Analyticity and Gevrey-class regularity for the Euler equations

The Euler equations are the classical model for the motion of an incompressible inviscid homogenous fluid. This thesis addresses geometric qualitative properties of smooth solutions to the Euler equations, namely the persistence of analyticity and Gevrey-class regularity on domains with smooth boundary.;The structure of the spectrum of solutions to the three-dimensional Euler equations is a problem of fundamental interest in turbulence theory. The size of the uniform real-analyticity radius of the solution provides an estimate for the scale below which the Fourier coefficients decay exponentially; it moreover gives the rate of this exponential decay. This thesis also addresses the problem of finding sharp lower bounds for the uniform real-analyticity and Gevrey-class regularity radius of the solutions. We prove that the rate of decay of the radius depends at most exponentially on the supremum of the velocity gradient, and algebraically on the higher Sobolev norms of the solution.