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On the global in time existence and uniqueness of solutions to the Boltzmann hierarchy

Ioakeim Ampatzoglou, Joseph K. Miller, Nataša Pavlović, Maja Tasković

Abstract

In this paper we establish the global in time existence and uniqueness of solutions to the Boltzmann hierarchy, a hierarchy of equations instrumental for the rigorous derivation of the Boltzmann equation from many particles. Inspired by available $L^{\infty}$-based a-priori estimate for solutions to the Boltzmann equation, we develop the polynomially weighted $L^\infty$ a-priori bounds for solutions to the Boltzmann hierarchy and handle the factorial growth of the number of terms in the Dyson's series by reorganizing the sum through a combinatorial technique known as the Klainerman-Machedon board game argument. This paper is the first work that exploits such a combinatorial technique in conjunction with an $L^{\infty}$-based estimate to prove uniqueness of the mild solutions to the Boltzmann hierarchy. Our proof of existence of global in time mild solutions to the Boltzmann hierarchy for admissible initial data is constructive and it employs known global in time solutions to the Boltzmann equation via a Hewitt-Savage type theorem.

On the global in time existence and uniqueness of solutions to the Boltzmann hierarchy

Abstract

In this paper we establish the global in time existence and uniqueness of solutions to the Boltzmann hierarchy, a hierarchy of equations instrumental for the rigorous derivation of the Boltzmann equation from many particles. Inspired by available -based a-priori estimate for solutions to the Boltzmann equation, we develop the polynomially weighted a-priori bounds for solutions to the Boltzmann hierarchy and handle the factorial growth of the number of terms in the Dyson's series by reorganizing the sum through a combinatorial technique known as the Klainerman-Machedon board game argument. This paper is the first work that exploits such a combinatorial technique in conjunction with an -based estimate to prove uniqueness of the mild solutions to the Boltzmann hierarchy. Our proof of existence of global in time mild solutions to the Boltzmann hierarchy for admissible initial data is constructive and it employs known global in time solutions to the Boltzmann equation via a Hewitt-Savage type theorem.
Paper Structure (14 sections, 16 theorems, 197 equations)

This paper contains 14 sections, 16 theorems, 197 equations.

Table of Contents

  1. Introduction
  2. Notation and main results
  3. The Boltzmann hierarchy
  4. The Functional Spaces
  5. Main results
  6. Uniqueness of solutions to the Boltzmann hierarchy
  7. A priori estimates
  8. Reorganization of the integral via the board game
  9. A priori estimate and combinatorial reorganization in action together
  10. Existence of Solutions to the Boltzmann equation and to the Boltzmann hierarchy
  11. The Boltzmann equation
  12. Global well-posedness of the Boltzmann equation for small space-velocity polynomially decaying initial data
  13. Global well-posedness of the Boltzmann hierarchy for admissible data
  14. Appendix

Key Result

Theorem 2.7

Consider the Boltzmann hierarchy BH with the cross section cross-section form. Let $T>0$, $p>1, q>\max\{d-1+\gamma,d-1\}$, and $\alpha,\beta>0$. Consider $\mu\in{\mathbb R}$ with $e^\mu> 4 C_{p,q,\alpha,\beta}$, where and $U_q$ is the constant of Lemma lemma on velocities weight. Let $F_0=(f_0^{(s)})\in \mathcal{X}_{p,q,\alpha,\beta,\mu}^\infty$, and assume $F=(f^{(k)})_{k=1}^\infty$ is a mild $\

Theorems & Definitions (47)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Theorem 2.7
  • Definition 2.8: Admissibility
  • Remark 2.9
  • Theorem 2.10
  • ...and 37 more