Estabilidade não/linear de soluções estacionárias das equações de Euler incompressíveis com simetria helicoidal

Abstract

In this work, we approach issues regarding weak solutions existences and nonlinear stability for the incompressible Euler equations. More precisely, we analyze two distinct issues within these topics. At Ąrst, we consider the Euler equations with helical symmetry and with no swirl. Then, we use the reduction through symmetry to extend the stability techniques developed by Burton and by Wan and Pulvirenti to the helical case. Consequently, for a simply connected, bounded in horizontal planes and smooth helical domain, we prove that the strict maximiser of kinetic energy relative to all rearrangement of an arbitrary helical function in 𝐿 𝑝 is a steady and 𝐿 𝑝 -stable helical vorticity. Furthermore, in a cylindrical domain, we also prove that there exists a steady and 𝐿 1 -stable helical vorticity which can be seen as an extension of the circular vortex patch. On the second issue, we consider the two-dimensional Euler equations in the domain R 2 and with initial data that do not decay at inĄnity. We show that initial vortex patches covered by Serfati Existence of Solutions Theorem (that is, solutions with bounded velocities and vorticities) cannot contain arbitrarily large balls. In addition, we construct a counterexample of a vortex patch for which there exists an associated bounded velocity and such that there exists a subset in which the vortex patch does not have any associated bounded velocity.