On the Stability of Steady-State Solutions to the Navier–Stokes Equations in the Whole Space

We prove asymptotic stability of steady-state solutions to the Navier–Stokes equations in the whole space. One of the novelties of this work consists in considering perturbations that show a distinct pointwise behavior (in space and time) and correspond to initial data only belonging to suitable weighted Lebesgue spaces.


Introduction
Stability of steady-state solutions to the Navier-Stokes equations in the exterior, Ω, of a three-dimensional bounded region is among the classical and most studied problems in mathematical fluid mechanics. Actually, the first, rigorous study goes back to more than half century ago, owing to the work of Heywood [10]. The main finding -as expected on physical ground-is that steady-state is asymptotically stable, provided its "magnitude" is sufficiently "small." Since then, many significant papers have been published, where the problem is addressed under different assumptions either on the steady-state or else the functional properties of the dynamical perturbations, or both. The list of contributors is too long to include here and we refer the reader to the review article [8], especially Section 10.4, and the bibliography therein.
In spite of the abundant literature, it seems that there is room left for further and interesting investigation. In fact, the class of perturbations to the basic state considered in the papers referred to as above, is contained in Lebesgue (or even "larger") spaces. Now, if, on the one hand, this allows for a rather general type of perturbations, on the other hand it may have less impact in the application and/or numerical computation, where the knowledge of a specific and controlled pointwise behavior would be more advantageous.
Objective of this paper is to perform a preliminary study in the latter direction. Actually, we will analyze the case when Ω is the whole three-dimensional space and prove an asymptotic stability theorem in a class of perturbations where, in particular, their pointwise behavior in space and time is entirely controlled by the initial data; see Theorem 2 and Corollary 1. In order to achieve this goal, we begin to define the class of steady-state solutions, whose asymptotic stability we intend to investigate. Such solutions are characterized by velocity fields decaying like |x| −2 at large distances, and their existence and uniqueness is proved for "small" and suitable corresponding body force; see Theorem 1. We next write the evolution equation for the perturbation field as an integral equation (see (5.4)) and study existence and uniqueness of corresponding solutions in appropriate weighted Lebesgue spaces, L p α , with a (spatial) weight that grows algebraically at large distances from the origin; see (2.1). We thus prove existence by standard successive approximations procedure, that we show to converge provided the magnitudes of the initial data and of the steady-state are restricted in suitable norms. A crucial tool for the above procedure to work is a result on convolutions in weighted spaces due to Kerman [11], and refined in [15]; see Lemma 2. Furthermore, we prove uniqueness by a duality argument. Our method is close to that used in the well-known Holmgren theorem, as proposed by Foias [6] in the Navier-Stokes setting, but suitably modified to fit in our context.
As mentioned earlier on, an important consequence of our main result -stated in Theorem 2-is that we are able to control the pointwise space-time behavior of perturbations. More precisely, we show that they decay in space as |x| −1+ 3 p and time as t − 3 2p , p > 3; see Corollary 1. It is worth remarking that these results are obtained without assuming any pointwise behavior on the initial data, but only that the latter belongs to a suitable weighted Lebesgue space (see (2.5)). Our finding should be contrasted with classical results (see [2,12]) and those ones obtained in the case of weak solutions [3,5], where an asymptotic pointwise behavior is imposed at the outset on the initial data. This behavior is preserved for t > 0 and it can also be converted into a time asymptotic behavior by means of parabolic scaling |x| ∼ t 1 2 . In fact, our results are more in the spirit of those obtained in [9], where the initial datum is given in a weighted L 2 space. The fundamental difference is that, unlike [9], a priori our solutions, at all t ≥ 0, are not requested to have finite energy and are perturbations not to the rest but to a non-trivial steady-state.
For the sake of the brevity, we omit the details concerning regularity properties of our solution. In fact, once we have established a (local) L ∞ -bound, they follow via standard arguments developed for the integral equation (5.3).
The plan of the paper is the following. In Sect. 2 we formulate the problem, and state and comment our main results. In Sect. 3 we prove existence and uniqueness of steady-state solutions in a suitable function class where the velocity field decays like |x| −2 for large |x|. In the following Sect. 4 we begin the study of the perturbation problem by presenting a number of preliminary lemmas. The main objective of Sect. 5 is to show several basic estimates in weighted spaces for convolutions of the Oseen fundamental solution with functions belonging to weighted spaces as well. In the last Sect. 6, we give a proof of our main result, Theorem 2, to get existence of solutions by applying successive approximation procedure to the integral equation (5.3) and, successively, discussing their uniqueness. In the same section, as a corollary to this finding, we furnish the desired pointwise space-time behavior.

