[Paper Review] Inhomogeneous Patlak-Keller-Segel models and Aggregation Equations with Nonlinear Diffusion in $\Real^d$

Jacob Bedrossian, Nancy Rodríguez|arXiv (Cornell University)|Aug 25, 2011

Mathematical Biology Tumor Growth28 references3 citations

TL;DR

This paper establishes a unified local and global existence theory for inhomogeneous Patlak-Keller-Segel models and aggregation equations with nonlinear diffusion in $\mathbb{R}^d$ for $d \geq 2$, using a direct regularization on $\mathbb{R}^d$ to overcome limitations in prior work. It proves small data global existence, uniform boundedness in supercritical regimes, and identifies the critical mass for logarithmic singular kernels, resolving the long-open $\mathbb{R}^2$ case.

ABSTRACT

Aggregation equations and Patlak-Keller-Segel (PKS) models for chemotaxis with nonlinear diffusion are popular models for nonlocal aggregation phenomenon and are a source of a number of interesting mathematical problems in nonlinear PDE. The purpose of this work is twofold. First, we continue our previous work, which focused on nonlocal aggregation, modeled with a convolution. The goal was to unify the local and global theory of these convolution-type models, including the identification of a sharp critical mass; however, some cases involving unbounded domains were left open. In particular, the biologically relevant case $\Real^2$ was not treated. In this paper, we present an alternative proof of local existence, which now applies to $\Real^d$ for all $d \geq 2$ and give global results that were left open. The proof departs from previous work in that it uses a more direct and intuitive regularization that constructs approximate solutions on $\Real^d$ instead of on sequences of bounded domains. Second, this work develops the local, subcritical, and small data critical theory for a variety of Patlak-Keller-Segel models with spatially varying diffusion and decay rate of the chemo-attractant.

Motivation & Objective

• To complete the local and global existence theory for Patlak-Keller-Segel models with nonlinear diffusion in $\mathbb{R}^d$, especially resolving the unresolved $\mathbb{R}^2$ case from prior work.
• To develop a direct regularization method on $\mathbb{R}^d$ instead of on bounded domains, enabling rigorous justification of homogeneous Sobolev embeddings in formal arguments.
• To extend the small data critical/supercritical theory and global existence results to variable-coefficient models with spatially varying diffusion and chemo-attractant decay.
• To identify the critical mass for kernels with logarithmic singularities in $\mathbb{R}^d$ using refined $L^p$ estimates and Calderón-Zygmund-type inequalities.
• To establish uniform boundedness and small data global existence in supercritical regimes through iteration techniques and Gagliardo-Nirenberg-type inequalities.

Proposed method

• Introduces a direct regularization on $\mathbb{R}^d$ for constructing approximate solutions, avoiding the need for approximation on bounded domains.
• Employs a modified Gagliardo-Nirenberg inequality to control $\|D^2 c\|_p$ in terms of $\|f\|_p$ and lower-order terms, crucial for handling non-constant coefficients.
• Uses Calderón-Zygmund estimates in balls to control second-order derivatives of the chemo-attractant $c$, with error terms controlled via interpolation and smallness assumptions.
• Applies iteration techniques similar to those in [20, 41, 3] to prove small data global existence and uniform boundedness in supercritical regimes.
• Derives $L^p$ estimates for $c$ and its derivatives via weighted $L^p$ norms and $\nabla a$-dependent constants, ensuring uniform bounds as $p \to \infty$.
• Relies on the free energy dissipation structure $\mathcal{F}(u) = S(u) - \mathcal{W}(u)$, with $S(u)$ strictly convex and $\mathcal{W}(u)$ representing interaction energy.

Experimental results

Research questions

• RQ1 What is the sharp critical mass for Patlak-Keller-Segel models with logarithmic singularity in $\mathbb{R}^d$?
• RQ2 How can local and global existence be established for $\mathbb{R}^2$ using a direct regularization on the whole space?
• RQ3 What conditions ensure small data global existence and uniform boundedness in the supercritical regime for nonlinear diffusion models?
• RQ4 How do variable coefficients in diffusion and chemo-attractant decay affect the regularity and long-time behavior of solutions?
• RQ5 Can homogeneous Sobolev embeddings be rigorously justified in the context of nonlocal aggregation equations via direct regularization?

Key findings

• The paper establishes local existence for $\mathbb{R}^d$, $d \geq 2$, using a direct regularization on $\mathbb{R}^d$, resolving the $\mathbb{R}^2$ case left open in prior work.
• Small data global existence and uniform boundedness are proven in the supercritical regime via iteration and $L^p$ estimates, with $C(a,p,d) \lesssim p$ as $p \to \infty$.
• The critical mass for kernels with logarithmic singularity at the origin is identified and estimated in $\mathbb{R}^d$, extending previous results to the biologically relevant 2D case.
• A new $L^p$ estimate for $D^2 c$ is derived: $\|D^2 c\|_p \lesssim \|f\|_p + \|f\|_{dp/(2p+d)}$, which controls lower-order terms uniformly in $R$.
• The method allows rigorous justification of homogeneous Sobolev embeddings in formal calculations, which are essential for proving supercritical global existence.
• The theory is extended to variable-coefficient models where $a(x)$ and $\gamma(x)$ are non-constant, with $\|D^2 c\|_p \leq C(a,p,d)\|f\|_p$ and $C(a,p,d) \lesssim p$ as $p \to \infty$.