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Global small-data solutions of a two-dimensional chemotaxis system with rotational flux terms
https://doi.org/10.1142/S0218202515500177Cited by:90 (Source: Crossref)
Abstract
We study non-negative solutions to the chemotaxis system
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 
, 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.AMSC: 35L45, 35L60, 92B05, 92B99, 92C15, 92C17