Formulation of the Problem and Statement of the Main Results
We introduce some notation. For given p > 3 and α ∈ R, by L p α = L p α (R 3 ) we denote the following weighted Lebesgue space endowed with the natural norm u L p α := u|x| α p . Moreover, by D 1,q (R 3 ), 1 < q < ∞, we denote the completion of C ∞ 0 (R 3 ) in the Lebesgue gradient norm ∇u q . We also set Finally, we indicate by C 0 the class of vector functions, ϕ ∈ C ∞ 0 (R 3 ) with div ϕ = 0. All other notation is rather standard.
Consider the steady-state Navier-Stokes equations in the whole space corresponding to a given body force f : Our first objective is to determine the class of solutions to (2.2) where our stability results apply. Precisely, we show the following theorem.
Theorem 1. Let f = div F , where F is a second-order tensor field such that Then, there is an absolute constant C 0 > 0 such that, if problem (2.2) has one and only one solution (w, π w ) with w satisfying The main goal of this paper is to investigate the asymptotic stability of the steady-state solutions of the theorem above, in the class of perturbations . We consider special perturbations u 0 of the kinetic field w, that is, such that u 0 − w := v 0 ∈ L p α , and look for sufficient conditions for the global existence of corresponding perturbation solutions (v, π), ensuring the stability in the L p α -norm and attractiveness in the L ∞ -norm of the given steady motion. To this end, we commence by observing that the perturbation (v, π) to the velocity w satisfies the following Cauchy problem The main result of our paper can be stated as follows.
There is an absolute constant c > 0 such that if where c is a constant independent of v.

Corollary 1.
Any solution of Theorem 2 has the following pointwise behavior: In particular, we achieve that for all R > 0 where c is a constant independent of v 0 and (t, x).
We would like to make a number of remarks concerning the above results.
Remark 1. Since, as is easily shown, all norms w (i) , i = 0, 1, 2, are bounded above by [|w|] 2 , in view of Theorem 2.2 the assumptions (2.6) can be restated by requiring that D and v 0 L p α are below a given absolute constant.
Remark 2. The L p -weighted approach used in Theorem 2 has two noteworthy aspects. On the one hand, the perturbation is allowed to be singular at a point x 0 (in our assumption the origin O). On the other hand, it provides a pointwise and uniform spatial behavior of the solution, for t > 0, without making a similar request on the initial data, but only requiring a weighted summability property.
is another example of scaling invariant norm (in the sense proposed in [1]) for the global existence with small data. In this regards we notice that for p = 2 the above space was considered in [4] in the different context of sufficient conditions for the existence of regular solutions. For p = 3 , L 3 α ≡ L 3 (that is α = 0) and the result goes back to the well known theory.

Remark 4.
In [4], the weighted L 2 1 2 -space is restricted to the so called Kato class. In this way the authors are able to give an estimate of the interval of existence (a priori finite) without employing auxiliary functions (density property). The interest of the Kato class is connected with the regularity criterium proposed in [1].

Remark 5.
We think that an analogous approach can be used to investigate the case p ∈ (2, 3) and furnish a new regularity criterium. This study will be object of future work.

Proof of Theorem 1
In this section we will provide a proof of Theorem 1. To this end, we need some preparatory results contained in the next lemma.

