Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Minimizing movements for quasilinear Keller--Segel systems with nonlinear mobility in weighted Wasserstein metrics

Kyogo Murai

TL;DR

This work develops a gradient-flow framework for a quasilinear Keller–Segel system with nonlinear mobility by formulating the problem as a minimizing-movement (JKO) scheme in a product of weighted Wasserstein space and $L^2$. The mobility nonlinearity is handled via an $\varepsilon$-regularization $m_\varepsilon(r)=(r+\varepsilon)^\alpha$ and a careful flow-interchange analysis to derive discrete Euler–Lagrange relations. Uniform a priori estimates and compactness arguments yield global weak solutions for $p\ge 1+\alpha-2/d$, including the subcritical regime and the critical case with small chemotactic sensitivity $\chi$. The paper also extends the existence theory to the broader range $1+\alpha-2/d<p<1+\alpha$ and to the critical exponent with small $\chi$, providing a robust variational approach to nonlinear mobility Keller–Segel systems and their gradient-flow structure.

Abstract

We prove the global existence of weak solutions to quasilinear Keller--Segel systems with nonlinear mobility by minimizing movements (JKO scheme) in the product space of the weighted Wasserstein space and $L^2$ space. While minimizing movements for Keller--Segel systems with linear mobility are adapted in the product space of the Wasserstein space and $L^2$ space, due to the nonlinearity of mobility, we need to use the weighted Wasserstein space instead of the Wasserstein space. Moreover, since the mobility function is not Lipschitz, we first find solutions to the Keller--Segel systems whose mobility is apporoximated by a Lipschitz function, and then we establish additional uniform estimates and convergences to derive solutions to the Keller--Segel systems.

Minimizing movements for quasilinear Keller--Segel systems with nonlinear mobility in weighted Wasserstein metrics

TL;DR

This work develops a gradient-flow framework for a quasilinear Keller–Segel system with nonlinear mobility by formulating the problem as a minimizing-movement (JKO) scheme in a product of weighted Wasserstein space and . The mobility nonlinearity is handled via an -regularization and a careful flow-interchange analysis to derive discrete Euler–Lagrange relations. Uniform a priori estimates and compactness arguments yield global weak solutions for , including the subcritical regime and the critical case with small chemotactic sensitivity . The paper also extends the existence theory to the broader range and to the critical exponent with small , providing a robust variational approach to nonlinear mobility Keller–Segel systems and their gradient-flow structure.

Abstract

We prove the global existence of weak solutions to quasilinear Keller--Segel systems with nonlinear mobility by minimizing movements (JKO scheme) in the product space of the weighted Wasserstein space and space. While minimizing movements for Keller--Segel systems with linear mobility are adapted in the product space of the Wasserstein space and space, due to the nonlinearity of mobility, we need to use the weighted Wasserstein space instead of the Wasserstein space. Moreover, since the mobility function is not Lipschitz, we first find solutions to the Keller--Segel systems whose mobility is apporoximated by a Lipschitz function, and then we establish additional uniform estimates and convergences to derive solutions to the Keller--Segel systems.
Paper Structure (17 sections, 39 theorems, 301 equations)

This paper contains 17 sections, 39 theorems, 301 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Notations
  4. Weighted Wasserstein distance
  5. Properties of the functional $\boldsymbol{U}_\varepsilon$ and a boundary estimate
  6. Existence of minimizers
  7. Boundedness from below of the functional $F_{\tau}$
  8. Compactness
  9. Lower semicontinuity
  10. Conclusion
  11. Euler--Lagrange equations
  12. Flow interchange lemma
  13. A discrete type of weak formulations
  14. Uniform estimates and convergences
  15. Proof of Theorem 1.1
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $\alpha \in (0,1),\ p=1+\alpha,\ \chi>0,\ d\geq2$ and $(u_0,v_0) \in X$ be a pair of nonnegative functions. Then for all $T>0$, there exists a nonnegative weak solution $(u,v)$ to peks on the time interval $[0,T]$ satisfying and for all $\varphi \in C^\infty(\overline{\Omega})$ with $\nabla \varphi\cdot\boldsymbol{n} = 0$ on $\partial\Omega$ and $\zeta \in H^1(\Omega)$.

Theorems & Definitions (80)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Definition 2.1
  • Definition 2.2: DNS, The action functional
  • Definition 2.3: DNS, Weighted Wasserstein distance
  • ...and 70 more