Existence of generalized solutions for Keller-Segel-Navier-Stokes equations with degradation in dimension three

Abstract: We construct generalized solutions for the Keller-Segel system with a degradation source coupled to Navier Stokes equations in three dimensions, in case that the power of degradation is smaller than quadratic. Furthermore, if the logistic type source is purely damping with no growing effect, we prove that solutions converge to zero in some norms and provide upper bounds of convergence rates in time.


Introduction
We consider a mathematical model to decribe the dynamics of biological organism influenced by chemical signal and living in fluid. The original Keller-Segel system was proposed to write the motion of biological individuals sensing gradient of a chemical substance and moving toward its higher concentration (see [9]). Such biological organisms often live in fluid, and thus their behaviors are influenced by motions of viscous fluid flows as well. There are, for example, the bacteria living in fluid such as Bacillus subtilus ( [1,2,7,11,18,24]) or Escherichia coli ( [12,22]) or phenomena of coral fertilization in sea resulting from the chemotatic behavior of sperm ( [4,6,10,24]).
(1. 5) In case that the Eq (1.1) has the logistic degradation, i.e., q = 2, Tao and Winkler [16] proved global existence and large time behavior of classical solutions to the system (1.1)-(1.3) in two dimensions. Such result was extended to the case of three dimensions, provided that the fluid equation is given by the Stokes system, instead of the Navier-Stokes equations, and µ is sufficiently large (see [15]). For the chemotaxis-Navier-Stokes system (1.1)-(1.3) with q = 2, the existence of generalized solutions was proved by Winkler [22].
To the best of our knowledge, if q < 2, it is not known whether or not classical solutions exist globally in time for general data and parameters. Instead of classical solutions, recently it was shown in [8] that generalized solutions to the chemotaxis-Stokes system exists globally in time for q ∈ (2 − 1 d , 2), where d is dimensions two or three, i.e., d = 2, 3. (the notion of generalized solutions is reminded in Definition 2). In the absence of fluid, i.e., u = 0, one can refer to [19,20,23] for generalized solutions.
The main objective of this note is to establish the existence of generalized solutions globally in time, in case that the degradation power q is less than 2, and the Navier-Stokes equations are coupled for the fluid equations in three dimensions.
For logistic coefficients ρ, µ and the potential function φ, we assume that ρ ∈ R, µ > 0 and φ ∈ C 1 (Ω). (1.8) We are now ready to state our main result.  [22], which showed the existence of the generalized solution in case that q = 2. Furthermore, it is also an extension to the result of [8], since the Navier-Stokes equations are considered instead the Stokes system. In such case, the range of q is, however, restrictive, compared to the case that q ∈ 5 3 , 2 in [8]. This is mainly due to the fact that the control of u is more difficult for the Navier-Stokes equations, which causes lower regularity of u · ∇c and, in turn, ∇c (see Lemma 3.6 for the details). Therefore, passing to the limit for regularized solutions, convergence to n∇c is well understood only for q ∈ 20 11 , 2 .
Next, in case that ρ 0, we can show that generalized solutions converge to zero in an appropriate sense, passing time to the limit. More precisely, we obtain the following: Theorem 1.2. Let (n, c, u) be the generalized solution established in Theorem 1.1. If ρ = 0, then (n, c, u) vanishes in L 1 (Ω) × L l (Ω) × L 2 (Ω) as time tends to infinity. Furthermore, (n, c, u) satisfies and Ω (c(·, t)) l dx Morerover, if ρ < 0, then (n, c, u) satisfies and C p is the Poincaré constant for Ω. Remark 2. The result of Theorem 1.2 can be extended to the case q = 2 and ρ = 0. In such case, in particular, estimates of c read as follows: This estimate of decay for c is slightly better, compared to those of [22,Section 8]. On the other hand, in case that q = 2 and ρ > 0, it was also shown in [22] that if µ > χ √ ρ/4, then This convergence is based on stabilization of a certain energy functional (see [22,Section 8]). Although similar results are expected, such a method doesn't seem to be valid unless q = 2. Therefore, we leave the asymptotic behaviors as an open question in case that ρ > 0 and q < 2.
This paper is organized as follows: In Section 2, we introduce an approximated system and recall some useful lemma for our purpose. Section 3 is devoted to obtaining estimates, independent of a regularizing parameter, of the approximated system.
We then discuss the convergence of approximated solutions to a generalized solution in Section 4. Finally, in Section 5, asymptotic estimates are provided.
Throughout this paper, we shall abbreviate f L p (Ω) as f p for simplicity. Further, we denote by C > 0 generic constants which may be different from line to line.

