Elliptic and Parabolic Equations - Abstract

Skalák, Zdeněk

Some recent results on the large-time energy concentration in the solutions to the Navier--Stokes equations

Let $Omega subseteq mathbfR^3$ be a uniformly regular domain of the class $C^3$ or $Omega = mathbfR^3$. Let $A$ denote the Stokes operator and $ E_lambda; lambda ge 0 $ be the resolution of identity of $A$. If $w$ is a nonzero global weak solution to the Navier--Stokes equations in $Omega$ satisfying the strong energy inequality, then there exists a nonnegative finite number $a=a(w)$ such that for every $varepsilon > 0$ begineqnarray nonumber && lim_t rightarrow infty frac (E_a+varepsilon-E_a-varepsilon) w(t) w(t) = 1, endeqnarray where we put $E_a-varepsilon = 0$ if $a-varepsilon < 0$. Therefore, every nonzero global weak solution satisfying the strong energy inequality exhibits asymptotic energy concentration in a particular frequency. \\The number $a$ is connected with the rate of the decay of $w$ in the following way: $a>0$ if and only if $w$ decreases exponentially (i.e. there exists some $lambda>0$ such that $lim_t rightarrow infty e^lambda t w(t) = 0$). beginthebibliography99 bibitemOk T. Okabe, Asymptotic energy concentration in the phase space of the weak solutions to the Navier--Stokes equations, J. Differential Equations bf 246 (2009), 895-908. bibitemSk1 Z. Skalák, On asymptotic dynamics of solutions of the homogeneous Navier--Stokes equations, Nonlinear Analysis bf 67 (2007), 981--1004. bibitemSk5 Z. Skalák, Large time behavior of energy in exponentially decreasing solutions of the Navier--Stokes equations, it Nonlinear Analysis bf 71 (2009), 593--603. endthebibliography