Exponential asymptotic stability of a class of dynamical systems with applications to models of turbulent flow in two and three dimensions

Abstract

We consider a class of dynamical systems of the form du/dt + Bu + F(u) = b on a Hilbert space H where the self-adjoint linear operator B is positive with a strictly positive first eigenvalue and b = b0 + b1 such that (b0, Bv) = 0 for all v ∈ H. Given two solutions u and v, we set u − v = w and show that if u(t) → 0 and v(t) → 0 as t → ∞, then in fact eventually w(t) → 0 at an exponential rate. We apply these results to the two-dimensional Navier–Stokes equations (NSEs), the three-dimensional hyperviscous NSEs and the three-dimensional NS-α equations on bounded domains and also establish stability in the sense of Lyapunov; for these systems we assume a condition on b1 to impose decaying turbulence. We also show for the case of decaying turbulence that Leray solutions of the three-dimensional NSEs on bounded domains eventually become regular in addition to decaying to zero. In particular, they eventually satisfy the conditions needed for the abstract stability results.