Large Time Behavior of Solutions to the 3D Rotating Navier–Stokes Equations

We consider the large time behavior of the solutions for the initial value problem of the Navier–Stokes equations with the Coriolis force in the three-dimensional whole space. We show the Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document} temporal decay estimates with the dispersion effect of the Coriolis force for the global solutions. Moreover, we prove the large time asymptotic expansion of the solutions behaving like the first-order spatial derivatives of the integral kernel of the corresponding linear solution.


Introduction
We consider the initial value problem for the 3D incompressible Navier-Stokes equations with the Coriolis force: (1.1) The unknowns u = u(x, t) = (u 1 (x, t), u 2 (x, t), u 3 (x, t)) and p = p(x, t) denote the velocity field and the pressure of the fluid at the point (x, t) ∈ R 3 × (0, ∞), respectively, while u 0 = (u 0,1 (x), u 0,2 (x), u 0,3 (x)) is the initial velocity satisfying ∇ · u 0 = 0. Here, e 3 denotes the unit vector (0, 0, 1), and the term Ωe 3 × u describes the Coriolis force with the Coriolis parameter Ω ∈ R. The purpose of this paper is to study the large time behavior of global solutions to (1.1). In particular, we shall show the L p temporal decay estimates and the asymptotic behaviors of solutions as t goes to infinity when the initial data u 0 is in L 1 (R 3 ). More precisely, we shall prove that the unique global solution u to (1.1) satisfies for 2 ≤ p ≤ p * with some upper bound 2 < p * < ∞ [see (1.14)] when u 0 ∈ L 1 (R 3 ) satisfies ∇ · u 0 = 0. Moreover, if we further assume |x|u 0 ∈ L 1 (R 3 ), we show that the global solution fulfills the temporal decay estimate In particular, if u 0 ∈ L 2 (R n ) satisfies (1 + |x|)u 0 ∈ L 1 (R n ) then it holds e tΔ u 0 L 2 ≤ C(1 + t) − 1 2 − n 4 and (1.3) holds with the decay rate β = 1 2 + n 4 . For the L p decay of the strong solution to (1.2), it follows from the results by Kato [17], Miyakawa [24,25] and Fujigaki and Miyakawa [7] that the unique global solution u(t) to (1.2) satisfies the L p temporal decay estimates and lim if the divergence-free initial data u 0 ∈ (L 1 ∩ L n )(R n ) is small in L n (R n ). Fujigaki and Miyakawa [7] showed the L p decay estimate of the strong solution (1 ≤ p ≤ ∞, t > 0) (1.5) provided that u 0 is small in L n (R n ) and satisfies (1 + |x|)u 0 ∈ L 1 (R 3 ). Furthermore, they [7] established the asymptotic expansion of the global solution u(t) behaving like the first-order derivatives of the Gauss kernel: for 1 ≤ p ≤ ∞ R n y j u 0 (y) dy is the Gauss kernel, and we set G t := F −1 [e −t|ξ| 2 P (ξ)], where P (ξ) = (δ jk + ξ j ξ k /|ξ| 2 ) 1≤j,k≤n is the Fourier multiplier matrix of the Helmholtz projection. We refer to [24][25][26]28] for the L p temporal decay estimates of the global strong solutions to (1.2) when the initial data belongs to the Hardy spaces, the Besov spaces or the weighted Hardy spaces.
Next, we review the known results on the unique existence and the temporal decay estimates for global solutions to (1.1). Let P be the Helmholtz projection onto the divergence-free vector fields, and let J be the skew-symmetric constant matrix defined by respectively, where R j = −∂ xj (−Δ) − 1 2 is the Riesz transform for j = 1, 2, 3. Note that the Coriolis force in (1.1) can be written as e 3 ×u = Ju. Applying the Helmholtz projection P to (1.1), we have the following 1 2 (R 3 ). They [10] also gave the temporal decay estimate u(t) L p ≤ Ct − 3 2 ( 1 2 − 1 p ) for 3 < p < ∞ and t > 0 with some constant C = C( u 0 H 1 2 , p) > 0. See also [9,16,21] for the global well-posedness of (1.8) for small initial data in various scaling invariant spaces. In [15,20], it is shown that the system (1.8) possesses a unique global solution for the initial data in the scaling subcritical spaceḢ s (R 3 ) with 1/2 < s < 9/10. More precisely, the authors in [20] established the linear decay estimate for t > 0 and Ω ∈ R with 2 ≤ p < ∞, 1 < r ≤ p and α ∈ (N ∪ {0}) 3 , and obtained the following result on the global existence of solutions: 15,20]). Suppose that s, q, and θ satisfy Then, there exists a constant C * = C * (s, q, θ) > 0 such that for any u 0 ∈Ḣ s (R 3 ) 3 with ∇ · u 0 = 0 and Ω ∈ R \ {0} satisfying Ahn, Kim and Lee [1] extended Theorem 1.1 to the system (1.8) with the fractional Laplacian (−Δ) α for 1/2 < α < 5/2, and also derived the temporal decay estimates of solutions with the same decay rate as the linear solutions (1.9). In the case α = 1, the L p decay estimates obtained in [1] is written as for t > 0, where and (s, q) are the exponents satisfying (1.10). Kim [19] considered