Lemma 1. Let the assumption of Theorem 1 be satisfied. Then, given
Proof. From [7, Theorem X.9.1] we know that, under the stated assumption on f there exists one and only one solution (w, where, here and hereafter, c denotes a positive absolute constant. We now recall the following classical representation for the velocity field, valid for all x ∈ R 3 [7, Theorem X.5.2] where (U, q) is the Stokes fundamental solution given by (in 3D) By [14, Lemma 2.5], we get Moreover, setting for ρ > 0 by (3.12) and (3.14), we have, on the one hand, by the inequality |x − y| and, on the other hand, by the Hölder inequality, We thus conclude where, by (3.15) and (3.16) Consider next the equation (3.19) in the unknown field w : Ω R0 → R 3 , and set By (3.17), w is a solution to (3.19), and, in fact, it is the only solution in the class of those w with n 0 (w) < ∞, provided D is sufficiently small. To show this, from (3.19) we see that the difference u := w−w satisfies which, by the properties of the tensor U , implies (3.20) Since (see e.g., [7, Lemma II.9.2]) with c = c(γ), by (3.20) and (3.12) we deduce from which the stated uniqueness property follows for D less than an absolute constant. Thus, in order to show the lemma it is enough to prove that, for any α ∈ (0, 1), under the stated assumption there exists a solution w to (3.19) with n α (w) < ∞. This property is, in turn, obtained by an elementary fixed point argument based on the contraction-mapping theorem. Actually, for δ > 0, let Combining (3.12), (3.18), (3.21), (3.22) and the assumption on z, we find Thus, if we take δ := 2n α (S), from this inequality we get n α (w) ≤ δ, provided c D ≤ 1/2. Moreover, by taking S = 0 in (3.22), and proceeding in analogous fashion, we find which proves that M is a contraction in E α,δ (Ω R0 ). Consequently, (3.19) has a solution w ∈ E α,δ (Ω R0 ), with n α (w) ≤ c D, and the proof of the lemma is completed.
Proof of Theorem 1. Fix β ∈ (1/2, 1). From the previous lemma, we know that In view of (3.15), in order to show the result, we only have to prove that the last integral on the right-hand side of (3.13) decays like 1/|x| 2 , as |x| → ∞. To this end, set |x| = 2R and split the above integral as follows Since |x − y| ≥ |x|/2 for y ∈ Ω R , by the properties of U and by (3.23) we immediately find (3.25) Moreover, again from (3.23), we deduce The theorem is then a consequence of this latter inequality and of (3.24), (3.25).

On the Resolution of the Perturbation Problem
In the present and next section, we shall collect a number of preparatory lemmas that we will eventually employ, in Sect. 6, to furnish a proof of Theorem 2 and its corollary. We begin with some results concerning convolution integrals. To this end, by K[g](x) we mean the convolution product with a similar notation for functions depending on (t, x). With the letter c we shall denote a constant whose value is independent of (t, x) and v. Only a finite number of constants will enter our computations, the greatest of which is the one involved in the statement of Theorem 2.
The following result is crucial to our aims.
if, and only if,

2)
and Proof. See Theorem A in [15]. Let Proof. Clearly H[ϕ](t, x) is a smooth solution to the heat equation, so that estimate (4.5) 1 easily follows from the equation after a straightforward integration by parts. Also, estimate (4.5) 2 follows at once from (4.1) in Lemma 2 with β : We next consider Poisson equation where a, b are given fields with ∇ · a = ∇ · b = 0, and " denotes transpose.

Lemma 4.
If a ∈ L ∞ ∩ L p α and b ∈ L p α , p > 3, α := 1 − 3 p , then there exists a unique solution to (4.6) such that Applying Hölder inequality, we get which leads to (4.7) provided that we pull out of the integral L ∞ -norm of |a(x)| p 1−ε ε .
Following [17] we show the following. > ξ 0 , via (4.8) for m = 1, we get By the induction procedure we arrive at the thesis.

