Global small-data solutions of a two-dimensional chemotaxis system with rotational flux terms

We study non-negative solutions to the chemotaxis system (u(t) = Delta u - del. (uS(x, u, v)del v), x is an element of Omega, t > 0, v(t) = Delta v - f(x, u, v), x is an element of Omega, t > 0, (star) under no-flux boundary conditions in a bounded planar convex domain with smooth boundary, where f and S are given parameter functions on Omega x [0, infinity)(2) with values in [0, infinity) and R-2x2, respectively, which are assumed to satisfy certain regularity assumptions and growth restrictions. Systems of type (star), in the special case S equivalent to (1/0 0/1) reducing to a version of the standard Keller-Segel system with signal consumption, have recently been proposed as a model for swimming bacteria near a surface, with the sensitivity tensor then given by S equivalent to (0/-1 1/0), reflecting rotational chemotactic motion. It is shown that for any choice of suitably regular initial data (u(0), v(0)) fulfilling a smallness condition on the norm of v(0) in L-infinity(Omega), the corresponding initial-boundary value problem associated with (star) possesses a globally defined classical solution which is bounded. This result is achieved through the derivation of a series of a priori estimates involving an interpolation inequality of Gagliardo-Nirenberg type which appears to be new in this context. It is next proved that all corresponding solutions approach a spatially homogeneous steady state of the form (u, v) = (mu, kappa) in the large time limit, with mu:= f(Omega)u(0) and some kappa >= 0. A mild additional assumption on the positivity of f is shown to guarantee that kappa = 0. Finally, numerical solutions are presented which suggest the occurrence of wave-like solution behavior.