Time almost-periodic solutions of the incompressible Euler equations

We construct time almost-periodic solutions (global in time) with finite regularity to the incompressible Euler equations on the torus $\T^d$, with $d=3$ and $d\in\N$ even.


Introduction
The goal of this paper is to construct time almost-periodic solutions (infinite dimensional invariant tori) of the Euler equations B t u `u ¨∇u `∇p " 0 , div u " 0 , u : R ˆTd Ñ R d , p : R ˆTd Ñ R , (1.1) on the d-dimensional torus T d , T :" R{2πZ, where either d " 3 or d ě 2 is any even positive integer.These solutions extend the works ot Crouseille & Faou [17] (in dimension 2) and Enciso, Peralta Salas & Torres de Lizaur [18] (in dimensions 3 or even) from time quasi-periodic to time almost-periodic solutions.In fact, the construction here follows closely the one in [18].We need to specify how a smooth solution of the Euler equations (1.1) is called almost-periodic in this paper.We need some preliminaries.
Let C s div pT d , R d q, with s P N Y t`8u, be the space of C s -smooth, divergence free d-dimensional vector fields on T d .This space is a Banach space if s ă 8 and a Fréchet space when s " 8.We endow it with the system of seminorms p} ¨}n,8 q nPt0,1,...,su defined by }f } n,8 :" sup |B α x f pxq| , n " 0, 1, ..., s ; throughout the paper, for sake of simplicity in the notation, | ¨| denotes the standard Euclidean norm, without specifying the dimension of the evaluated object, which will be clear from the context each time.We denote by ℓ 8 pN, Nq the set of sequences in N that are bounded.Let pJ k q kPN P ℓ 8 pN, Nqzt0u be given and, for a fixed m P t1, ..., d ´1u, we define the sequence pN k q kPN P ℓ 8 pN, Nq , N k :" pd ´mqJ k P N , k P N . (1.2) We define the infinite dimensional torus pT N k q kPN and its "tangent space" pR N k q kPN as pT N k q kPN :" θ " pθ k q kPN : θ k P T N k , |θ| 8 ă 8 ( , where we defined |ν| 8 :" sup kPN |ν k |.Note that, since pN k q kPN is bounded, then |θ| 8 ď p2πq }pN k q} ℓ 8 ă 8 for any sequence θ " pθ k q kPN .Definition 1.1.Let s P N Y t`8u.We say that upt, xq is time almost-periodic if there exists a sequence of vectors ν P pR N k q kPN and a C 1 -smooth embedding U : pT N k q kPN Ñ C s div pT d , R d q such that the velocity field upt, xq can be written as upt, ¨q " Upϑq| ϑ"θ`νt , for some θ P pT and the sequence of frequency vectors ν " pν k q kPN is non-resonant, meaning that where, for a fixed η ą 0, we define |ℓ| η :" ř kPN k η |ℓ k |.Note that |ℓ| η ă 8 implies that ℓ k ‰ 0 P Z N k only for finitely many k P N. Definition 1.2.By saying that the map U : pT N k q kPN Ñ C s div pT d , R d q is C 1 b (C 1 and bounded), we mean that, U is a Frechet-differentiable map with continuous Frechet derivative and for any n P t0, 1, . . ., su, there exists a constant C n ą 0 such that where the linear operator d ϑ Upϑq : With this definition of a C 1 b embedding, we have that the function upt, ¨q is C 1 with respect to t P R, by (1.3) and B t upt, ¨q " d ϑ Upθ `νtqrνs P C s div pT d , R d q.We will look for solutions where the embedding U is non-symmetric, or nontraveling, in the sense that, for any ϑ P pT N k q kPN , the divergence-free vector field Upϑq is not invariant under any 1-parameter group of translations on T d .In this way, we ensure that the solution upt, xq depends effectively on all d coordinates and we do not have any reduction to solutions of lower dimensions by traveling directions.
The statement of the main result is the following.
Theorem 1.3.(Time almost-periodic solutions of the Euler equations).Assume that the dimension d is either 3 or even.Let S P N be fixed.There exists ε 0 P p0, 1q small enough such that, for any ε P p0, ε 0 q and for any sequence of frequencies ν P pR N k q kPN zt0u satisfying sup there exists a non-symmetric C 1 b embedding U : pT N k q kPN Ñ C S div pT d , R d q and a family of initial data u θ P C S div pT d , R d q, θ P pT N k q kPN , such that upt, ¨q " Upθ `νtq, with up0, ¨q " u θ , is a solution of (1.1) with pressure ppt, ¨q " P pθ `νtq, where As a consequence, if the sequence ν " pν k q kPN is non-resonant, namely it satisfies (1.4), then the solution upt, xq is time almost-periodic.
