[Paper Review] Renormalized solutions to a chemotaxis system with consumption of chemoattractant

Hengling Wang, Yuxiang Li|arXiv (Cornell University)|Mar 14, 2018

Mathematical Biology Tumor Growth16 references3 citations

TL;DR

This paper establishes the global existence of renormalized solutions to a high-dimensional Keller-Segel-type chemotaxis system with chemoattractant consumption in bounded convex domains for dimensions $ n \geq 4 $. By introducing a truncation-based renormalization framework and leveraging entropy estimates, the authors prove that for initial data $ u_0 \in C^0(\overline{\Omega}) $, $ v_0 \in W^{1,q}(\Omega) $ with $ q > n $, a global renormalized solution exists despite the lack of uniform boundedness in standard weak solution frameworks.

ABSTRACT

This paper investigates a high-dimensional chemotaxis system with consumption of chemoattractant \begin{eqnarray*} \left\{\begin{array}{l} u_t=Δu- abla\cdot(u abla v), v_t=Δv-uv, \end{array} ight. \end{eqnarray*} under homogeneous boundary conditions of Neumann type, in a bounded convex domain $Ω\subset\mathbb{R}^n~(n\geq4)$ with smooth boundary. It is proved that if initial data satisfy $u_0\in C^0(\overlineΩ)$ and $v_0\in W^{1,q}(Ω)$ for some $q>n$, the model possesses at least one global renormalized solution.

Motivation & Objective

• To resolve the open problem of global existence for renormalized solutions in high-dimensional chemotaxis systems with chemoattractant consumption.
• To overcome the lack of uniform boundedness in $ u\nabla v $, a key obstacle in classical weak solution theory.
• To extend the concept of renormalized solutions to chemotaxis models with consumption, particularly in dimensions $ n \geq 4 $.
• To establish existence under general initial data conditions: $ u_0 \in C^0(\overline{\Omega}) $, $ v_0 \in W^{1,q}(\Omega) $ for $ q > n $.

Proposed method

• Introduce a family of smooth, bounded truncation functions $ \varphi_E $ satisfying (E1)–(E7) to regularize the solution $ u $.
• Apply the renormalized solution framework from Fischer [8], using $ \xi(\varphi_E(u)) $ as test functions to derive approximate equations.
• Use entropy estimates and energy methods to control the nonlinearities and derive uniform bounds independent of $ \varepsilon $.
• Pass to the limit $ E \to \infty $ using Fatou’s lemma and dominated convergence, relying on the decay of $ \varphi_E'' $ in $ L^\infty $.
• Establish convergence of terms involving $ \varphi_E'(u) $ and $ \varphi_E''(u) $ by exploiting compact support of $ \xi $'s derivatives.
• Leverage the fact that $ \varphi_E(u) = u $ for $ u < E $, ensuring pointwise convergence and consistency in the limit.

Experimental results

Research questions

• RQ1 Can global renormalized solutions be constructed for the chemotaxis system with chemoattractant consumption in dimensions $ n \geq 4 $, even for arbitrarily large initial data?
• RQ2 How can the lack of uniform $ L^s $ bounds on $ u\nabla v $ be overcome in the weak solution framework?
• RQ3 Does the renormalized solution concept provide a viable alternative to classical solutions when global boundedness fails?
• RQ4 What role do entropy estimates and truncation techniques play in ensuring existence beyond the classical framework?

Key findings

• The model admits at least one global renormalized solution for all $ n \geq 4 $, even with arbitrarily large initial data.
• The solution satisfies $ u \in L^\infty(0,T; L^\infty(\Omega)) $ and $ \nabla \sqrt{u} \in L^2(0,T; L^2(\Omega)) $, ensuring sufficient regularity for renormalization.
• The renormalized solution framework successfully handles the cross-diffusive term $ \nabla \cdot (u\nabla v) $ via truncation and limit passage.
• The limit of the measure $ \mu^E $ vanishes as $ E \to \infty $, confirming the consistency of the renormalized formulation.
• The convergence of the nonlinear terms in the weak formulation is guaranteed by the compact support of $ \xi' $ and $ \xi'' $, ensuring the limit yields an exact equation.
• The result extends the applicability of renormalized solutions to chemotaxis systems beyond low dimensions and small initial data.