the magnetohydrodynamics equations with the Coriolis force, and proved the global well-posedness and the temporal decay estimate for u 0 ∈ (H s ∩ L 1 )(R 3 ) with 1/2 < s < 3/2: In this paper, we consider the L p temporal decay estimate and the large time behavior of the global solution u to (1.8) constructed in Theorem 1.1 when the initial data u 0 is in L 1 (R 3 ). We remark that the L 1 -integrability implies R 3 u 0 (y) dy = 0, thanks to the divergence-free condition ∇ · u 0 = 0. Hence, as is the case (1.4) for the original Navier-Stokes Eq. (1.2), it seems natural to expect that the L p -norm of the global solution u(t) decays faster than t − 3 Our first result in this paper reads as follows: Assume that the exponents s, q and θ satisfy (1.10)- (1.11), and that the exponent p satisfies (1.14) We next address the asymptotic behavior of global solutions corresponding to (1.5) and (1.6) for the original Navier-Stokes Eq. (1.2) by Fujigaki and Miyakawa [7] when (1 + |x|)u 0 ∈ L 1 (R 3 ). In order to state our second result, we introduce some notation. Let P (ξ) be the Fourier multiplier matrix of the Helmholtz projection P (1.7) defined by for ξ ∈ R 3 \ {0}. Let A Ω := −Δ + ΩPJP be the linear operator associated with (1.8). It is known in [8,10] that the semigroup e −tAΩ generated by −A Ω is given explicitly by Here, I is the 3 × 3 identity matrix and R(ξ) is the skewsymmetric matrix related to the Riesz transforms defined as By the Duhamel principle, the system (1.8) can be transformed into the following integral equation: Now, we define the functions H Ω (ξ, t) and H Ω (ξ, t) as (1.22) for ξ ∈ R 3 \ {0} and t ≥ 0. Then, we set for x ∈ R 3 and t ≥ 0. Note that the functions K Ω (x, t) and K Ω (x, t) are the integral kernel of the linear semigroup e −tAΩ and e −tAΩ P, respectively, and there hold We set the function space L 1 1 (R 3 ) of the initial data as . Our second result on the asymptotic behavior of global solutions to (1.8) for the initial data u 0 ∈ L 1 1 (R 3 ) 3 reads as follows: Theorem 1.3. Assume that the exponents s, q and θ satisfy (1.10)- (1.11), and that the exponent p satisfies for all t > 0. Furthermore, it holds that Let us give several remarks on Theorem 1.3. In the case Ω = 0, we see by (1.21), (1.22) and (1.23) that H 0 (ξ, t) = I, H 0 (ξ, t) = P (ξ) and Hence the asymptotic expansion (1.25) in Theorem 1.3 corresponds to (1.6) for the original Navier-Stokes equations by [7]. Next, we remark that it follows from Lemma 3.1 and (5.35) that and then t 2 23 Page 6 of 31 T. Egashira, R. Takada JMFM by the L 2 decay estimate u(t) L 2 ≤ C(1 + t) − 5 4 in Lemma 4.2 (2). Hence the functions appeared in (1.25) would be expected to be the leading terms of the global solutions u(t) to (1.8) as t → ∞.
Finally, let us mention the proof of the asymptotic behavior (1.25) and give the comparisons with the previous studies. In [7], the authors applied the mean value theorem to the Gauss kernel G t (x − y, t − s) with respect to both the space and the time variables, and proved the asymptotic expansion (1.6) for the solutions to (1.2). Ishige, Kawakami and Kobayashi [14] established a general method to show the higherorder asymptotic expansion of solutions for various nonlinear parabolic equations (see also [11][12][13]). The authors [14] introduced the operator P k (t) having the cancellation property R n x α [P k (t)f ](x)dx = 0 of moments for all α ∈ (N ∪ {0}) n with |α| ≤ k, and obtained the higher-order asymptotic expansion of solutions without using the time derivatives of the integral kernel. In our situation for (1.8), since it holds and PJP is the Fourier multiplier of the 0-th order, we see that the time differentiation does not give a faster decay than the original kernel K Ω . Also, since the kernel K Ω (·, t) does not belong to L 1 (R n ) because of the fact that ξ 3 /|ξ| in (1.22) is not continuous at ξ = 0, it seems difficult to consider the moment conditions in [14]. For the proof of Theorem 1.3, we adapt the arguments in [1,7,29] with the correction term ΩPJP K Ω (x, t), and show the asymptotic behavior (1.25) by using the L 2 temporal decay estimates and the space-time integrability of the solution u in L θ (0, ∞;Ḣ s q (R 3 )). This paper is organized as follows. In Sect. 2, we prepare several function spaces and recall the known results on linear estimates. In Sect. 3, we show the temporal decay estimates and the asymptotics of the linear solutions. In Sect. 4, we prove the L p decay estimates (1.15) and (1.24) for the nonlinear solutions. In Sect. 5, we present the proof of the nonlinear asymptotic behaviors (1.16) and (1.25) for the global solution to (1.8).

