GLOBAL WEAK SOLUTIONS TO A CHEMOTAXIS-NAVIER-STOKES SYSTEM IN R-3

초록

The Cauchy problem in R3 for the chemotaxis-Navier-Stokes system {n(t) + u. & nabla; n = delta n - & nabla; .(n & nabla; c), ct + u.& nabla; c = delta c - nc, u(t) + (u.& nabla; )u = delta u + & nabla; P + n & nabla; phi, & nabla; . 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)|(t=0) = (n(0), c(0), u(0)) is constructed. Apart from that, assuming that moreover integral(R3) n(0)(x) ln(1 + |x|(2))dx is finite, it is shown that a weak solution exists which enjoys further regularity features and preserves mass in an sense.

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