Linear stability of blowup solution of incompressible Keller–Segel–Navier–Stokes system

In this paper, we consider the linear stability of blowup solution for incompressible Keller–Segel–Navier–Stokes system in whole space R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}^{3}$\end{document}. More precisely, we show that, if the initial data of the three dimensional Keller–Segel–Navier–Stokes system is close to the smooth initial function (0,0,us(0,x))T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(0,0,\textbf{u}_{s}(0,x) )^{T}$\end{document}, then there exists a blowup solution of the three dimensional linear Keller–Segel–Navier–Stokes system satisfying the decomposition (n(t,x),c(t,x),u(t,x))T=(0,0,us(t,x))T+O(ε),∀(t,x)∈(0,T∗)×R3,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \bigl(n(t,x),c(t,x),\textbf{u}(t,x) \bigr)^{T}= \bigl(0,0, \textbf{u}_{s}(t,x) \bigr)^{T}+\mathcal{O}(\varepsilon ), \quad \forall (t,x)\in \bigl(0,T^{*}\bigr) \times \mathbb{R}^{3}, $$\end{document} in Sobolev space Hs(R3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H^{s}(\mathbb{R}^{3})$\end{document} with s=32−5a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$s=\frac{3}{2}-5a$\end{document} and constant 0<a≪1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0< a\ll 1$\end{document}, where T∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T^{*}$\end{document} is the maximal existence time, and us(t,x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\textbf{u}_{s}(t,x)$\end{document} given in (Yan 2018) is the explicit blowup solution admitted smooth initial data for three dimensional incompressible Navier–Stokes equations.


Introduction and main results
The Keller-Segel system coupled to the incompressible Navier-Stokes equations arises from a biological process in which cells move towards a chemically more favorable environment [20]. In this paper, we consider the blowup analysis for the three dimensional Keller-Segel-Navier-Stokes system ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ n t + u · ∇n = n -χ∇ · (n∇c) + σ nμn 2 , c t + u · ∇c = cc + n, u t + u · ∇u = u -∇P + n∇ , ∇ · u = 0, (1.1) where ∀(t, x) ∈ R + × R 3 , σ ∈ R and μ > 0 are given parameters, the constant χ is the chemotactic sensitivity. We assume χ > 0 presuming that cells move toward increasing concentrations of the signal substance which is produced by themselves. The unknown scalar functions n(t, x) : R + × R 3 → R and c(t, x) : R + × R 3 → R denote the population density and the signal concentration, respectively, the vector function u(t, x) : R + × R 3 → R 3 denotes the 3 D velocity field of the fluid, P(t, x) : R + × R 3 → R stands for the pressure in the fluid. Moreover, the pressure P(t, x) is determined by the formula (1.2) (x) is a given potential function accounting for the effects of external forces such as gravity. The divergence free condition in the last equation of (1.1) guarantees the incompressibility of the fluid.
Chemotaxis, as a mechanism of the partially-oriented movement of cells in response to a chemical signal, plays an important role in various biological process [3,11]. Tuval et al. [26] observed large-scale convection patterns in a water drop sitting on a glass surface containing oxygen-sensitive bacteria, oxygen diffusing into the drop through the fluid-air interface. They established a mathematical model, the so-called chemotaxis-fluid system, to describe the dynamics of swimming bacteria, Bacillus subtilis. It has the form ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ n t + u · ∇n = n -∇ · (nχ(c)∇c), c t + u · ∇c = cnκ(c), u t + κ(u · ∇u) = u -∇P + n∇ , where D is a smooth bounded domain or R 3 . Lorz [19] first showed the local existence of weak solutions for the two and three dimensional system (1.1) with nonflux and inhomogeneous Dirichlet boundary conditions, respectively. Chae-Kang-Lee [6] gave the local existence of the smooth solution in the cases of two and three dimensions and global existence of classical solution for the two dimensional case for system (1.1). Winkler [31] asserted that a solution of the two dimensional chemotaxis-Navier-Stokes system in a bounded convex domain stabilizes to the spatially uniform equilibrium ( 1 | | n 0 (x)dx, 0, 0) T in L ∞ ( ). Zhang-Zheng [40] used a microlocal analysis to obtain the global existence and uniqueness of weak solutions for a two dimensional chemotaxis-Navier-Stokes system in R 3 for a large class of initial data. One can also refer to [7,18,28,29] for more results on this two dimensional model. For the spatially three dimensional case, Tao-Winkler [25] constructed locally bounded global solutions in a chemotaxis-Stokes system with nonlinear diffusion. Winkler [33] established the global weak existence theory to a chemotaxis-Navier-Stokes system in a smooth bounded convex domain under a homogeneous Neumann boundary condition. With some relaxation time, Winkler [34] has shown this model admitted eventual energy solutions, meanwhile, he got such eventual energy solutions (n, c, u) → (n 0 , 0, 0) uniformly in a smooth bounded convex domain after the waiting time. Due to spatially limitation, we do not list all of the interesting papers on this kind of models.
Since the Keller-Segel-Navier-Stokes system plays an important role in bioconvection processes, it has attracted great interest also at the level of mathematical theory [2,3].
Wang [27] showed the global existence of weak solutions for a three dimensional Keller-Segel-Navier-Stokes system with subcritical sensitivity. Winkler [35] proved that the three dimensional model with logistic source admits global weak solutions and asymptotic stabilization in a smooth bounded convex domain with homogeneous Dirichlet boundary conditions. At present, there are extreme difficulties to study the properties of solutions for system (1.1). One of the reasons is the question of finite time singularity/global regularity for three dimensional incompressible Navier-Stokes equations, one of the most important open problems in mathematical fluid mechanics [9]. On the one hand, even if we get rid of the effect of a fluid (u ≡ 0), the question whether blowup may occur in the three dimensional Keller-Segel system with small positive constant μ is still open [35]. Here is a smooth bounded domain in R 3 . Lankeit [15] got the global existence of weak solutions for system (1.3). Winkler [30] proved that system (1.3) can lead to a finite time explosion even for the case σ = μ = 0.
One can refer to [12][13][14]32] for more results on this kind of models.
On the other hand, if we set n(t, x) = c(t, x) ≡ 0, then the Keller-Segel-Navier-Stokes system (1.1) is reduced to the 3D incompressible Navier-Stokes equations For these equations, Leray [16] showed there is global-forward-in-time weak solution of the initial value problem. After that, non-existence of finite energy self-similar blowup solutions in L 3 (R 3 ) was obtained in [22]. As is well known, Eq. (1.4) admits a simplest nontrivial stationary solution with infinite energy, i.e. the Couette flow (y, 0, 0) T . Bedrossian-Germain-Masmoudi [1] proved the nonlinear stability of this flow. One can refer to [4,5,8,10,17,21,24,39] for more related results. Yan [36,37] found that the three dimensional Navier-Stokes equation (1.4) admits a family of stable explicit blowup solutions with infinite energy, with the smooth initial data where we have the constants a, k ∈ R/{0}, and the positive constant T * is the maximal existence time. Moreover, the pressure P is determined by the formula We supplement the 3 D incompressible Keller-Segel-Navier-Stokes equations (1.1) with the initial data (1.6) We now state the main result.
Throughout this paper, we denote the usual norm of L 2 (R 3 ) and Sobolev space H s (R 3 ) by · L 2 and · H s for s ∈ R + , respectively. In fact, the Sobolev space can be equivalent to be defined via Fourier transformation, thus the fractional case is contained. The norm of is equipped with the norm We also introduce the function spaces with the norm respectively. The symbol a b means that there exists a positive constant C such that a ≤ Cb. (a, b, c) T denotes the column vector in R 3 . The letter C with subscripts to denote dependencies stands for a positive constant that might change its value at each occurrence.

