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