Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Analiticity of the Flow for the Aggregation Equation

J. M. Burgués, J. Mateu

Abstract

Let $Ω$ be a bounded domain in $\R^n$ whose boundary is $\ka{1,\,γ}$ for $γ\in(0,\,1)$. Consider the aggregation equation in the case of the initial condition being a positive multiple of the characteristic function of $Ω$. In this paper we prove global in time analyticity of the flow generated by the velocity field which propagates the density solution of this equation.

Analiticity of the Flow for the Aggregation Equation

Abstract

Let be a bounded domain in whose boundary is for . Consider the aggregation equation in the case of the initial condition being a positive multiple of the characteristic function of . In this paper we prove global in time analyticity of the flow generated by the velocity field which propagates the density solution of this equation.
Paper Structure (26 sections, 31 theorems, 426 equations)

This paper contains 26 sections, 31 theorems, 426 equations.

Table of Contents

  1. Introduction
  2. Historical
  3. Idea of the proof
  4. Plan of the paper
  5. Preliminaries
  6. Differential forms
  7. The fundamental operators.
  8. The patch problem for the aggregation equation.
  9. The flow in terms of the density.
  10. Analyticity of the flow: Proof of Theorem \ref{['Main']}.
  11. Global analyticity in time.
  12. Construction of an analytic solution for the flow-density equation (\ref{['eqP']}).
  13. A priori relationships
  14. Extension of the recurrence to the boundary.
  15. Estimates
  16. ...and 11 more sections

Key Result

Theorem 1

Let $\Omega\subset\mathbb R^n$ be a bounded domain with $\partial\Omega\in\mathcal{C}^{1,\, \gamma}$ for $\gamma\in(0,\, 1)$. Assume $\psi(x,\, t)$ be the flow corresponding to the solution $(v,\, \rho)$ of the equation eq(A) when the initial condition $\rho_0(x)$ is a constant multiple of the chara

Theorems & Definitions (61)

  • Theorem 1
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • Remark
  • proof
  • Theorem 2
  • Proposition 3
  • ...and 51 more