Resumen de Instabilities in fluid mechanics and convex integration

Francisco Mengual Bretón

• The version of the convex integration method introduced in Hydrodynamics by De Lellis and Székelyhidi has been proved to be a very powerful tool in the last decade. In this dissertation we apply this method to solve several problems related to unstable interfaces by creating a "turbulence zone'' around the unstable region of the interface.

  Firstly, jointly with Ángel Castro and Daniel Faraco, we present a quantitative version of the homotopy principle for a class of evolution equations. This version allows us to recover microscopic and macroscopic information of the solutions from the "subsolution'' inside the turbulence zone. In spite of the lack of uniqueness and regularity of the solutions inside the turbulence zone, we show that these solutions are essentially indistinguishable from the subsolution at a macroscopic level.

  Secondly, we study the vortex sheet problem for the incompressible Euler equation. We construct, jointly with László Székelyhidi, infinitely many admissible solutions starting from vorticities without fixed sign and concentrated on non-analytic curves. Moreover, we show the relation between the dissipation of energy and the growth of the turbulence zone.

  Thirdly, we study the Muskat problem for the incompressible porous media equation. For the case of different densities and viscosities, we prove an h-principle which allows us to relate the subsolution of the planar interface with the Lagrangian relaxed solution of Otto. For the case of different densities and constant viscosity, we construct, jointly with Ángel Castro and Daniel Faraco, mixing solutions starting from the partially unstable regime. To do this, we combine the parabolic analysis in the stable region with the convex integration method by localizing the mixing zone around the unstable region.