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

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.

💡 Research Summary

This paper establishes the global existence of weak solutions for a class of quasilinear Keller–Segel systems in which the chemotactic mobility is nonlinear, namely (m(u)=u^{\alpha}) with (0<\alpha<1). The authors adopt the minimizing‑movement (JKO) scheme, but instead of the classical Wasserstein‑2 distance they work in a weighted Wasserstein space (W_{m,\Omega}) that incorporates the mobility function as a weight. This choice overcomes the difficulty that the mobility is not Lipschitz and becomes singular at zero, which prevents the direct use of the standard Wasserstein framework.

The paper proceeds as follows. First, the authors recall the weighted Wasserstein distance introduced in previous work and collect its key properties: lower semicontinuity, a continuity‑equation formulation, and a suitable action functional. They then introduce a regularisation of the mobility, (m_{\varepsilon}(r)=(r+\varepsilon)^{\alpha}), which is smooth and Lipschitz. For each (\varepsilon>0) they consider the regularised Keller–Segel system and formulate the JKO minimisation problem \