Proof of Theorem 1.1
The linear stability of a blowup solution is equivalent to the well-posedness of linearized equations in R 3 . For any (t, x) ∈ (0, T * ) × R 3 , we recall the perturbation equations and the boundary condition where (t, x) ∈ (0, T * ) × R 3 , the pressure P satisfies and we have Let R ∈ (0, 1) be a fixed constant. We define with a constant s > 0. Assume that fixed functions (n(t, x), c(t, x), v(t, x)) T ∈ B R . We linearize the nonlinear equations (2.1) around fixed functions (n(t, x), c(t, x), v(t, x)) T to get the linearized equations on the unknown variables ( (t, x), (t, x), h(t, x)) T with an external force x)) T as follows: In order to get some suitable prior estimates, we rewrite the linearized equations (2.5) as a coupled system, with the incompressibility condition We introduce the similarity coordinates The linearized coupled system (2.6)-(2.10) under these coordinates is transformed into

16)
with the incompressibility condition We supplement the linearized system (2.13)-(2.17) with the initial data 19) and the boundary condition (2.20) We first derive prior estimates of the linearized coupled system (2.13)-(2.17) with the initial data (2.19) and condition (2.20).
Proof Multiplying both sides of (2.13)-(2.17) by , , h 1 , h 2 and h 3 , respectively, then integrating by parts (using the boundary condition (2.20)), we have (2.25) Summing up (2.21)-(2.25), then We now estimate each coupled nonlinear term in (2.26). Note that (n, c, v) T ∈ B R and . We use Young's inequality to derive where C R , C a,R , C κ,R are three positive constants depending on R, a, R and κ, R, respectively.
On the other hand, by (2.18) and the standard Calderon-Zygmund theory, i.e. for the Riesz operator R, we have Rw L p ≤ w L p with 1 < p < ∞, we also use Young's inequality to get and where C R is a positive constant depending on R, which can be very small if constants R is small. There exists a sufficiently small positive constant b ∈ (0, 1) such that Hence, applying Gronwall's inequality to (2.31), there exists a positive constant C such In what follows, we plan to carry out higher order derivative estimates to the solutions of linearized system (2.8)-(2.10). For a fixed constant s > 0, applying ∇ s = ∂ s y i to both sides of (2.22)-(2.10), we obtain We now have the following higher order derivatives estimates.

Lemma 2.2
Let 0 < a 1 8 and 0 < s < 3 2 -5a be constants. Assume that where C R,T * is a positive constant depending on constants R, T * .
Proof Taking the inner product of both sides of (2.32)-(2.36) by ∇ s y , ∇ s y , ∇ s y h 1 , ∇ s y h 2 and ∇ s y h 3 , respectively, then integrating by parts, we have We now estimate each nonlinear term in (2.45). On the one hand, note that (n, c, v) T ∈ B R . We employ Young's inequality, H 5 2 (R 3 ) ⊂ L ∞ (R 3 ) and integrating by parts to derive where the C R are a positive constants depending on R, which are small constants as R is small.
On the other hand, by (2.18), we know the highest order derivatives on h i of ∂ y 1 ∇ s y f is s. So we can use the standard Calderon-Zygmund theory, Young's inequality and integrating by parts to derive where C R,T * is a positive constant depending on R, T * , which is a small constant as R small.