Invariant Manifolds and the Long-Time Asymptotics of the Navier-Stokes and Vorticity Equations on R2

By Thierry GallayUniversit. 2001

236 citations recorded by the source corpus.

Abstract We construct finite-dimensional invariant manifolds in the phase space of the Navier-Stokes equation on R2 and show that these manifolds control the long-time behavior of the solutions. This gives geometric insight into the existing results on the asymptotics of such solutions and also allows us to extend those results in a number of ways.

Subfields: Applied Mathematics, Computational Mechanics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9cecab71b6cc8fdd39f6c3ed8ea3ccfbf6bef336

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In Applied Mathematics

In Computational Mechanics

In Mathematical Physics

Citations in the corpus

ASYMPTOTIC BEHAVIOR OF THE ENERGY AND POINTWISE ESTIMATES FOR SOLUTIONS TO THE NAVIER-STOKES EQUATIONS (2004, methodology): “For a detailed discussion on this point, and the description of a method (very different from ours) to overcome this type of difficulties, we refer to [13], [14].”
On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation (2005, background): “Related problems have been solved, in a much simpler but already delicate setting, by Gallay and Wayne [28] in the context of mathematical fluid mechanics.”
Hypocoercivity (2006, methodology): “Oseen vortices are certain self-similar solutions to the two-dimensional incompressible Navier–Stokes equation, in vorticity formulation [24, 25].”
Global Stability of Vortex Solutions of the Two-Dimensional Navier-Stokes Equation (2004, methodology): “in L1(R2) behave asymptotically as the solutions of the linear heat equation ∂ tω= ∆ωwith the same initial data. For small solutions, this result has been obtained by Giga and Kambe in [17], see also [14]. As was observed by Carpio [10] (see also [16]), there is a deep connection between the asympto”
Existence, Uniqueness and Lipschitz Dependence for Patlak-Keller-Segel and Navier-Stokes in R2 with Measure-valued Initial Data (2012, methodology): “With the convention we are taking, we have that in L(2)0(m), σess(S(τ )) ⊂ { λ ∈ C : |λ| eτ(1−m)/2} [35].”
The Navier-Stokes Equations in ℝn with Linearly Growing Initial Data (2005, methodology): “Notice also that in the special case where M = Id, (L–L )-smoothing estimates as well as gradient estimates for e−tA were obtain by Gallay & Wayne [8].”