Remark 1.4.As it will be clear from the construction in the following section, the embedding Upϑq is determined as a combination of infinitely many embedding U k pϑ k q, with ϑ k P T N k , which coincides with the embedding constructed in [18], with the size of the embedding U k becoming smaller and smaller as k Ñ 8.The major difference in the analysis with respect to [18] is that we have to effectively prove the smoothness of the embedding and the regularity of the vector field.This is not trivial.
Remark 1.5.The condition (1.4) of irrationality for the sequence of frequencies ν P pR N k q kPN zt0u is not necessary in the construction of the embedding U. Depending on relations between all the frequencies, we may obtain embedding for lower dimensional tori, either finite dimensional (quasi-periodic or periodic) or still infinite dimensional (that is, almost-periodic).On the other hand, the control on the frequency vectors in (1.5) is required to ensure that the solution upt, xq is indeed a finitely smooth vector field and a simpler control on the norm |ν| 8 is not enough.At the physical level, is also implies that we obtain solutions whose leading order frequencies of oscillations are only finitely many and the almost-periodicity in time is due to the presence of infinitely oscillations with smaller and smaller frequencies.
Related results.In the last years, there has been a discrete surge of works proving the existence of time quasi-periodic waves for PDEs arising in fluid dynamics.With the exception of the aforementioned works [17] and [18], there type of results in literature are proved by means of KAM for PDEs techniques, to deal with the presence of small divisors issues and consequent losses of regularity.For the two dimensional water waves equations, we mention Berti & Montalto [7], Baldi, Berti, Haus & Montalto [2] for time quasi-periodic standing waves and Berti, Franzoi & Maspero [4], [5], Feola & Giuliani [19] for time quasi-periodic traveling wave solutions.Recently, the existence of time quasi-periodic solutions was proved for the contour dynamics of vortex patches in active scalar equations.We mention Berti, Hassainia & Masmoudi [6] for vortex patches of the Euler equations close to Kirchhoff ellipses, Hmidi & Roulley [27] for the quasi-geostrophic shallow water equations, Hassainia, Hmidi & Masmoudi [24] for generalized surface quasi-geostrophic equations, Roulley [31] for Euler-α flows, Hassainia & Roulley [26] for Euler equations in the unit disk close to Rankine vortices and Hassainia, Hmidi & Roulley [25] for 2D Euler annular vortex patches.Time quasi-periodic solutions were also constructed for the 3D Euler equations with time quasi-periodic external force [3] and for the forced 2D Navier-Stokes equations [20] approaching in the zero viscosity limit time quasi-periodic solutions of the 2D Euler equations for all times.
The existence of other non-trivial invariant structures is also a topic of interest in fluid dynamics.In particular, for the Euler equations in two dimension close to shear flows, we mention the works by Lin & Zeng [28] and Castro & Lear [11] for periodic traveling waves close the Couette flow, by Coti Zelati, Elgindi & Widmayer [16] for stationary waves around non-monotone shears, by Franzoi, Masmoudi & Montalto [21] for quasi-periodic traveling waves close to the Couette flow, and the recent work by Castro & Lear [12] for time periodic rotating solutions close to the Taylor-Couette flow.
Concerning the existence of almost periodic solutions by means of KAM methods, we mention Pöschel [30], Bourgain [10], Biasco-Massetti-Procesi [8], [9] and Corsi-Gentile-Procesi [14].In all these results the authors consider semilinear NLS type equations with external parameters.For PDEs with unbounded perturbations (with external parameters as well) we mention Montalto-Procesi [29] and Corsi-Montalto-Procesi [15].We remark that our result is the first one concerning existence of almost-periodic solutions for an autonomous quasi-linear PDEs in higher space dimension and it is obtained with non-KAM techniques.

Acknowledgments.
The ‚ a À b stands for a ď Cb, for some constant C ą 0; ‚ a À n b stands for a ď C n b, for some constant C n ą 0 depending on n.

Proof of Theorem 1.3
The scheme follows essentially the one proposed in [18], with the required adaptations.
