Daniel Peralta-Salas

Instituto de Ciencias Matemáticas, Madrid

Existence of stationary Euler flows with compact support

The dynamics of an inviscid and incompressible fluid flow is described by the Euler equations. In this talk I will be concerned with the existence of compactly supported stationary solutions, which can be easily constructed in 2D using radial stream functions. In 3D the problem turns out to be much harder, and in fact it has been open for decades. In a surprising paper published in 2019, A.V. Gavrilov constructed a smooth stationary 3D Euler flow with compact support. An intriguing (and key) property of Gavrilov’s solution is its “localizability”, i.e., the hydrodynamic pressure is constant along the stream lines of the velocity field. In this talk I will present a new method to construct large families of stationary weak solutions of the 3D Euler equation with compact support. These solutions, which are piecewise smooth and discontinuous across a surface, are not localizable. The proof is based on the construction of solutions to an overdetermined elliptic boundary value problem where one prescribes both Dirichlet and (nonconstant) Neumann data. This is based on joint work with Miguel Dominguez and Alberto Enciso.