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

TL;DR

This paper establishes the global existence of weak solutions for quasilinear Keller--Segel systems with nonlinear mobility using a novel minimizing movement approach in weighted Wasserstein spaces, addressing degenerate diffusion and sub-linear sensitivity.

Contribution

It introduces a new method employing weighted Wasserstein spaces for systems with nonlinear mobility, extending the existence theory to critical cases with degenerate diffusion.

Findings

Proved global existence of weak solutions for systems with nonlinear mobility.

Developed a new minimizing movement scheme in weighted Wasserstein spaces.

Established convergence from Lipschitz approximations of mobility functions.

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. In particular, we newly show the global existence of weak solutions to the Keller--Segel system with the degenerate diffusion and the sub-linear sensitivity in the critical case. The advantage of our approach is that we can connect the global existence of weak solutions to the Keller--Segel systems with the boundedness from below of a suitable functional. 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 approximated by a Lipschitz function, and then we establish additional uniform estimates and convergences to derive solutions to the Keller--Segel systems.