The key starting point is the existence of smooth, compactly supported stationary solutions of the Euler equations.In d " 3, this is celebrated result by Gravilov [22] (see also [13]), whereas in even dimension it has been proved in [18].We recall the statement of the result of the latter.Proposition 2.1.(Smooth stationary Euler flows with compact support -Proposition 2, [18]).If d " 3 or d P N is even, there exists a smooth, compactly supported solution vpxq P C 8 div pR d , R d q, with pressure p v pxq P C 8 pR d q, of v ¨∇v `∇p v " 0 , div v " 0 , x P R d . (2.1) Without any loss of generality, we assume that sptpvq, sptpp v q Ď B d,1 p0q Ă R d .Then, given S P N and for any k P N, we define the rescaled functions, for any ε P p0, 1q small enough, v k pxq :" ε pS`1qpk´1q vpε ´kxq , p v k pxq :" ε 2pS`1qpk´1q p v pε ´kxq .
A straightforward computation shows that v k pxq P C 8 div pR d , R d q is also a solution of (2.1) with pressure p v k pxq P C 8 pR d q with compact support and we have the control on the seminorms, for any n P N 0 , for some constant C n ą 0 independent of ε P p0, 1q and k P N. We remark that, as soon as n ą S `1, the seminorms }v k } n,8 start to diverge with respect to k Ñ 8 as ε ´pn´S´1qk for ε P p0, 1q, whereas the seminorms }p v k } n,8 start to diverge when n ą 2S `2.
Moreover, for ε P p0, 1q small enough and independent of k P N, we define the periodicized versions We recall the sequence pN k q kPN P ℓ 8 pN, Nq in (1.2) is determined by a fixed m P t1, ..., d´1u and a fixed sequence pJ k q kPN P ℓ 8 pN, Nq.The choice of m P t1, ..., d´1u induces the splitting T d " T m ˆTd´m and we write We select a sequence of points py k,j q kPN, j"1,..,J k Ă T m with the properties that: (A) For any k 1 , k 2 P N, j 1 " 1, ..., J k 1 , j 2 " 1, ..., J k 2 , with pk 1 , j 1 q ‰ pk 2 , j 2 q, we have The existence of such sequence of points with these desired properties is proved in the following lemma.Lemma 2.2.There exist ε 0 " ε 0 `m, }pJ k q} ℓ 8 ˘P p0, 1q small enough and a choice of infinitely many distinct points py k,j q kPN, j"1,..,J k Ă T m , such that the following holds.For ε ą 0, we define iteratively the sets Then, for any ε P p0, ε 0 q and for any k P N, we have: As a consequence, conditions (A) and (B) are satisfied.
We now assume that the claims piq, piiq and piiiq are satisfied for some k P N and we prove them for k `1.We set (2.4) By (2.4), there exist J k`1 distinct points y k`1,1 , ..., y k`1,J k`1 P T m zE k , with J k`1 ď }pJ k q} ℓ 8 , such that, for any ε P p0, ε 0 q we have that the J k`1 balls B m,2ε k`1 py k`1,1 q, ..., B m,2ε k`1 py k`1,J k`1 q are contained in T m zE k and they are disjoint, namely they satisfy items piq and piiq at the step k `1.This follows from the fact that, by the induction assumption on piiiq, we have that , whereas the measure of the finite union of closed disjoint balls is estimated, for any ε P p0, ε 0 q with ε 0 as in (2.4), by which implies the existence of the J k`1 points y k`1,1 , ..., y k`1,J k`1 in the open and bounded set T m zE k with the desired properties.Therefore, let E k`1 be defined as in (2.3).Clearly, E k`1 is closed, which also implies that T m zE k`1 is open.By (2.5) and item piiq at the step k `1, we also deduce that, which is indeed the estimate in item piiiq at the step k `1.This closes the induction argument and concludes the proof.
As a last preliminary, we take a sequence of frequency vectors ν " pν k q kPN P pR N k q kPN , where ν k " pν k,1 , ..., ν k,J k q P R N k , with ν k,j P R d´m , recalling (1.2).We now define the vector field upt, xq :" where v k,j pt, xq :" v k px 1 ´yk,j , x 2 ´νk,j tq , p k,j pt, xq :" p v k px 1 ´yk,j , x 2 ´νk,j tq , (2.8) and w k pxq " p0, F k px 1 qq , F k : T m Ñ R d´m . (2.9) Note that, no matter the choice of F k px 1 q sufficiently smooth is, the vector field w k pxq is a stationary solution with constant pressure of the Euler equations (1.1), namely we have w k ¨∇w k " 0 , div w k " 0 .