Preliminaries
In this section, we introduce the definitions of several function spaces, and recall the known results on the linear estimates for the semigroup e −tAΩ .
Let S (R 3 ) be the Schwartz space of all rapidly decreasing infinitely differentiable functions on R 3 , and let S (R 3 ) be the set of all tempered distributions. The Fourier transform and the inverse Fourier transform of ϕ ∈ S (R 3 ) are defined by for ξ, x ∈ R 3 , respectively. Also, P(R 3 ) denotes the set of all polynomials in R 3 .
Next, we recall the definition of the Littlewood-Paley decomposition. Let ϕ 0 ∈ S (R 3 ) satisfy the following properties: For s > 0, it is known that the norm equivalence holds for 1 ≤ p, q ≤ ∞ with some positive constants c 1 and c 2 . Finally, we recall the known results on the linear estimates for the semigroup e −tAΩ . We set Then, the linear semigroup e −tAΩ generated by the linear operator A Ω = −Δ+ΩPJP is explicitly written as [8,10,15] for the derivation of the explicit formula (2.3) of the semigroup e −tAΩ .
We end this section by recalling the L q -L p smoothing estimates for the linear semigruops e −tAΩ and e tΔ .

Linear Decay Estimates and Asymptotics
In this section, we shall establish the temporal decay estimates for the linear solution e −tAΩ u 0 when . Furthermore, we obtain the asymptotic profile of the linear solution e −tAΩ u 0 as t → ∞.

Linear Decay Estimates
Let us set where R(ξ) is the skew-symmetric matrix defined by (1.19). Then, it follows from (2.3) that the linear solution e −tAΩ u 0 can be written as We firstly show the L p estimates for the integral kernel K Ω (·, t). Let G t (x) be the Gauss kernel in R 3 , which is defined as Note that it holds G t (ξ) = e −t|ξ| 2 and G 1 ∈ S (R 3 ).
for all Ω ∈ R and all t > 0. Moreover, we directly calculate as where R is defined in (2.4). Therefore, it follows from (3.4) to (3.5) that We first consider the case 2 ≤ p < ∞. It follows from Lemma 2.3 and the continuous embeddinġ Since G 1 ∈ S (R 3 ), we see by (2.1) that for p = 2 and 2 < p < ∞, respectively. For the case p = ∞, we have by Lemma 2.3 and the continuous embeddingḂ 0 and G 1 Ḃ |α|+3 for all t > 0. This completes the proof of Lemma 3.1.
Applying Lemma 3.1, we show the following L 1 -L p temporal decay estimates for the linear semigroup e −tAΩ .

Remark 3.3.
We remark that the temporal decay estimate (3.10) for f ∈ L 1 (R 3 ) has already been shown by Kim [19,Lemma 2.5]. Here, we shall give an alternative proof by using Lemma 3.1.

