Global Existence of Chemotaxis-Navier–Stokes System with Logistic Source on the Whole Space R 2

: In this article, we study the Cauchy problem of the chemotaxis-Navier–Stokes system with the consumption and production of chemosignals with a logistic source. The parameters χ ̸ = 0, ξ ̸ = 0, λ > 0 and µ > 0. The system is a model that involves double chemosignals; one is an attractant consumed by the cells themselves, and the other is an attractant or a repellent produced by the cells themselves. We prove the global-in-time existence and uniqueness of the weak solution to the system for a large class of initial data on the whole space R 2 .


Introduction
The present paper is concerned with the following chemotaxis-Navier-Stokes system with the consumption and production of chemosignals with logistic source where x ∈ R 2 , t > 0. The terms n = n(x, t), c = c(x, t), v = v(x, t), u = u(x, t) = (u 1 (x, t), u 2 (x, t)) and P denote the unknown density of amoebae, the unknown oxygen concentration, the unknown concentration of the chemical attractant, and the unknown fluid velocity field and the unknown pressure, respectively.The parameters χ ̸ = 0, ξ ̸ = 0 and κ ≥ 0. The terms λ ≥ 0 and µ ≥ 0 reflect the rate of reproduction and death, respectively.We impose the following intial data (u(x, 0), n(x, 0), v(x, 0), c(x, 0)) = (u 0 , n 0 , v 0 , c 0 ).
The time-independent function ϕ = ϕ(x) denotes the potential function produced by different physical mechanisms, e.g., the gravitational force or centrifugal force.
In some biological processes, chemotactic cells often interact with multiple chemotactic cues, either of which may be an attractant or a repellant, to produce variety of intricate patterns.It was pointed out in [1][2][3][4] that this phenomenon is widely present in many prototypical biological situations.As compared to the chemotaxis-fluid models involving just one chemical signal that is consumed or produced by the species themselves as mentioned above, chemotaxis-fluid models incorporating at least two different chemical signals seem much less understood.When χ > 0, ξ < 0, λ = µ = 0 and κ = 1, the system (1) becomes an attraction-attraction Navier-Stokes system, and the corresponding Cauchy problem admits global mild solutions with small initial data in some scaling invariant space [5].When χ > 0, ξ > 0 and λ = κ = µ = 0, the system (1) becomes an attraction-repulsion-Stokes system; the global bounded classical solution and the large time behavior of the solution have been established in a smoothly bounded planar domain [6,7].When κ = 1 and χ > 0, ξ > 0, λ = µ = 0, in [8] the corresponding attraction-repulsion Navier-Stokes system is proved to possess a unique global classical solution; however, the uniform boundedness and large time behavior of the solutions to this attraction-repulsion Navier-Stokes system can be achieved simultaneously in [9].We refer to [10][11][12][13][14][15][16][17][18][19][20][21][22] for more details concerning some properties of chemotaxis-fluid models.
In this work, we shall focus on the Cauchy problem (1) with the logistic term in two dimensional.Here, the parameters χ ̸ = 0, ξ ̸ = 0, λ > 0, µ > 0 and κ = 1.Precisely, we shall consider the global-in-time existence and uniqueness of the weak solution to the system for a large class of initial data on the whole space R 2 .
We first state the assumptions on the initial data and introduce the following notation: Now, we state our main theorem.
Then, the solution of (1) possesses a unique global-in-time weak solution satisfying We mention that our results may be generalized to a bounded domain by slightly modifying our proof and adding reasonable boundary conditions.Compared with [5,9], we can obtain the existence and uniqueness of weak solutions to (1) in the whole space R 2 .
Notation.We will set ∂ t = ∂ ∂t and ∂ i = ∂ ∂x i for i = 1, 2 and denote all the partial derivatives ∂ β with multi-index β satisfying |β| = k by ∇ k (k ≥ 0).We adopt the convention that the nonessential constant C may change from line to line, and C(a 1 , a 2 , ..., a k ) means a constant C depending on a 1 , a 2 , ..., a k .Given two quantities A and B, we denote B ≲ A as B ≤ CA.We often label ∥(a, b)∥ X = ∥a∥ X + ∥b∥ X .

Preliminaries
In the following, we would like to present some preliminaries.We begin with recalling the well-known estimate for the product of two functions.

Lemma 1 ([22]
).Let s > 0.Then, there exists a constant C > 0 such that for all u, v ∈ A key point of obtaining direct compactness results is the so-called Aubin-Lions lemma.
Let ( f * g)(x) = R 2 f (x − y)g(y)dy.Weak solutions to (1), in the sense of Definition 1, will be constructed as limit objects from a family of appropriately regularized systems as follows: where ρ ϵ (x) is defined by the standard mollifier ρ(x) satisfying R 2 ρ(x)dx.We now state the global classical solution for the regularized system to the Cauchy problem (3).

