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Global well-posedness and relaxation for solutions of the Fokker-Planck-Alignment equations

R. Shvydkoy

Abstract

In this paper we prove global existence of weak solutions, their regularization, and global relaxation to Maxwellian for a broad class of Fokker-Planck-Alignment models which appear in collective dynamics. The main feature of these results, as opposed to previously known ones, is the lack of regularity or no-vacuum requirements on the initial data. With a particular application to the classical kinetic Cucker-Smale model, we demonstrate that any bounded data with finite energy, $(1+ |v|^2) f_0 \in L^1$, $f_0 \in L^\infty$, and finite higher moment $|v|^q f\in L^2$, $q \gg 2$, gives rise to a global instantly smooth solution, satisfying entropy equality and relaxing exponentially fast. The results are achieved through the use of a new thickness-based renormalization, which circumvents the problem of degenerate diffusion in non-perturbative regime.

Global well-posedness and relaxation for solutions of the Fokker-Planck-Alignment equations

Abstract

In this paper we prove global existence of weak solutions, their regularization, and global relaxation to Maxwellian for a broad class of Fokker-Planck-Alignment models which appear in collective dynamics. The main feature of these results, as opposed to previously known ones, is the lack of regularity or no-vacuum requirements on the initial data. With a particular application to the classical kinetic Cucker-Smale model, we demonstrate that any bounded data with finite energy, , , and finite higher moment , , gives rise to a global instantly smooth solution, satisfying entropy equality and relaxing exponentially fast. The results are achieved through the use of a new thickness-based renormalization, which circumvents the problem of degenerate diffusion in non-perturbative regime.
Paper Structure (18 sections, 29 theorems, 327 equations, 1 table)

This paper contains 18 sections, 29 theorems, 327 equations, 1 table.

Table of Contents

  1. Introduction
  2. Environmental averaging models, description of the results
  3. Environmental averaging models and their regularity
  4. Existence of weak solutions for regular models of Schur's class
  5. Gain of positivity and parabolic regularization
  6. Global relaxation
  7. Global existence of weak solutions
  8. Approximation
  9. Averaging Lemma
  10. Obtaining a weak solution
  11. Fisher information
  12. Energy law
  13. Renormalization
  14. Entropy law
  15. Weak maximum principle and Proposition \ref{['p:Gauss']}
  16. ...and 3 more sections

Key Result

Theorem 1.1

If the kernel is fat tail, $\int_0^\infty \phi(r) \, \hbox{d}r = \infty$, then solutions to e:CS align exponentially fast to the conserved mean velocity $\bar{v} = \frac{1}{ \sum_{j=1}^N m_j} \sum_{j=1}^N m_j v_j$, while the flock remains bounded where $C,\delta,\bar{D}$ depend only on the initial condition and parameters of the kernel.

Theorems & Definitions (53)

  • Theorem 1.1: CS2007aCS2007bHT2008HL2009
  • Theorem 1.2
  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3: S-EA
  • Remark 2.4
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Remark 2.7
  • ...and 43 more