Space and time analyticity for inviscid equations of fluid dynamics

TL;DR

This paper proves that solutions to various inviscid fluid equations are complex-analytic in time and space, extending the Gevrey class techniques used for Navier-Stokes to a broader class of models with explicit estimates.

Contribution

It extends the Gevrey class analyticity results from Navier-Stokes to a wide class of inviscid fluid equations, providing explicit bounds on the domain of analyticity.

Findings

Solutions are holomorphic in time with values in a Gevrey class.

Includes equations like Euler, quasi-geostrophic, Boussinesq, MHD.

Provides explicit estimates for the domain of analyticity.

Abstract

We show that solutions to a large class of inviscid equations, in Eulerian variables, extend as holomorphic functions of time, with values in a Gevrey class (thus space-analytic), and are solutions of complexified versions of the said equations. The class of equations we consider includes those of fluid dynamics such as the Euler, surface quasi-geostrophic, Boussinesq and magnetohydrodynamic equations, as well as other equations with analytic nonlinearities. The initial data are assumed to belong to a $\mathit{Gevrey\:class}$, i.e., analytic in the space variable. Our technique follows that of the seminal work of Foias and Temam (1989), where they introduced the so-called Gevrey class technique for the Navier-Stokes equations to show that the solutions of the Navier-Stokes equations extend as holomorphic functions of time, in a complex neighborhood of $(0,T)$, with values in a Gevrey class of functions (in the space variable). We show a similar result for a wide class of inviscid models, while obtaining an $\mathit{explicit \: estimate \: of \: the \: domain \: of \: analyticity}$.