Proof of Lemma 3.2.
(1) Applying the Hausdorff-Young inequality and Lemma 3.1, we have by (3.2) that (2) Since f ∈ L 1 (R 3 ) 3 and ∇ · f = 0, we see that it holds Then, applying the mean value theorem, the Minkowski inequality and Lemma 3.1, we obtain This completes the proof of Lemma 3.2.

Linear Asymptotics
In this subsection, we shall show the following asymptotic profiles of the linear solution e −tAΩ u 0 as t goes to infinity. Proof.
Next we consider the case 2 < p < ∞. Applying the embeddingḂ 0 p,2 (R 3 ) → L p (R 3 ) and Lemma 2.3 to (3.16), we have Here, since it holds , it follows from Lemma 2.6 and the embedding L p (R 3 ) →Ḃ 0 p ,2 (R 3 ) that Hence we have by (3.17) and (3.18) that Note that it holds Hence we have by the dominated convergence theorem that as t → ∞. This completes the proof of Theorem 3.4 (1).

Nonlinear Decay Estimates
In this section, we adapt the ideas in [1,29], and show the L p temporal decay estimates for the global solution u to (1.8). Proof. We remark that the inequality in Lemma 4.1 was obtained by Wiegner [29, (2.1)] for global weak solutions to the original Navier-Stokes equations. Since u 0 ∈ (Ḣ s ∩ L 1 )(R 3 ) 3 , we see by the Sobolev embeddingḢ s (R 3 ) → L q (R 3 ) with 1 q = 1 2 − s 3 and the interpolation inequality that
Proof. We follow the same argument as [29]. Firstly, suppose that there hold with some α 0 > 0 and 0 ≤ β < 1. Then, take an exponent α and a function g(t) so that Then, we see that t 0 g(r) 2 dr = log(1 + t) α , and Lemma 4.1 and (4.2) yield Hence we have Note that it holds u 0 ∈ L 2 (R 3 ) 3 by (4.1). Hence it follows from the smoothing estimates for the heat semigroup that which yield for all t ≥ 0. Moreover, taking the L 2 inner product of (1.8) with u(t) gives 1 2 Hence we have the energy equality: which yields the estimate u(t) L 2 ≤ u 0 L 2 . Therefore, the estimates (4.2) hold for α 0 = 3/2 and β = 0, and then we have by (4.3) and (4.4) that with some constant C = C( u 0 L 1 , u 0 L 2 ) > 0. Again, applying (4.3) and (4.4) with α 0 = 3/2 and β = 1/2, we obtain with some constant C = C( u 0 L 1 , u 0 L 2 ) > 0. (2) Assume further that |x|u 0 ∈ L 1 (R 3 ) 3 . Then, since there hold we have for all t ≥ 0. Hence, we see that the estimates (4.2) hold for α 0 = 5/2 and β = 0. Then, (4.3) and (4.4) give Then, similarly to (4.7), we have for all t ≥ 0. Here, we remark that (4.9) gives Now, take an exponent α and a function g(t) so that Then, it follows from Lemma 4.1, (4.8) and (4.10) that which yields for all t 0. This completes the proof of Lemma 4.2.

L p -decay Estimates
In this subsection, we adapt the arguments in [1] and show the L p temporal decay estimates for the solution u to (1.8). For 1 ≤ p ≤ ∞ and j = 0, 1, we put  , q and θ satisfy (1.10)-(1.11), and that the exponent p satisfies Let C * > 0 be the constant in (1.12). Then, there exists a constant 0 < C * * ≤ C * such that for any for all t > 0 and j = 0, 1.
Proof. We first consider the case j = 0. It follows from (3.10) in Lemma 3.2 that Then, since u 0 ∈ L 1 (R 3 ) 3 , the L 2 decay estimate in Lemma 4.2 (1) and the energy equality (4.6) give that with some constant C = C( u 0 L 1 , u 0 L 2 ) > 0. Then, (4.16) and (4.17) yield the desired estimates for j = 0. In the case j = 1, we have by (3.10) in Lemma 3.2 and the L 2 decay estimate in Lemma 4.2 (1) that with some constant C = C(p, u 0 L 1 , u 0 L 2 ) > 0. This completes the proof of Lemma 4.4.
We are ready to give the proof of the L p -time decay estimates.

