Methods and Applications of Analysis

Volume 22 (2015)

Number 4

Gevrey regularity for a class of dissipative equations with analytic nonlinearity

Pages: 377 – 408

DOI: http://dx.doi.org/10.4310/MAA.2015.v22.n4.a3

Authors

Hantaek Bae (Department of Mathematical Sciences, Ulsan National Institute of Science and Technology (UNIST), Korea)

Animikh Biswas (Department of Mathematics and Statistics, University of Maryland, Baltimore County, Md., U.S.A.)

Abstract

In this paper, we establish Gevrey class regularity of solutions to a class of dissipative equations with an analytic nonlinearity in the whole space. This generalizes the results of Ferrari and Titi in the periodic space case with initial data in $L^2$-based Sobolev spaces to the $L^p$ setting and in the whole space. Our generalization also includes considering rougher initial data, in negative Sobolev spaces in some cases including the Navier-Stokes and the subcritical quasi-geostrophic equations, and allowing the dissipation operator to be a fractional Laplacian. Moreover, we derive global (in time) estimates in Gevrey norms which yields decay of higher order derivatives which are optimal. Applications include (temporal) decay of solutions in higher Sobolev norms for a large class of equations including the Navier-Stokes equations, the subcritical quasi-geostrophic equations, nonlinear heat equations with fractional dissipation, a variant of the Burgers’ equation with a cubic or higher order nonlinearity, and the generalized Cahn-Hilliard equation. The decay results for the last three cases seem to be new while our approach provides an alternate proof for the recently obtained $L^p (1 \lt p \lt 2)$ decay result for the Navier–Stokes equations by Bae, Biswas and Tadmor. These applications follow from our global Gevrey regularity result for initial data in critical spaces with low regularity.

Keywords

dissipative equations, gevrey regularity, temporal decays

2010 Mathematics Subject Classification

Primary 35Q35, 76D03, 76D05. Secondary 35J60, 76F05.

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Published 20 January 2016