GLOBAL WEAK SOLUTIONS TO A CHEMOTAXIS-NAVIER-STOKES SYSTEM IN R3

The Cauchy problem in R-3 for the chemotaxis-Navier-Stokes system {n(t) + u . del n = Delta n - del . (n del c), c(t) + u . del c = Delta c - nc, u(t) + (u . del)u = Delta u + del P + n del phi, del . u = 0, is considered. Under suitable conditions on the initial data (n(0), c(0), u(0)), with regard to the crucial first component requiring that n(0) is an element of L-1 (R-3) be nonnegative and such that (n(0) + 1) ln(n(0) + 1) is an element of L-1 (R-3), a globally defined weak solution with (n, c, u)vertical bar(t=0) = (n(0), c(0), u(0)) is constructed. Apart from that, assuming that moreover integral(R3) n(0)(x) ln(1 + vertical bar x vertical bar(2))dx is finite, it is shown that a weak solution exists which enjoys further regularity features and preserves mass in an appropriate sense.