Global Strong Solutions to a Coupled Chemotaxis-Fluid Model with Subcritical Sensitivity
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Global Strong Solutions to a Coupled Chemotaxis-Fluid Model with Subcritical Sensitivity Jishan Fan1 · Fucai Li2
Received: 4 July 2019 / Accepted: 25 February 2020 © Springer Nature B.V. 2020
Abstract In this paper we prove the global existence of strong solutions to a coupled chemotaxis-fluid model with subcritical sensitivity in a bounded domain Ω ⊂ R2 without small assumptions on initial data. Keywords Chemotaxis · Navier-Stokes system · Strong solutions Mathematics Subject Classification 92C17 · 35B65 · 35Q35
1 Introduction In this paper we first consider the following chemotaxis-fluid model, which is a coupling of Keller-Segel system and incompressible Navier-Stokes system, with subcritical sensitivity in a bounded domain Ω ⊂ R2 (see [6, 14, 19]): n , p, q · ∇p ∂t n + u · ∇n − n = −∇ · nS x, 1 + n n − ∇ · nS x, , p, q · ∇q + ∇ · (n∇φ), (1.1) 1 + n ∂t p + u · ∇p − p = −np,
(1.2)
∂t q + u · ∇q − q + q = n,
(1.3)
∂t u + u · ∇u − u + ∇π = n∇φ + nS(x, n, p, q) · ∇p + nS(x, n, p, q) · ∇q,
(1.4)
div u = 0,
(1.5)
B F. Li
[email protected] J. Fan [email protected]
1
Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, P.R. China
2
Department of Mathematics, Nanjing University, Nanjing 210093, P.R. China
J. Fan, F. Li
with initial and boundary conditions: ∂n ∂p ∂q = = = 0, ∂ν ∂ν ∂ν
u=0
on ∂Ω × (0, ∞),
(n, p, q, u)(·, 0) = (n0 , p0 , q0 , u0 )(·)
in Ω ⊂ R2 ,
(1.6) (1.7)
where n, p and q denote the density of amoebae, oxygen and chemical attractant, respectively; u is the velocity of the fluid, and π is the pressure; φ = φ(x) denotes a potential which is a smooth function; and S = S(x, n, p, q) respects the chemotactic sensitivity. 0 < < 1 is a positive constant. Ω ⊂ R2 is a bounded convex domain with smooth boundary ∂Ω, and ν is the unit outward normal vector to ∂Ω. For more details on the background and derivation of (1.1)–(1.5), see [2, 6, 13, 14, 19]. When u ≡ 0, the system (1.1)–(1.5) is reduced to the Keller-Segel type system and there are many studies on it in different situations, among others, we mention [3, 5, 8–12, 16, 20, 22]. The are some results on the special case of the system (1.1)–(1.5). Kozono-MiuraSugiyama [14] proved the existence and uniqueness of global small mild solutions to the system (1.1)–(1.5) in the whole space RN (N ≥ 2) with S = 0 and φ = 0 in Lp setting. Di Francesco-Lorz-Markowich [6] studied the full chemotaxis-fluid coupled system (q = 0) with degenerate nonlinear diffusion and obtain the global solution for general initial data and the asymptotic behavior of the solution in two dimension case. Fan-Zhao [7] established some regularity criteria for the system (1.1)–(1.5) when q = 0. Very recently, WangWinkler-Xiang [19] showed the global existence of strong solutions to the system (1.1)–(1.5) with p = 0 and without the last terms on the right hand side of (1.1) and (1.4). For more results on the case p = 0, see [4, 17, 18, 21] and the references cited therein. The first aim of this paper is to generalize the res
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