We prove the existence of small amplitude, time-quasi-periodic solutions (invariant tori) for the incompressible Navier-Stokes equation on the d-dimensional torus 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 ->+infinity, with an exponential rate of convergence O(e(-alpha t)) for any arbitrary alpha is an element of(0, 1).

The Navier–Stokes Equation with Time Quasi-Periodic External Force: Existence and Stability of Quasi-Periodic Solutions / R. Montalto. - In: JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS. - ISSN 1040-7294. - 33:3(2021 Sep), pp. 1341-1362. [10.1007/s10884-021-09944-w]

The Navier–Stokes Equation with Time Quasi-Periodic External Force: Existence and Stability of Quasi-Periodic Solutions

R. Montalto
2021-09

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 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 ->+infinity, with an exponential rate of convergence O(e(-alpha t)) for any arbitrary alpha is an element of(0, 1).
Asymptotic and orbital stability; Fluid dynamics; Navier–Stokes equation; Quasi-periodic solutions
Settore MAT/07 - Fisica Matematica
Settore MAT/05 - Analisi Matematica
27-gen-2021
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2434/859766
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