Asymptotic behavior of global mild solutions to the Keller-Segel-Navier-Stokes system in Lorentz spaces

1.1 Existence of global solutions to the chemotaxis(-fluid) system

While chemotaxis is one of the significant features in the biological field, the Keller-Segel system (1.2) also has a special property in the mathematical sense, critical mass phenomena. One of the typical problems for evolution equations may be the construction of solutions. However, in general, we do not know whether we may construct global solutions of nonlinear evolution equations for arbitrarily large initial data. Concerning this problem, in the 2D Keller-Segel system (1.2), it is known that solutions exist globally in time if the initial density n 0 satisfies ∫ R 2 n 0 ( x ) d x < 8 π . Conversely, for any M > 8 π , we may take some initial density n 0 such that ∫ R 2 n 0 ( x ) d x = M and corresponding solutions of (1.2) blow up in finite time. We refer to [2–5,14,16,18,39,40] for such results and related topics. In addition, similar results hold even if we consider the case of the Keller-Segel-Navier-Stokes system [23,29,34]. We also note that there have been a lot of studies on the existence and asymptotic behavior of global solutions to the 2D Keller-Segel-Navier-Stokes system in various domains. For instance, we refer to [6,8–10,15,35,43,46,47,49].

As mentioned above, mathematical models of chemotaxis in the 2D case have the critical mass phenomenon, and thus, the global existence of solutions may be characterized by the mass of the initial density n 0 . However, we do not know the corresponding results in the higher dimensional case N ≥ 3 . Therefore, to construct global solutions in such a case, we have to consider weak solutions or we have to assume that all of the initial data are sufficiently small. In the former case, there have been various kinds of the existence results of global weak solutions to the 3D Keller-Segel-Navier-Stokes system. In fact, Liu and Lorz [35] considered the case where the term Δ n appearing in (1.1) is replaced by Δ n 4 ⁄ 3 , nonlinear diffusion type. Winkler [46] studied the case where the domain is bounded and the Navier-Stokes system is replaced by the Stokes system. Chae et al. [8] considered the case where some nonlinear terms have special structures. Tao and Winkler [44] treated the case where the domain is bounded and Δ n is replaced by Δ n m , m > 8 ⁄ 7 . In addition, Winkler [48] considered the case of a bounded domain and achieved obtaining global weak solutions without any restriction, such as considering nonlinear diffusion and Stokes system and assuming structures of nonlinear terms. Kang et al. [25] also treated the whole space case.

Next, we mention another way to construct global solutions, assuming the smallness conditions. A starting point to obtain such a result may be the article by Duan et al. [20]. In fact, they considered the 3D case and constructed global classical solutions for small initial data in the Sobolev spaces H 3,2 ( R 3 ) framework. After that, several results were obtained by following the well-known method of Kato [26]. Here, let us consider the scaling invariant spaces to system (1.1). Suppose that ( n , c , u , p ) is a global classical solution of (1.1) with a gravitational potential φ . Then, by setting

for ( t , x ) ∈ ( 0 , ∞ ) × R N and 0 < λ < ∞ , we observe that ( n λ , c λ , u λ , p λ ) also satisfies (1.1) with φ replaced by φ λ ( x ) ≔ φ ( λ x ) for all 0 < λ < ∞ . Since it holds that

for all 0 < λ < ∞ , we see that

is one of the scaling invariant spaces to (1.1). Kozono et al. [28] considered a more involved system by adding a new unknown function v to system (1.1), where v stands for the concentration of the chemo-attractant. From the viewpoint of the scaling invariance, they obtained global mild solutions by assuming that the given data ( n 0 , c 0 , v 0 , u 0 ) and φ are sufficiently small in the following spaces:

where P denotes the Helmholtz projection operator. Note that their result is valid for all N ≥ 3 and that the case of N = 2 is also valid if n 0 ∈ L N ⁄ 2 , ∞ ( R N ) is replaced by n 0 ∈ L 1 ( R 2 ) . Cao and Lankeit [6] treated the case of a bounded domain Ω ⊂ R N for N ∈ { 2,3 } and assumed that the initial data satisfy either ‖ n 0 − n ¯ 0 ‖ L q ( Ω ) + ‖ c 0 ‖ L ∞ ( Ω ) + ‖ u 0 ‖ L N ( Ω ) ≪ 1 with N ⁄ 2 < q < ∞ or ‖ n 0 ‖ L q ( Ω ) + ‖ ∇ c 0 ‖ L N ( Ω ) + ‖ u 0 ‖ L N ( Ω ) ≪ 1 with N ⁄ 2 < q < N and some suitable assumptions, where n ¯ 0 ≔ ∣ Ω ∣ − 1 ∫ Ω n 0 ( x ) d x . Then, they constructed global classical solutions. Jiang [24] considered a similar system to (1.1) in a bounded domain with N ≥ 2 by replacing c with the concentration v of the chemo-attractant. He assumed that the initial data satisfy ‖ n 0 − n ¯ 0 ‖ L N ⁄ 2 ( Ω ) + ‖ ∇ v 0 ‖ L N ( Ω ) + ‖ u 0 ‖ L N ( Ω ) ≪ 1 with certain assumptions to show the existence of global classical solutions. Moreover, concerning more involved scaling invariant spaces, Choe et al. [13] and Choe and Lkhagvasuren [12] considered the 3D case in the homogeneous Besov spaces framework; n 0 ∈ B ˙ q , 1 − 2 + 3 ⁄ q ( R 3 ) , c 0 ∈ B ˙ q , 1 3 ⁄ q ( R 3 ) , u 0 ∈ P ( B ˙ q , 1 − 1 + 3 ⁄ q ( R 3 ) ) 3 , and φ ∈ B ˙ q , 1 3 ⁄ q ( R 3 ) for 1 ≤ q < 3 . For the case of the other function spaces based on Besov-type spaces, we refer to, e.g., [17,22,42,51–55].

