GLOBAL VERY WEAK SOLUTIONS TO A CHEMOTAXIS-FLUID SYSTEM WITH NONLINEAR DIFFUSION

We consider the chemotaxis-fluid system given by n(t) + u.del n - Delta n(m) - del.(n del c), c(t) + u.del c = Delta c - c + n, u(t) + (u.del) u =Delta u + del P + n del phi, and del.u = 0, for x is an element of Omega and t > 0, where subset of Omega R-3 is a bounded domain with smooth boundary and m > 1. Assuming m > 4/3 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 > 5/3 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 integral(Omega)n(m-1) + integral Omega c(2) to obtain a spatio-temporal L-2 estimate on del 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 > 5/3 will yield sufficient regularity to pass to the limit in the integrals appearing in the weak formulation, while for m > 4/3 we have to rely on milder regularity requirements making only very weak solutions attainable.