To make sure that upt, xq is indeed a solution of (1.1), we need to specify the functions F k px 1 q.In particular, we choose where χ k prq P C 8 pRq is an even cut-off function satisfying for some constant C n ą 0. For each k P N, we have that F k P C 8 pT m , R d´m q and that the vector field w k pxq is locally equal to p0, ν k,j q when x P sptpv k,j q.Note that each pair pu k pt, xq, p u k pt, xqq defined above by (2.6)-(2.10)has actually the form of a quasi-periodic solution of the Euler equations (1.1) as provided in [18], which has been reproduced here on supports of scale ε k .Moreover, by construction and by (A), the support in space of pu k pt, xq, p u k pt, xqq is in `ŤJ k j"1 B m,2ε k py k,j q ˘ˆT d´m and it is disjoint from the one of pu k 1 pt, xq, p u k 1 pt, xqq for any k 1 ‰ k.We use these properties to check that the pair pupt, xq, p u pt, xqq in (2.6)-(2.7) is indeed a solution of (1.1) as well.
First, we prove that each pair pu k pt, xq, p u k pt, xqq is a solution of (1.1) and we provide estimates on the seminorms.Lemma 2.3.Assume that ν " pν k q kPN satisfies (1.5).For each k P N, the vector field u k pt, xq is in C 8 div pT d , R d q, with pressure p u k pt, xq in C 8 pT d q, is a solution of the Euler equations (1.1), namely compactly supported in space in Ť J k j"1 B m,2ε k py k,j q ˆTd´m .Moreover, we have the estimates, for any integer n ě 0, ν k,j ¨∇2 v k px 1 ´yk,j , x 2 ´νk,j tq , (2.14) Proposition 2.5.Assume that ν " pν k q kPN satisfies (1.5).Then the embedding U : This implies the claimed estimate and concludes the proof.
work of the authors Luca Franzoi and Riccardo Montalto is funded by the European Union, ERC STARTING GRANT 2021, "Hamiltonian Dynamics, Normal Forms and Water Waves" (HamDyWWa), Project Number: 101039762.Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Council.Neither the European Union nor the granting authority can be held responsible for them.The work of the author Riccardo Montalto is also supported by PRIN 2022 "Turbulent effects vs Stability in Equations from Oceanography", project number: 2022HSSYPN.Riccardo Montalto is also supported by INDAM-GNFM.Notations.In this paper, we use the following notations: ‚ B d,ρ ppq :" tx P R d : |x ´p| ă ρu, with p P R d and ρ ą 0 sup tPR }u k pt, ¨q} n,8 ď C n ε kpS`1´nq´S´1 , sup tPR }B t u k pt, ¨q} n,8 ď C n ε kp2S`2´pn`1qq´2S´2 , sup tPR }p u k pt, ¨q} n,8 ď C n ε kp2S`2´nq´2S´2 , (2.13)for some constant C n ą 0 independent of ε P p0, 1q and of k P N.Proof.By (2.6)-(2.10),Proposition 2.1 and by (A), we computeB t u k " ´Jk ÿ j"1 to Definition 1.2, with estimatessup ϑPpT N k q kPN }Upϑq} n,8 ď C n ε ´S´1 , 0 ď n ď S ,(2.18)sup ϑPpT N k q kPN }d ϑ Upϑqr p ϑs} n,8 ď C n ε ´S´1 | p ϑ| 8 @ p ϑ P pR N k q kPN , 0 ď n ď S .Proof.The first estimate in (2.18) follows by (2.17) and(2.16).We now prove the second estimate in(2.18).By (2.17), we compute, for any ϑ P pT N k q kPN and p ϑ P pR N k q kPN , ¨∇2 v k px 1 ´yk,j , x 2 ´ϑk,j q .(2.19) Therefore, by (2.2), using the fact that each term in the series in(2.19) is supported in space on the cylinder B m,2ε k py k,j q ˆTd´m , that these supports are disjoint one from the other, and that | p ϑ k | ď | p ϑ| 8 , we obtain, for any ϑ P pT N k q kPN , p ϑ P pR N k q kPN and for any n " 0, 1, ..., S, }d ϑ Upϑqr p ϑs} n,8 ď sup kPN sup j"1,...,J k }∇ 2 v k p ¨´y k,j , ¨´ϑ k,j q} n,8 | p ϑ| 8 ď sup kPN sup j"1,...,J k }v k p ¨´y k,j , ¨´ϑ k,j q} n`1,8 | p ϑ| 8 ď C n sup kPN ε kpS`1´pn`1qq´S´1 | p ϑ| 8 ď C n ε ´S´1 | p ϑ| 8 .