1.2 Our motivations in this article

In this article, we consider the case where the given data ( n 0 , c 0 , u 0 ) and φ satisfy

where N ≥ 3 and 1 ≤ ρ < ∞ . In particular, we focus on scaling invariant Lorentz spaces. As stated before, Kozono et al. [28] have already shown the existence of global solutions to a more involved system under the condition ρ = ∞ . One of the reasons why we consider the scaling invariant spaces is to enlarge the spaces of the given data that ensure the existence of global solutions. Although our assumption (1.4) does not seem to provide an advantage from this viewpoint, what we want to do is to reveal the detailed properties of solutions of system (1.1) by focusing on a difference of the index 1 ≤ ρ < ∞ . More precisely, we show that mild solutions of (1.1) have higher regularities than those of [28]. We also reduce a part of the smallness conditions of the given data. In addition, we improve the asymptotic behavior of the global mild solutions to (1.1) by relying on the condition 1 ≤ ρ < ∞ .

Before stating our main results, we shall mention the key ideas of the analysis in this article. The method to obtain regularities of solutions is mainly based on the real interpolation. We recall that the Lorentz spaces are characterized by the real interpolation spaces of the usual Lebesgue spaces. Kozono and Shimizu [33, Proposition 2.1] focused on the smoothing estimates of the heat semigroup defined on homogeneous Besov spaces, which were shown by Kozono et al. [30, Lemma 2.2 (ii)], and relied on the real interpolation technique to obtain the space-time estimates of the heat semigroup:

From this result, we may expect that a similar estimate is valid in the case of the Lorentz spaces. By showing the corresponding estimate, we will observe that mild solutions of (1.1) have higher regularities. It should be noted that Yamazaki [50, Corollary 2.3] has obtained almost the same estimate as ours. We also refer to [31,32] for related papers. Concerning the reduction of the smallness condition of the given data, we note that the term n ∇ φ appearing in (1.1) is a linear term with respect to n . From the viewpoint of perturbation theory, we have to assume that ∇ φ is small to obtain solutions. However, we may reduce such a condition by a simple modification of the usual method in the construction of solutions. We remark that Choe and Lkhagvasuren [12] have used such a method. Moreover, for the initial datum c 0 ∈ L ∞ ( R N ) with ∇ c 0 ∈ ( L N , ρ ( R N ) ) N , we only assume that ‖ ∇ c 0 ‖ L N , ρ ( R N ) is small to obtain global solutions, while it is not necessary to assume that ‖ c 0 ‖ L ∞ ( R N ) is small. In other words, we may construct global solutions even if c 0 is close to a large constant. We shall refer to the result of Cao and Lankeit [6, Theorem 1.2], who have obtained a similar result through a different approach. Finally, let us mention the asymptotic behavior; by following the approach due to Kato [26], we may show that the global solutions decay as t → ∞ with suitable rates. For instance, Kozono et al. [28] obtained the decay of global solutions n such that

as t → ∞ for some q > N ⁄ 2 . Compared with this result, we show that

as t → ∞ for some q > N ⁄ 2 and 1 ≤ ρ < ∞ . Such a result is obtained by the method of Kato [26, Note]. Mention that the set C 0 ∞ ( R N ) of smooth functions with compact support is dense in L N ⁄ 2 , ρ ( R N ) for 1 ≤ ρ < ∞ but not dense in L N ⁄ 2 , ∞ ( R N ) . This is an advantage of considering the case of 1 ≤ ρ < ∞ . However, we should remark that our condition (1.4) contains c 0 ∈ L ∞ ( R N ) , and thus, the problem of a lack of density still remains. To overcome this problem, we show the decay of the global solutions without any approximation for the initial datum c 0 . In particular, we approximate only the initial data n 0 and u 0 .

1.3 Main results

In what follows, we shall state our main results. To this end, we consider the integral forms of system (1.1). Let P denote the Helmholtz projection operator defined on R N , namely, we set P ≔ I + ∇ ( − Δ ) − 1 ∇ ⋅ . Then, since P ( ∇ p ) = 0 and P u = u under the condition ∇ ⋅ u = 0 , we observe that the third equation of (1.1) is given by ∂ t u − Δ u = − P ( ( u ⋅ ∇ ) u − n ∇ φ ) . We also note that u ⋅ ∇ n = ∇ ⋅ ( n u ) and ( u ⋅ ∇ ) u = ∇ ⋅ ( u ⊗ u ) due to ∇ ⋅ u = 0 . As the Laplacian − Δ generates the heat semigroup e t Δ ≔ G t ⁎ for 0 < t < ∞ , where G t ( x ) ≔ ( 4 π t ) − N ⁄ 2 e − ∣ x ∣ 2 ⁄ ( 4 t ) for ( t , x ) ∈ ( 0 , ∞ ) × R N , we obtain the following integral forms of system (1.1):

A solution ( n , c , u ) of the integral forms (1.5) may be called a mild solution of the original system (1.1). Throughout this article, we assume that N ≥ 3 .

Our first result is the existence and uniqueness theorem of local mild solutions of (1.1) with initial data in scaling invariant Lorentz spaces.

Our second result states that if the initial data ( n 0 , c 0 , u 0 ) are sufficiently small, then we may obtain global mild solutions of (1.1). In addition, the asymptotic behavior of the global mild solutions is obtained.