Skip to main content Accessibility help
×
Home

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

  • Joel Avrin (a1)

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 vH. Given two solutions u and v, we set uv = 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.

Copyright

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed