ABSENCE OF COLLAPSE INTO PERSISTENT DIRAC-TYPE SINGULARITIES IN A KELLER-SEGEL-NAVIER-STOKES SYSTEM INVOLVING LOCAL SENSING

摘要

The chemotaxis-Navier-Stokes system #xf8f1; #xf8f2; #xf8f3; nt + u BULL; Vn = increment (n phi;(c)), ct + u BULL; Vc = increment c c + n, ut + (u BULL; V)u = increment u + VP + nV phi;, V BULL; u = 0, is considered along with homogeneous boundary conditions of no-flux type for n and c, and of Dirichlet type for u, in a smoothly bounded planar domain 52, where phi; E W2, INFIN;(52) is given. A concept of generalized solvability is introduced, and in this framework a statement on global existence is derived for all reasonably regular initial data (n0, c0, u0), provided that phi; E C3((0, oo)) is positive and bounded on ( xi;0, oo) for each xi;0 > 0, and such that xi; phi;( xi;) -+oo and xi; phi; PRIME;2( xi;) phi;( xi;) -0 as xi; -oo. These solutions particularly satisfy fTf OHM; n ln(n + 1) 0 as well as f OHM; n( BULL;, t) = f 0 OHM; n0 for a.e. t > 0.