Integral Equation Formulation and Preliminary Results
The first objective of this section is to rewrite the perturbation problem (2.4) as an integral equation in the space L p α . Successively, we will prove some preparatory results on the involved convolution integrals. We begin to recall that the Oseen tensor, E(s, z), fundamental solution to the Stokes system, is defined by ] is the fundamental solution to the heat equation, and E (z) fundamental solution to Laplace equation. The following estimates are well known (cf. [12], or [16]): where D h z is the symbol of the partial derivatives with respect to z i -variable α i times, i = 1, 2, 3, and h = α 1 + α 2 + α 3 .
A solution to problem (2.4) can be searched as a solution to the following nonlinear integral equation where E[v · ∇v] denotes the convolution of the fundamental Oseen tensor with the convective term on the cylinder (0, T ) × R 3 and, we recall, H(t, z) is given in (4.4). Setting, for i = 1, 2, 3, we deduce, after an integration by parts over R 3 of the nonlinear term, that (5.3) can be rewritten as follows  for all (t, x) ∈ (0, T ) × R 3 . The function G is the solution to the heat equation corresponding to the initial where L s w denotes the weak L s space. By the properties of the solution to the heat equation in Lorentz spaces (e.g., [13,19]) we thus deduce that which, combined with (5.6), leads to (5.5).
Proof. For 0 ≤ αp < 1, p ≤ 2 conjugate exponent of p, estimate (5.7) is an immediate consequence of Lemma 3. By duality one deduce the same for p ≥ 2. The property (5.8) follows from the weak convergence in t = 0 of H[a](t) to a, and the estimate (5.7).
Then there exists a constant c independent of a and of b such that By our hypotheses we get Applying Hölder inequality, we get The following hold: These jointly with the one for I 1 complete the estimate for the right-hand side of (5.9).
Proof. With the help of (5.1) we get which, in turn, by Hölder inequality and (3.21), furnishes Summing the estimates of I 1 and I 2 , we arrive at (5.10).  a(t, x) and b(t, x) such that Proof. By virtue of estimates (5.1), for arbitrary ε > 0, we get where we set Applying to this last term Lemma (2) with we get We readily show whereas, by Hölder inequality, Collecting the last two estimates, we arrive at Therefore from (5.12) and the generalized Minkowski inequality, we infer which furnishes (5.11).
In the proof of Theorem 2, we will employ the following lemma with a ≡ w(x). However, for the sake of generality, we will show it under the more general assumption a = a(t, x). Then there exists a constant c independent of a(t, x) and b(t, x) such that for all t ∈ (0, T ) . (5.14) Proof. We shall use the arguments employed in the previous lemma. In fact, we may show (5.12) with I(t, τ, x) substituted by Making the same estimate employed for (5.13), we obtain As for J 2 , by employing the assumptions on a, we get which furnishes (5.14).

Proof of Theorem 2 and Corollary 1
The proof of the theorem will be achieved by the following approximation scheme with m ≥ 1: x) , (6.1) whose investigation requires the proof of a number of preliminary lemmas. Set , x). Then, there exists a constant c, independent of v 0 and m ∈ N, such that for the sequence defined by (6.1) the following estimate holds for all t > 0 .
Proof of Thorem 2. In view of the previous lemma, in order to conclude the proof of the theorem, it remains to show the existence of a pressure field satisfying the stated properties, the way in which v assumes the initial data and, finally, the uniqueness of the obtained solution. We begin to associate to the solution v determined in the previous lemma, a pressure field defined by By the theory of singular integrals with weight [18], one shows ||π(t)|| L p α ≤ c|||v| 2 + 2|v||w||| L p α , for all t > 0 , which implies , for all t > 0 . (6.8) Since ||w|| L p α ≤ c [|w|] 2 (6.9) the inequality (2.9) for the pressure follows from (2.8), ((6.8)) and ((6.9)). We also observe that, by virtue of classical arguments, the uniform convergence of the sequence of continuous functions {v m } ensures that the limit function v is continuous in (t, x) ∈ (η, T ) × R 3 , for all η > 0, and also differentiable together with the associated pressure field π defined by (6.7). The existence proof of the pressure is thus completed. We next show the property (2.7). This will be done by using the representation (5.3). In view of (5.8), we only have to show that the integral involving the Oseen term tends weakly to zero in t = 0. Multiplying by ϕ ∈ C 0 this integral and integrating over R 3 , we get (with obvious meaning of the symbols) we get the following estimate: Hence, applying Hölder inequality, we get Analogously, from estimate (6.9) and Hölder inequality, we get