Space and time analyticity for inviscid equations of fluid dynamics

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 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 𝒆𝒙𝒑𝒍𝒊𝒄𝒊𝒕 𝒆𝒔𝒕𝒊𝒎𝒂𝒕𝒆 𝒐𝒇 𝒕𝒉𝒆 𝒅𝒐𝒎𝒂𝒊𝒏 𝒐𝒇 𝒂𝒏𝒂𝒍𝒚𝒕𝒊𝒄𝒊𝒕𝒚.