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

We study non-negative solutions to the chemotaxis system [Formula: see text] under no-flux boundary conditions in a bounded planar convex domain with smooth boundary, where f and S are given parameter functions on Ω × [0, ∞)2 with values in [0, ∞) and ℝ2×2, respectively, which are assumed to satisfy certain regularity assumptions and growth restrictions. Systems of type (⋆), in the special case [Formula: see text] 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 [Formula: see text], 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∞(Ω), 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 := fΩu0 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.