Large time periodic solution to the coupled chemotaxis‐Stokes model

Abstract

In this paper, we deal with the time periodic problem for the coupled chemotaxis‐fluid model with logistic growth term. We prove the existence of large time periodic solution in spatial dimension N=3. Furthermore, we also show that if the time periodic source g and the potential force ∇φ belong to Cα,α2false(Ω¯×double-struckRfalse), the solution is also a classical solution.

Periodic pattern formation in the coupled chemotaxis-(Navier–)Stokes system with mixed nonhomogeneous boundary conditions

“…Similar to the proof of [9], we have n ∈ C β,β/2 (Ω × R + ). Recalling (3.1), by the standard parabolic regularity theory, we successively obtainc ∈ C 2+β,1+β/2 (Ω × R + ), u ∈ C 2+β,1+β/2 (Ω × R + ), n ∈ C 2+β,1+β/2 (Ω × R + ).…”

“…Combining with (4.2), and we have u ∈ C β,β/2 (Ω × R + ) for some β ∈ (0, α], see for example [9]. To use the Neumann heat semigroup theory for the homogeneous Neumann boundary problem, we letñ = ne −χce g 1 , then ∂ñ/∂ν| ∂Ω = 0, and we havẽ…”

“…While, the study for the 3-D case is much more difficult. In 2010, Di Francesco [3] obtained the existence of a global bounded weak solution for m ∈ ((7 + √ 217)/12, 2]; a locally bounded global weak solution was then obtained for m ∈ ( 8 7 , +∞) in 2013 [23]; the uniform boundedness of solutions was subsequently supplemented for m ∈ ( 7 6 , +∞) [29]; further extension was made by Winkler for m > 9 8 in [31] to a convex domain; recently, we also [10] improved the results to the case m > 11 4 − √ 3 (approximating to 56 55 ). However, if a logistic term reflecting the cell proliferation is added to this model, Jin [7] established the existence of global bounded weak solutions for any m > 1 to the fluid-free case in dimension 3.…”

Time periodic solutions for a two-species chemotaxis-Navier-Stokes system

“…where Q = Ω × R + , Ω ⊂ R N is a bounded domain. Jin employed a fix-point method to obtain that when N = 2, the problem (1.3) admits a time periodic solution (see also [4]).…”

“…It follows from (3.16)- (3.19) and the proof of Lemma 2.3-Lemma 2.5 that (3.13)-(3.15) hold. 4. The existence of the periodic solution.…”