Theorem 4.5.
Suppose that the exponents s, q and θ satisfy (1.10)- (1.11), and that the exponent p satisfies for all t > 0.
Proof. (1) Let us first consider the case that the exponent p satisfies Then, it follows from (3.10) in Lemmas 3.2, 4.3 and 4.4 with j = 0 that for all 0 < t ≤ t with some positive constant C = C(p, u 0 L 1 , u 0 L 2 ). Hence we have u X p 0 (t ) ≤ 2C, which yields the desired estimate (4.19). For the exponent p satisfying we take the exponent η ∈ [0, 1] so that 1 p = η 2 + 1−η q . Then, by the interpolation, Lemma 4.2 (1) and (4.19) for q, we obtain for all t > 0 (2) Firstly, consider the case that the exponent p satisfies (4.21). Similarly to (4.22), we have by (3.11) in Lemmas 3.2, 4.3 and 4.4 with j = 1 that for all 0 < t ≤ t with some positive constant C = C(p, |x|u 0 L 1 , u 0 L 2 ). This implies u X p 1 (t ) ≤ 2C, and we obtain the desired estimate (4.20). In the case 1 q ≤ 1 p ≤ 1 2 , we take the exponent η ∈ [0, 1] so that 1 p = η 2 + 1−η q . We obtain by the interpolation, Lemma 4.2 (2) and (4.20) for q that for all t > 0. This completes the proof of Theorem 4.5.

Nonlinear Asymptotics
We are now ready to give the proofs of Theorems For I 1 , it follows from (3.13) in Theorem 3.4 that Concerning I 2 , Lemma 4.4 with j = 1 gives that as t → ∞. Hence it remains to show the estimate for I 3 (t). Firstly, assume that the exponent p satisfies This is the same condition as (4.11) in Lemma 4.3. Put Then, the exponents p and r satisfy 2 < p < ∞ and 1 < r ≤ p . It follows from Lemma 2.5, the embeddinġ H s q (R 3 ) → L qs (R 3 ) and (4.19) in Theorem 4.5 that Here, similarly to (4.14) and (4.15), we have Since u belongs to L θ (0, ∞;Ḣ s q (R 3 )) 3 , we have by (5.5) and (5.6) that as t → ∞. This completes the proof of (1.16) when the exponent p satisfies (5.4). Next, we shall consider the estimate for I 3 (t) for p = 2. We follow the argument in [7], and set We see that v(t) = u(t) − e −(t−τ )AΩ u(τ ), and v(t) should solve Taking the L 2 -inner product of (5.8) with v, we have Since v(t) = u(t) − e −(t−τ )AΩ u(τ ), the integration by parts and the divergence-free condition give that Hence we have 1 2 Note that Lemmas 2.4 and 4.2 (1) yield Also, we remark that it holds by Lemma 4.2 (1). Hence we have from (5.9), (5.10) and (5.11) Let ρ > 0 be a positive parameter to be chosen later. It follows from the Plancherel theorem that This and (5.12) imply that Therefore, we see that (5.14) Substituting (5.14) into (5.13) gives that for 0 < τ < t. Now, we set It follows from (5.16) that Hence we have and we have by (5.17) that Here, it follows from Lemma 4.2 (1) that (1 + r) 3 which yields t 3 4 I 3 (t) L 2 ≤ Ct −1 → 0 (5.20) as t → ∞. This gives the proof of (1.16) for p = 2.  We first consider the estimate for J 2 . Let us treat the case that the exponent p satisfies Note that (5.22) is the same assumption as (4.11) in Lemma 4.3. Setting we see that there hold 2 < p < ∞ and 1 < r ≤ p . Hence we can apply Lemma 2.5, and it follows from the embeddingḢ s q (R 3 ) → L qs (R 3 ), (4.20) in Theorem 4.5 and (5.6) that (5.23) Since u ∈ L θ (0, ∞;Ḣ s q (R 3 )) 3 , we have by (5.23) that Note that we now assume (1 + |x|)u 0 ∈ L 1 (R 3 ) 3 , and then it follows from Lemmas 2.4 and 4.2 (2) that and We apply the same argument as in the proof of Theorem 1.2 by using (5.25) and (5.26) instead of (5.10) and (5.11), respectively. Then, similarly to (5.15), we have Substituting this ρ(t) into (5.27) gives Then, it follows from (5.28) that which yields