflisch1980a} and Ukai-Asano\cite{Ukai1982} for the case of soft potential, where they obtained the solution without requiring any velocity deviation. By generalizing the estimate on linearized collision operator to the case of very soft potential, we obtain a similar global existence result for . For soft potential, the spectrum structure of the linearized Boltzmann operator couldn't give spectral gap, so we use the method of integration by parts and consider a weighted velocity space in order to obtain algebraic decay in time.

Paper Structure

This paper contains 20 sections, 37 theorems, 353 equations.

Table of Contents

  1. Introduction
  2. Properties of the Linearized Collision Operator
  3. Estimate on the Linearized Boltzmann Operator
  4. Eigenvalue Structure near $y=0$
  5. Eigenvalue Structure of $D$
  6. The Eigenvalue Equation.
  7. The Eigenvalues of $D$.
  8. Asymptotic Behavior of the Eigenvalues to $D$.
  9. The Eigenvalue Projection of $D$.
  10. Eigenvalue Structure of $P(\lambda I -\widehat{B_0}(y))^{-1}P$.
  11. Estimate on the Semigroup $e^{t\widehat{B}(y)}$
  12. $y$ away from origin
  13. $y$ near origin
  14. Result
  15. Estimate on the Semigroup $e^{tB}$ and Global Existence
  16. ...and 5 more sections

Key Result

Theorem 1.1

Assume the cross-section $q$ satisfies the angular cut-off assumption I_cutoffassumption. Assume $d\ge 3$, $\gamma\in[0,d)$, $l>\frac{d}{2}$, $\beta>\frac{d}{2}$, $p\in[1,2)$ such that $\frac{d}{4}(\frac{2}{p}-1)>1/2$. Let $\alpha\in [\frac{1}{2},\min(\frac{d}{4}(\frac{2}{p}-1),1))$. There exists co Let $X=\{f\in L^\infty_{\alpha}(L^\infty_{\beta}(H^l)): \|f\|_{L^\infty_{\alpha}(L^\infty_{\beta}(H

Theorems & Definitions (75)

  • Theorem 1.1
  • Theorem 2.1
  • proof
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.4
  • proof
  • Theorem 2.5
  • Remark 3.1
  • Theorem 3.2
  • ...and 65 more