Mathematical modelling of chemotaxis (the movement of biological cells or organisms in response to chemical gradients) has developed into a large and diverse discipline, whose aspects include its mechanistic basis, the modelling of specific systems and the mathematical behaviour of the underlying equations. The Keller-Segel model of chemotaxis (Keller and Segel in J Theor Biol 26:399–415, 1970; 30:225–234, 1971) has provided a cornerstone for much of this work, its success being a consequence of its intuitive simplicity, analytical tractability and capacity to replicate key behaviour of chemotactic populations. One such property, the ability to display “auto-aggregation”, has led to its prominence as a mechanism for self-organisation of biological systems. This phenomenon has been shown to lead to finite-time blow-up under certain formulations of the model, and a large body of work has been devoted to determining when blow-up occurs or whether globally existing solutions exist. In this paper, we explore in detail a number of variations of the original Keller–Segel model. We review their formulation from a biological perspective, contrast their patterning properties, summarise key results on their analytical properties and classify their solution form. We conclude with a brief discussion and expand on some of the outstanding issues revealed as a result of this work.

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A user’s guide to PDE models for chemotaxis

Convex bodies: the brunn–minkowski theory

This article summarizes various aspects and results for some general formulations of the classical chemotaxis models also known as Keller-Segel models. It is intended as a survey of results for the most common formulation of this classical model for positive chemotactical movement and offers possible generalizations of these results to more universal models. Furthermore it collects open questions and outlines mathematical progress in the study of the Keller-Segel model since the first presentation of the equations in 1970.

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From 1970 until present: the Keller-Segel model in chemotaxis and its consequences

Preface.- Preface to the 2nd edition.- Notational conventions.- 1 Preliminary general material.- I Steady-state problems.- 2 Pseudomonotone or weakly continuous mappings.- 3 Accretive mappings.- 4 Potential problems: smooth case.- 5 Nonsmooth problems variational inequalities.- 6. Systems of equations: particular examples.- II Evolution problems.- 7 Special auxiliary tools.- 8 Evolution by pseudomonotone or weakly continuous mappings.- 9 Evolution governed by accretive mappings.- 10 Evolution governed by certain set-valued mappings.- 11 Doubly-nonlinear problems.- 12 Systems of equations: particular examples.- References.- Index.

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Nonlinear partial differential equations with applications

We consider nonnegative solutions of the Neumann boundary value problem for the chemotaxis system in a smooth bounded convex domain Ω ⊂ ℝ n , n ≥ 1, where τ > 0, χ ∈ ℝ and f is a smooth function generalizing the logistic source It is shown that if μ is sufficiently large then for all sufficiently smooth initial data the problem possesses a unique global-in-time classical solution that is bounded in Ω × (0, ∞). Known results, asserting boundedness under the additional restriction n ≤ 2, are thereby extended to arbitrary space dimensions.

Boundedness in the Higher-Dimensional Parabolic-Parabolic Chemotaxis System with Logistic Source

We investigate point singularities of Willmore surfaces, which for example appear as blowups of the Willmore flow near singularities, and prove that closed Willmore surfaces with one unit density point singularity are smooth in codimension one. As applications we get in codimension one that the Willmore flow of spheres with energy less than 8π exists for all time and converges to a round sphere and further that the set of Willmore tori with energy less than

http://annals.math.princeton.edu/wp-content/uploads/annals-v160-n1-p07.pdf

Removability of point singularities of Willmore surfaces

We consider the L2 gradient flow for the Willmore functional. In [5] it was proved that the curvature concentrates if a singularity develops. Here we show that a suitable blowup converges to a nonumbilic (compact or noncompact) Willmore surface. Furthermore, an L∞ estimate is derived for the tracefree part of the curvature of a Willmore surface, assuming that its L2 norm (the Willmore energy) is locally small. One consequence is that a properly immersed Willmore surface with restricted growth of the curvature at infinity and small total energy must be a plane or a sphere. Combining the results we obtain long time existence and convergence to a round sphere if the total energy is initially small.

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The Willmore Flow with Small Initial Energy

We consider curves in ${\mathbb R}^n$ moving by the gradient flow for elastic energy, i.e., the L2 integral of curvature. Long-time existence is proved in the two cases when a multiple of length is added to the energy or the length is fixed as a constraint. Along these lines, a lower bound for the lifespan of solutions to the curve diffusion flow is observed. We derive algorithms for both the elastic flows and the curve diffusion equation. After a numerical test we compute several examples, including cases of curve diffusion in which a singularity develops.

Evolution of Elastic Curves in $\Rn$: Existence and Computation

We consider two-dimensional, compact immersed surfaces in R moving by the gradient of the L integral of their curvature. It is not known whether solutions to this fourth-order, geometric evolution equation may develop singularities in finite time. We give a lower bound on the lifespan of a smooth solution, which depends only on how much the curvature of the initial surface is concentrated in space.

https://www.intlpress.com/site/pub/files/_fulltext/journals/cag/2002/0010/0002/CAG-2002-0010-0002-a004.pdf

Gradient flow for the Willmore functional

We study the Γ-convergence of functionals arising in the Van der Waals–Cahn–Hilliard theory of phase transitions. Their limit is given as the sum of the area and the Willmore functional. The problem under investigation was proposed as modification of a conjecture of De Giorgi and partial results were obtained by several authors. We prove here the modified conjecture in space dimensions n  =  2,3.

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Reiner Schätzle

Papers

Removability of point singularities of Willmore surfaces

The Willmore Flow with Small Initial Energy

Evolution of Elastic Curves in $\Rn$: Existence and Computation

Gradient flow for the Willmore functional

On a Modified Conjecture of De Giorgi