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

We study non-negative solutions to the chemotaxis system {equation presented} under no-flux boundary conditions in a bounded planar convex domain with smooth boundary, where f and S are given parameter functions on × [0,)² with values in [0,) and R^2×2, respectively, which are assumed to satisfy certain regularity assumptions and growth restrictions. Systems of type (∗), in the special case {equation presented} 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{equation presented}, reflecting rotational chemotactic motion. It is shown that for any choice of suitably regular initial data (u₀, v₀) fulfilling a smallness condition on the norm of v₀ in L(), the corresponding initial-boundary value problem associated with (∗) 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) (μ, κ) in the large time limit, with μ:= fu₀ and some κ ≥ 0. A mild additional assumption on the positivity of f is shown to guarantee that κ = 0. Finally, numerical solutions are presented which suggest the occurrence of wave-like solution behavior.