Preliminaries
In the following proposition we define an appropriate approximated system to (1.1)-(1.3), for which global classical solutions can be verified. The approximated system is given by Here ∈ (0, 1), κ > 2 and Y is the Yosida approximation defined by , where A is the realization of the stokes operator in D(A) = W 2,2 (Ω) ∩ W 1,2 0,σ (Ω) ⊂ L 2 σ (Ω). Following method of proofs developped in [8] and [22], one can prove the existence of classical solution of the approximated system (2.1). Since its verification is similar to thoes of [8] and [22], we skip its proof.
We recall an effective inequality in Sobolev spaces called the Gagliardo-Nirenberg interpolation inequality. Here we only consider a version of bounded Lipschitz domain Ω in R 3 . The proof can be found in [3, Theorem 1.5.2] and [13].
Lemma 2.1. Let 1 p, r ∞ and 0 n < m ∈ N. Then there exist constants C 1 and C 2 > 0 such that 1], and s > 0 is arbitrary. The following two Lemmas named maximal estimates are crutial to obtain a regularity of approximated solutions (see [5,8,14]).
Furthermore, v attains the following estimate.
. Then for every v 0 ∈ W 1,∞ (Ω) and h ∈ L p (Ω × (0, T )), the following heat equation with Neumann boundary condition [14] and [19, where · 1− 1 p ,p stands for the norm in the real interpolation space (X, Remark 3. For the purpose of our analysis, we consider only the case p ∈ (1, 2] in Lemma 2.3. One can refer to [21] for more general cases, in particular p 3, where the interpolation space (X, Next, we present a compactness theorem called Aubin-Lions Lemma [17, Theorem 2.1] that will be used to give convergence results for the approximated solution (n , c , u ).
Lemma 2.4. Let T > 0, 1 α 0 , α 1 < ∞ and X 0 , X, X 1 be Banach spaces with X 0 ⊂ X ⊂ X 1 . Suppose further that the embedding X 0 → X is compact and the embedding X → X 1 is continuous. Let Then the embedding W → L α 0 (0, T ; X) is compact.

Regularized solutions
The following basic properties of these solutions are well-known. and Proof. Integrating the first equation in (2.1) over Ω, employing the divergence theorem, and using the Hölder inequality yield that, for all t > 0, An ODE comparison implies (3.1). Integrating (3.3) with respect to time and then using (3.1), we have The following estimate is easily obtained by (3.1). We recall a useful result shown in [22,Lemma 3.4].
The following lemma is a variant of the result with q = 2 in [22, Lemma 3.6].
Lemma 3.4. Let T > 0 and q ∈ ( 5 3 , 2). Then there exists C > 0 such that for any ∈ (0, 1) we obtain where r = 3q 5−2q . Proof. Multiplying the equation for c in (2.1) by c r−1 and integrating over Ω, we have for all t > 0, where the Hölder inequality is used. Using the Gagliardo-Nirenberg inequality and (3.4), we note that Let y(t) := c (t) r r + 1 and h(t) := n (t) q q + 1, which is in L 1 locally in time. Then, dividing (3.9) by y(t) yields that d dt We use again the Gagliardo-Nirenberg inequality to obtain that for all t > 0 which leads that ∇c where we use the trivial inequality ln y y k for k > 0. Putting the above inequality (3.11) into (3.10), we have d dt ln y + C ln y h.
By Lemma 3.3, we can conclude that there exists C > 0 satisfying y(t) C for all t > 0 which proves (3.5) as required. Integrating (3.10) with respect to time and exploiting the boundedness of y(t), guaranteed by (3.5), yield that We adopt well-known energy estimate for the Navier-Stokes system to gain a bound for u in energy class. We can estimate the right hand side of (3.14) using the Hölder inequality, the Sobolev embedding W 1,2 0,σ → L 6 , and the interpolation inequality for n that Ω n u ∇φ C n 6 5 u 6 C n 2 6 5 for all t > 0, (3.15) where we used that q q. Thus, with the aid of (3.15) and the Poincaré inequality, we have for some C d dt (3.12) is proved if we use (3.2) and Lemma 3.3, and then (3.13) can be calculated by integrating (3.14) with respect to time and using (3.15).
A direct consequence of Lemma 3.5 is the following. Since u only belong to energy class, we have lower regularity of ∇c , due to difficulties of controlling convective term u · ∇c, than the case that the Stokes sysem is coupled. Nevertheless, using the divergence free condtion, we obtain a certain integrability of ∇c by the following decompsition, which makes computations easier. More precisely, let w be a solution satisfying Now we setw := c − w . Then, due to the divergence free condition for u , it follows thatw solves w (x, 0) = 0, x ∈ Ω.
The estimate for the time derivative of u is obtained by the simple calculation.

Convergence
We are now ready to prove the convergence property for (n , c , u ). u → u a.e. in Ω × (0, ∞) , which asserts that n → n in L p loc (Ω × [0, ∞)) due to uniform convexity of L p -space for p > 1. This proves (4.3).
Proof of Theorem 1.1. This is the combination of Lemma 4.2, Lemma 4.3 and Lemma 4.4.

Asymptotic behavior
The following Lemma is elementary, but for clarity, we give its detail.
Lemma 5.1. Let a > 1 and f ∈ L 1 ([0, ∞)). Suppose there is t 0 > 0 such that f (t) Nt −a for sufficiently large t t 0 . Assume further that a non-negative measurable function y(t) satisfies Then, y(t) Ct −a for sufficiently large t.
Proof. Firstly we note that y(t) is bounded uniformly in time. Then, using the integrating factor, we have for t t 0 e 2t y(2t) − e t y(t) 2t t e τ f (τ) dτ, which yields, using integration by parts, 2t t e τ τ −a dτ Proof of Theorem 1.2. • (The case ρ = 0) Noting that ρ = 0, we integrate the equation for n in (2.1) over Ω to get for all t > 0.