Annihilation of slowly-decaying terms of Navier-Stokes flows by external forcing

Post-refereeing version. To appear in Nonlinearity

Lorenzo Brandolese (1, 2) , Takahiro Okabe

1 ICJ - Institut Camille Jordan
2 EDPA - Équations aux dérivées partielles, analyse

The goal of this paper is to provide an algorithm that, for any sufficiently localised, divergence-free small initial data, explicitly constructs a localised external force leading to a rapidly dissipative solutions of the Navier-Stokes equations R n : namely, the energy decay rate of the flow will be forced to satisfy \( \|u(t)\|_ 2^2 = o(t^{ −(n+2)/2}) \) as \( t\to\infty \), which is beyond the usual optimal rate. An important feature of our construction is that this force can always be taken compactly supported in space-time, and its profile arbitrarily prescribed up to a spatial rescaling. Since the forcing term vanishes after a finite time interval, our result suggests that nontrivial interactions between the linear and nonlinear parts occur, annihilating all the slowly decaying terms contained in Miyakawa and Schonbek's asymptotic profiles.

Mots clés: Navier-Stokes equations; Energy decay; Asymptotic profiles. MSC(2010): 35Q30; 76D05

Domaines: Equations aux dérivées partielles [math.AP]

Lorenzo Brandolese, Takahiro Okabe. Annihilation of slowly-decaying terms of Navier-Stokes flows by external forcing. Nonlinearity, 2021, 34 (3), pp.1733-1757. ⟨10.1088/1361-6544/abdbbf⟩. ⟨hal-03113073⟩