Abstract
We prove the existence of small amplitude, time-quasi-periodic solutions (invariant tori) for the incompressible Navier–Stokes equation on the d-dimensional torus mathbb T^d, with a small, quasi-periodic in time external force. We also show that they are orbitally and asymptotically stable in H^s (for s large enough). More precisely, for any initial datum which is close to the invariant torus, there exists a unique global in time solution which stays close to the invariant torus for all times. Moreover, the solution converges asymptotically to the invariant torus for t rightarrow + infty , with an exponential rate of convergence O( e^{- alpha t }) for any arbitrary alpha in (0, 1).
Highlights
Introduction and Main ResultsWe consider the Navier–Stokes equation for an incompressible fluid on the d-dimensional torus Td, d ≥ 2, T := R/2π Z,∂t u − u + u · ∇u + ∇ p = ε f (1.1)div u = 0 where ε ∈ (0, 1) is a small parameter, the frequency ω = (ω1, . . . , ων) ∈ Rν is a νdimensional vector and f : Tν × Td → Rd is a smooth quasi-periodic external force
The purpose of the present paper is to show the existence and the stability of smooth quasiperiodic solutions of the Eq (1.1)
We show the existence of smooth quasi-periodic solutions of small amplitude and we prove their orbital and asymptotic stability in H s for s large enough
Summary
We consider the Navier–Stokes equation for an incompressible fluid on the d-dimensional torus Td , d ≥ 2, T := R/2π Z,. We show the existence of smooth quasi-periodic solutions (which are referred to as invariant tori) of small amplitude and we prove their orbital and asymptotic stability in H s for s large enough (at least larger than d/2 + 1). For the Navier–Stokes equation, unlike in the aforementioned papers on KAM for PDEs, the existence of quasi-periodic solutions is not a small divisors problem and it can be done by using a classical fixed point argument. Since the eigenvalues of Lω are iω · + | j |2, ∈ Zν , j ∈ Zd \{0}, the inverse of Lω gains two space derivatives, see Lemma 3.2 This is suffcient to perform a fixed point argument on the map defined in (3.13) from which one deduces the existence of smooth quasi-periodic solutions of small amplitude. We prefer in this paper to focus on the Navier–Stokes equation for clarity of exposition and since it is a very important physical model
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