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Fractional Keller-Segel Equation: Global Well-posedness and Finite Time Blow-up

Laurent Lafleche, Samir Salem

Abstract

This article studies the aggregation diffusion equation \[ \partial_tρ= Δ^\fracα{2} ρ+ λ\,\mathrm{div}((K*ρ)ρ), \] where $Δ^\fracα{2}$ denotes the fractional Laplacian and $K = \frac{x}{|x|^β}$ is an attractive kernel. This equation is a generalization of the classical Keller-Segel equation, which arises in the modeling of the motion of cells. In the diffusion dominated case $β< α$ we prove global well-posedness for an $L^1_k$ initial condition, and in the fair competition case $β= α$ for an $L^1_k\cap L\ln L$ initial condition with small mass. In the aggregation dominated case $β> α$, we prove global or local well-posedness for an $L^p$ initial condition, depending on some smallness condition on the $L^p$ norm of the initial data. We also prove that finite time blow-up of even solutions occurs under some initial mass concentration criteria.

Fractional Keller-Segel Equation: Global Well-posedness and Finite Time Blow-up

Abstract

This article studies the aggregation diffusion equation where denotes the fractional Laplacian and is an attractive kernel. This equation is a generalization of the classical Keller-Segel equation, which arises in the modeling of the motion of cells. In the diffusion dominated case we prove global well-posedness for an initial condition, and in the fair competition case for an initial condition with small mass. In the aggregation dominated case , we prove global or local well-posedness for an initial condition, depending on some smallness condition on the norm of the initial data. We also prove that finite time blow-up of even solutions occurs under some initial mass concentration criteria.

Paper Structure

This paper contains 16 sections, 11 theorems, 217 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Proof of Theorem \ref{['thm:welpos']} and Theorem \ref{['thm:behaviour']}
  4. A Priori estimates.
  5. Step 1. Differential inequality for the $L^p$ norm.
  6. Step 2. Conditions on $\varepsilon$.
  7. Step 3. Case $\beta < \alpha$.
  8. Step 4. Case $\beta > \alpha$.
  9. Step 5. Case $\beta=\alpha$.
  10. Tightness and coupling.
  11. Step 1. Proof of \ref{['eq:estim_d_wk_lineaire']}.
  12. Step 2. Proof of \ref{['eq:estim_d_wk_log']}.
  13. Step 1. Tightness.
  14. Step 2. A priori properties of the limit point.
  15. Step 3. Uniqueness of the limiting point.
  16. ...and 1 more sections

Key Result

Theorem 1

Let $(\alpha,\beta)\in [0,2)\times [0,d)$ be such that $\beta + \alpha > 1$ and $k\in [(1-\beta)_+,\alpha)$. $\bullet$ When $\beta<\alpha$ and $\rho^\mathrm{in}\in L^1_k$, there exists a unique and global weak solution to the eq:FKS equation. $\bullet$ When $\beta=\alpha$, if $\rho^\mathrm{in}\in L^ then there exists a unique and global weak solution to the eq:FKS equation. $\bullet$ When $\beta>\

Figures (3)

  • Figure 1: Existing results for the \ref{['eq:FKS']} equation.
  • Figure 2: Range of application of Theorems \ref{['thm:welpos']} and \ref{['thm:BU']}. We emphasize that for $d>2$ the results extend to the segment $(\alpha,\beta)\in \{2\}\times(0,d)$.
  • Figure 3: Lower bound of the threshold of condition \ref{['eq:condSM']} for $d=2,3,4$ and $\beta\in(0,2)$. For the case $\beta\leq \frac{1}{2}$ see Remark \ref{['rq:TriBer']}.

Theorems & Definitions (28)

  • Definition 2.1
  • Theorem 1
  • Remark 2.1
  • Remark 2.2
  • Theorem 2
  • Remark 2.3
  • Theorem 3
  • Proposition 3.1: Propagation of weight
  • proof
  • Lemma 3.1: General estimate
  • ...and 18 more