The Navier-Stokes equation with time quasi-periodic external force:\n existence and stability of quasi-periodic solutions

2020/05/27 by Riccardo Montalto, Montalto, Riccardo
Mathematics · Engineering · #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations #Computational Fluid Dynamics and Aerodynamics

paper · pdf · doi:10.48550/arxiv.2005.13354

Abstract

We prove the existence of small amplitude, time-quasi-periodic solutions\n(invariant tori) for the incompressible Navier-Stokes equation on the\nd-dimensional torus T^d, with a small, quasi-periodic in time external\nforce. We also show that they are orbitally and asymptotically stable in H^s\n(for s large enough). More precisely, for any initial datum which is close to\nthe invariant torus, there exists a unique global in time solution which stays\nclose to the invariant torus for all times. Moreover, the solution converges\nasymptotically to the invariant torus for t \→ + \∞, with an exponential\nrate of convergence O( e^{- \α t} ) for any arbitrary \α \∈ (0,\n1).\n

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