On the radius of analyticity of solutions to the three-dimensional Euler equations

Abstract:

We address the problem of analyticity of smooth solutions $u$ of the incompressible Euler equations. If the initial datum is real–analytic, the solution remains real–analytic as long as $\int _{0}^{t} \left \Vert {\nabla u(\cdot ,s)}\right \Vert _{L^\infty } ds< \infty$. Using a Gevrey-class approach we obtain lower bounds on the radius of space analyticity which depend algebraically on $\exp {\int _{0}^{t} \left \Vert {\nabla u(\cdot ,s)}\right \Vert _{L^\infty }}ds$. In particular, we answer in the positive a question posed by Levermore and Oliver.