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Tesis Doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de lectura: 23-06-2017

Tesis doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de lectura: 16-07-2021, In this dissertation we analyze several problems related to unstable interfaces in uid mechanics: the vortex sheet problem and both the fully unstable and partially unstable Muskat problem. Moreover, we present a quantitative homotopy principle with applications to evolution equations within the framework of the technique we will employed, the convex integration method. In chapter 1 we introduce the partial di erential equations that we will study, the problems that have motivated this dissertation, the results we have obtained in [26, 103, 102, 27] and, nally, the convex integration method and the homotopy principle. The chapter 2 is devoted to gather some common aspects of the problems we analyze. This allows us to introduce the \turbulence zone" where the uid behaves wildly. This zone starts to grow around the unstable region of the interface. Moreover, we present some lemmas related to the Birkho -Rott and Muskat operators, which will be crucial within the proofs of the results in chapters 4-6. These lemmas can be found in [103], joint work with L aszl o Sz ekelyhidi, and in [27], joint work with Angel Castro and Daniel Faraco. In chapter 3 we present a quantitative version of the homotopy principle for a class of evolution equations. In the previous versions ([52]) weak solutions are recovered from a \subsolution" through a convex integration scheme. This quantitative version recovers also microscopic and macroscopic information. On the one hand, it shows that the uid can behave wildly inside the turbulence zone. On the other hand, it measures the proximity, in terms of weak -continuous quantities, between the weak solutions obtained and the subsolution. This allows to select those which retain more information from the subsolution, thus emphasizing the role of the subsolution as the candidate for the macroscopic solution. These results are based on [26], joint work with Angel Castro and Daniel Faraco. In chapter 4 we study the vortex sheet problem for the incompressible Euler equation. We construct in nitely many admissible solutions starting from vorticities without xed sign and concentrated on non-analytic curves. The existence of weak solutions for vortex sheet data with mixed sign was an open problem from the work of Delort ([56]) and was only known for particular initial data ([94]). These weak solutions are smooth outside a turbulence zone which grows linearly in time around the vortex sheet. Furthermore, this approach shows how the growth of the turbulence zone is controlled by the local energy inequality, and measures the maximal initial dissipation rate in terms of the vortex sheet strength. These results appear in [103], joint work with L aszl o Sz ekelyhidi. In chapter 5 we prove a homotopy principle for the incompressible porous media (IPM) equation with density-viscosity jump. As a rst example, non-trivial weak solutions with compact support in time are obtained (see [39] for the case of constant viscosity). Secondly, we construct mixing solutions to the unstable Muskat problem with initial at interface. As a byproduct, we check that the connection, established by Sz ekelyhidi for the case of constant viscosity ([126]), between the subsolution and the Lagrangian relaxed solution of Otto ([116]), holds for the case of viscosity jump as well. In this case we show how a pinch singularity in the relaxation prevents the two uids from mixing wherever there is neither Rayleigh-Taylor instability nor vorticity at the interface. In chapter 6 we construct mixing solutions to the incompressible porous media equation starting from Muskat type data in the partially unstable regime. In particular, we consider bubble and turned type interfaces with Sobolev regularity. As a by-product, we prove the continuation of the evolution of IPM after the Rayleigh-Taylor and smoothness breakdown exhibited in [24, 23]. At each time slice the space is split into three evolving domains: two non-mixing zones and a mixing zone which is localized in a neighborhood of the unstable region. In this way, we show the compatibility between the classical Muskat problem and the convex integration method. These results appear in [27], joint work with Angel Castro and Daniel Faraco

Tesis Doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de lectura: 22-02-2019