[Paper Review] An interpolation inequality and its application in Keller-Segel model

Xinru Cao|arXiv (Cornell University)|Jul 28, 2017

Mathematical Biology Tumor Growth14 references3 citations

TL;DR

This paper establishes a new Ehrling-type interpolation inequality that improves upon the Gagliardo-Nirenberg inequality, which is then applied to the Keller-Segel chemotaxis model in bounded domains. The key result shows that global existence and boundedness of solutions are guaranteed if the $ L^{N/2} $-norm of the cell density $ u $ is uniformly bounded and the family $ \{u^{N/2}\} $ is equi-integrable—conditions that are strictly weaker than previous global existence criteria.

ABSTRACT

In this paper, we first prove an interpolation inequality of Ehrling-type, which is an improvement of a special case to the well known Gargliardo-Nirenberg inequality. Then we apply it to study the classical Keller-Segel system \begin{equation} \left\{ \begin{array}{llc} u_t=Δu- abla\cdot(u abla v), \\[6pt] \displaystyle v_t=Δv-v+u, \end{array} ight. \end{equation} in a bounded domain $Ω\subset\mathbb{R}^N$ ($N\ge 2$) with smooth boundary. It is known that for any $δ>0$, if $\int_Ωu^{\frac N2+δ}(\cdot,t)$ is bounded, then the solution is global and bounded. Here we show that the same conclusion holds for a weaker assumption: the equi-integrability of $\{\int_Ωu^\frac N2(\cdot,t)|~t\in(0,T_{\max})\}$ can prevent blow up.

Motivation & Objective

• To establish a refined interpolation inequality of Ehrling-type that strengthens the classical Gagliardo-Nirenberg inequality in a special case.
• To investigate the role of equi-integrability in preventing blow-up in the Keller-Segel system, particularly in dimensions $ N \geq 2 $.
• To weaken the standard $ L^p $-boundedness condition ($ p > N/2 $) for global existence, replacing it with a weaker equi-integrability assumption on $ u^{N/2} $.
• To extend known global existence criteria by showing that $ \int_\Omega u^{N/2} \log u $ boundedness suffices, even when $ \|u\|_{L^{N/2}} $ is bounded but not uniformly integrable.
• To provide a new sufficient condition for global existence and boundedness that applies even when the $ L^{N/2} $-norm is bounded but lacks uniform integrability.

Proposed method

• Derives a new interpolation inequality via a refined analysis of the Gagliardo-Nirenberg framework, tailored to the critical exponent $ p = N/2 $.
• Applies the new inequality to the energy estimates of the Keller-Segel system, particularly analyzing the evolution of $ \int_\Omega u^p $ and $ \int_\Omega |\nabla v|^{2q} $.
• Uses the De la Vallée-Poussin criterion to characterize equi-integrability of $ \{u^{N/2}\} $, linking it to the growth of a function $ f(s) $ such that $ f(s)/s^{N/2} \to \infty $ as $ s \to \infty $.
• Constructs a reflection-based extension operator for functions on a domain with boundary, using a partition of unity and $ C^1 $-diffeomorphisms to control norms in the extension.
• Employs change-of-variables and norm estimates in local coordinates to bound the $ L^p $-norm of the extended function, ensuring the extension preserves equi-integrability.
• Combines the refined inequality with the equi-integrability condition to derive a priori bounds on $ u $ and $ v $, leading to global existence and boundedness.

Experimental results

Research questions

• RQ1 Can the classical $ L^p $-boundedness condition for global existence in the Keller-Segel model be weakened when $ p = N/2 $?
• RQ2 Is equi-integrability of $ \{u^{N/2}(\cdot,t)\} $ sufficient to prevent finite-time blow-up in the Keller-Segel system for $ N \geq 2 $?
• RQ3 Does the condition $ \sup_{t \in (0,T_{\max})} \|u(\cdot,t)\|_{L^{N/2}(\Omega)} < \infty $ combined with equi-integrability of $ u^{N/2} $ imply global boundedness?
• RQ4 Can the growth condition $ f(s)/s^{N/2} \to \infty $ as $ s \to \infty $ be used to characterize a broader class of functions that prevent blow-up?
• RQ5 How does the new interpolation inequality improve upon the Gagliardo-Nirenberg inequality in the critical case $ p = N/2 $?

Key findings

• The paper establishes a new interpolation inequality that improves the Gagliardo-Nirenberg inequality in the critical case $ p = N/2 $, providing a sharper control on $ L^p $-norms.
• Global existence and boundedness of solutions to the Keller-Segel system are proven under the weaker condition that $ \{u^{N/2}(\cdot,t)\}_{t \in (0,T_{\max})} $ is equi-integrable, even if $ \|u(\cdot,t)\|_{L^{N/2}} $ is bounded.
• The condition $ \int_\Omega u^{N/2} \log u \, dx < \infty $ uniformly in time is shown to be sufficient for global existence and boundedness, which is not covered by classical $ L^p $-boundedness criteria.
• The De la Vallée-Poussin criterion is applied to show that equi-integrability of $ \{u^{N/2}\} $ is equivalent to the uniform boundedness of $ \int_\Omega f(u) \, dx $ for any $ f(s) $ with $ f(s)/s^{N/2} \to \infty $ as $ s \to \infty $.
• The proof relies on a novel extension operator for functions on domains with boundary, constructed via reflection and partition of unity, to control the $ L^p $-norm of the extension.
• The authors demonstrate that the equi-integrability of $ \{u^{N/2}\} $ ensures the existence of a uniform $ L^p $-bound for the extended function, which is essential for the a priori estimates.