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Existence of weak solutions for the kinetic models of motion of myxobacteria with alignment and reversals

Patrick Murphy, Oleg Igoshin, Misha Perepelitsa, Ilya Timofeyev

TL;DR

This paper considers three non-linear kinetic partial differential equations that emerge in the modeling of motion of rod-shaped cells such as myxobacteria, characterized by nematic alignment with neighboring cells, orientation reversals from cell polarity switching, orientation diffusion, and transport driven by chemotaxis.

Abstract

In this paper, we consider three non-linear kinetic partial differential equations that emerge in the modeling of motion of rod-shaped cells such as myxobacteria. This motion is characterized by nematic alignment with neighboring cells, orientation reversals from cell polarity switching, orientation diffusion, and transport driven by chemotaxis. Our primary contribution lies in establishing the existence of weak solutions for these equations. Our analytical approach is based on the application of the classical averaging lemma from the kinetic theory, augmented by a novel version where the transport operator is substituted with a uni-directional diffusion operator.

Existence of weak solutions for the kinetic models of motion of myxobacteria with alignment and reversals

TL;DR

This paper considers three non-linear kinetic partial differential equations that emerge in the modeling of motion of rod-shaped cells such as myxobacteria, characterized by nematic alignment with neighboring cells, orientation reversals from cell polarity switching, orientation diffusion, and transport driven by chemotaxis.

Abstract

In this paper, we consider three non-linear kinetic partial differential equations that emerge in the modeling of motion of rod-shaped cells such as myxobacteria. This motion is characterized by nematic alignment with neighboring cells, orientation reversals from cell polarity switching, orientation diffusion, and transport driven by chemotaxis. Our primary contribution lies in establishing the existence of weak solutions for these equations. Our analytical approach is based on the application of the classical averaging lemma from the kinetic theory, augmented by a novel version where the transport operator is substituted with a uni-directional diffusion operator.
Paper Structure (7 sections, 4 theorems, 69 equations)

This paper contains 7 sections, 4 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Models of cell motion with nematic alignment and reversals
  3. Model I: liquid crystal-type alignment and reversals
  4. Model II: macroscopic diffusion limit
  5. Model III: uni-directional chemotaxis
  6. Existence theory: model I
  7. Existence theory: model II

Key Result

Theorem 1

Let $f_0\in L^2_{x,\theta}\cap L^1_{x,\theta},$ periodic in $(x,\theta),$$f_0\geq0$ a.e. and have unit mass. There is a unique, global in time, periodic, weak solution of eq:FK3. The solution is non-negative, has unit mass for all $t>0,$ and such that for any $T>0,$ and eq:energy1 holds.

Theorems & Definitions (10)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Lemma 1
  • Remark 1
  • proof
  • Theorem 3
  • Remark 2
  • proof