Regularity Properties of the Solution Map of the Incompressible Euler Equations

We study two properties of the data-to-solution map associated to the incompressible Euler equations of fluid motion. First, we show that the maximum norm of the vorticity controls the breakdown of the solution in the Eulerian coordinates for the “little” Hölder space. As a consequence we obtain an extension result for local solutions to the incompressible Euler equations. Second, we prove the non-uniform continuity of the data-to-solution map from bounded subsets of the Besov space into the space of continuous curves landing in the Besov space. In the periodic case two sequences of bounded solutions are shown to converge at time zero and to remain apart at later times. In the non-periodic case we use the initial values of two sequences of bounded approximate solutions (which converge at time zero and remain apart at later times) to construct two sequences of exact solutions to the incompressible Euler equations. Using standard energy estimates for solutions to the incompressible Euler equations and estimates for a solution to a linear Transport equation we show that the distance between the exact and approximate solutions is negligible in the Besov norm. Finally, we show that the exact solutions initially converge and remain separated at later times.