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

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.