Dynamics of Partial Differential Equations

Volume 11 (2014)

Number 3

Log-Lipschitz continuity of the vector field on the attractor of certain parabolic equations

Pages: 211 – 228

DOI: http://dx.doi.org/10.4310/DPDE.2014.v11.n3.a1

Authors

Eleonora Pinto de Moura (Instituto de Matemática, Universidade Federal do Rio de Janeiro, Brazil)

James C. Robinson (Mathematics Institute, University of Warwick, Coventry, United Kingdom)

Abstract

We discuss various issues related to the finite-dimensionality of the asymptotic dynamics of solutions of parabolic equations. In particular, we study the regularity of the vector field on the global attractor associated with these equations. We show that if the linear term associated with certain dissipative partial differential equations is log-Lipschitz continuous on the global attractor $\mathcal{A}$, then $\mathcal{A}$ lies within a small neighbourhood of a smooth manifold, given as a Lipschitz graph over a finite number of Fourier modes. In this case, $\mathcal{A}$ can be shown to have zero Lipschitz deviation and, therefore, there are linear maps $L$ into finite-dimensional spaces, whose inverses restricted to $L\mathcal{A}$ are Hölder continuous with an exponent arbitrarily close to one. Finally, we use an argument due to Kukavica (2007; Proc. Amer. Math. Soc. 135 2415-2421) to prove that the linear term associated with a class of parabolic equations, that includes the 2D Navier-Stokes equations, is 1-log-Lipschitz continuous.

Keywords

log-Lipschitz continuity, attractor, parabolic equations

2010 Mathematics Subject Classification

34-xx, 35-xx, 37-xx

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