Large-data global generalized solutions in a chemotaxis system with tensor-valued sensitivities

TL;DR

This paper establishes the existence of global generalized solutions for a chemotaxis system with tensor-valued sensitivities in arbitrary dimensions, overcoming the lack of gradient structure through new a priori bounds and compactness arguments.

Contribution

It introduces a generalized solution concept for chemotaxis systems with tensor sensitivities and proves global existence in any dimension under mild conditions, extending previous 2D results.

Findings

Proved global existence of solutions for tensor-valued sensitivities.

Developed new a priori bounds based on gradient estimates.

Extended results to higher dimensions beyond 2D.

Abstract

A chemotaxis system possibly containing rotational components of the cross-diffusive flux is studied under no-flux boundary conditions in a bounded domain $\Omega\subset R^n$, $n\ge 1$, with smooth boundary, where the evolution of the signal is determined by consumption through cells. In contrast to related Keller-Segel-type problems with scalar sensitivities, in presence of such tensor-valued sensitivities this system in general apparently does not possess any useful gradient-like structure. Accordingly, its analysis needs to be based on new types of a priori bounds. Using a spatio-temporal $L^2$ estimate for the gradient of the logarithm of the cell density as a starting point, we derive a series of compactness properties of solutions to suitably regularized versions of the system. Motivated by these, we develop a generalized solution concept which requires solutions to satisfy very mild regularity hypotheses only. On the basis of the above compactness properties, it is finally shown that within this framework, under a mild growth assumption on the sensitivity matrix and for all sufficiently regular nonnegative initial data, the corresponding initial-boundary value problem possesses at least one global generalized solution. This extends known results which in the case of such general matrix-valued sensitivities provide statements on global existence only in the two-dimensional setting and under the additional restriction that the initial signal concentration be suitably small.