A Priori Estimates for a Regularized System
In what follows, we let C denote some different constants, which depend at most on If there are no special explanations, they are independent of ϵ and t.
Proposition 1. Assume that (u 0 , c 0 , m 0 , n 0 ) ∈ X 0 , and let (u ϵ , n ϵ , c ϵ ) be a unique classical solution to the system (3).Then, a positive constant C exists such that λ + 1 as well as Proof.We first show some priori estimates of n ϵ , v ϵ , and c ϵ .By a direct integrating for, we have From ( 4), with the aid of Gronwall's inequality, we have From ( 5) and ( 7), we have From ( 6), the weak maximum principle gives rise to Testing the third equation in (3) by n ϵ and integrating it over R 2 , we obtain from which, with aid of (8), Testing the second equation in (3) by c ϵ and integrating it over R 2 , we obtain On the other hand, we multiply u ϵ with the fourth equation of (3) and apply the Gagliardo-Nirenberg inequality to deduce that 1 2 From the Gronwall inequality and (7), we have It follows from Hölder's inequality, Gagliardo-Nirenberg's inequality, and Young's inequality that Applying ∇ to the third equation of (3) gives Taking the L 2 inner product for above equality with ∇v ϵ and applying (14), we obtain from which, from (7), from (13), and from the Gronwall inequality, we have Combining this with ( 7), ( 8), ( 9), ( 10), (11), and (13) directly result.
Proposition 2. Suppose v 0 , n 0 , c 0 ≥ 0 and that (u 0 , c 0 , v 0 , n 0 ) ∈ X 0 and ∇ϕ ∈ L ∞ (R 2 ).Let (n ϵ , v ϵ , c ϵ , u ϵ ) be the solutions to the model (3).Then, there a constant C > 0 exists independently of ϵ such that Proof.Multiplying Equation (3) 2 by ln n ϵ and integrating over R 2 and by prats, we have where ⟨x⟩ = Since we work in whole space R 2 , here we have to bound R 2 n ϵ ln n ϵ from below.In order to do that, we have to control the behavior of n ϵ as |x| → ∞, similarly to [15,25].To perform this task, we multiply (3) 1 by the smooth function ⟨x⟩; then, by integrating it over R 2 , by Young's inequality, we have Multiplying the above inequality by 2 and using ( 16), one obtains By the pointwise identity Multiplying the above equation by −∆ √ c ϵ , by Young's inequality, we integrate by parts to obtain 1 2 where For I 2 , we have that from which, by Young's inequality, it follows that Plugging ( 21) into ( 19), we have 1 2 , we have by multiplying ( 22) by 4 that Summing up ( 17) and ( 23), we have from which, let By Proposition 1, one has By same reasoning for obtaining (2.27) in [15], we can easily obtain from ( 26) and ( 27) and applying Grönwall's inequality to (25), we have The proof of Proposition 2 is completed.
With the preparations of Propositions 1 and 2 at hand, we can further obtain a uniform estimate for the high regularity of (n ϵ , v ϵ , c ϵ , u ϵ ).
Proof.Now, by the Cauchy-Schwarzed inequality, Propositions 1 and 2, we have Using identity Using Proposition 1, we have Then, owing to (29)-(31), we have Testing the first equation in (3) against n ϵ and using Young's inequality yields 1 2 on the basis of which, (32), Proposition 1, and Grönwall's inequality, it follows that Collecting the above inequality with (32), we can thereby complete the proof of Proposition 3.
Furthermore, using the regularized equations, and the uniform estimates obtained above, we can directly obtain the following proposition.
From Definition 1, we have for any t > 0, Using Hölder's inequality, Young's inequality, and the Gagliardo-Nirenberg interpolation inequality, by virtue of ∇ • u 2 = ∇ • U = 0, we deduce and as well as Substituting ( 35), (36), and (37) into (34), one has from which we have Next from Definition 1, we have Taking ψ = C, we have from which we have Letting ψ = −∂ i C and summming over i, we infer that (48) From Definition 1, we have for any t > 0, Taking ψ = V, we have When i = 1, 2, we also have Letting ψ = ∂ i V and summming over i, we infer that from which we conclude that (U, V, N, C) = 0, and we thus complete the proof of uniqueness.

Conclusions
We introduce the notion of a weak solution and establish both the existence and uniqueness of such a weak solution for a large class of initial data on the whole space R 2 .