Global Very Weak Solutions to a Chemotaxis-Fluid System with Nonlinear Diffusion

We consider the chemotaxis-fluid system given by nₓ+u\! n= nᵐ-\! (n c), cₓ+u\! c= c-c+n, uₓ+ (u) u= u+ P+n, and u=0, for x and t>0, where ³ is a bounded domain with smooth boundary and m>1. Assuming m>43 and sufficiently regular nonnegative initial data, we ensure the existence of global solutions to the no-flux-Dirichlet boundary value problem for this system under a suitable notion of very weak solvability, which in different variations has been utilized in the literature before. Comparing this with known results for the fluid-free setting of the system above the condition appears to be optimal with respect to global existence. In the case of the stronger assumption m>53 we moreover establish the existence of at least one global weak solution in the standard sense. In our analysis we investigate a functional of the form _\! n^m-1+_\! c² to obtain a spatio-temporal L² estimate on n^m-1, which will be the starting point in deriving a series of compactness properties for a suitably regularized version of the system above. As the regularity information obtainable from these compactness results vary depending on the size of m, we will find that taking m>53 will yield sufficient regularity to pass to the limit in the integrals appearing in the weak formulation, while for m>43 we have to rely on milder regularity requirements making only very weak solutions attainable.