Global existence and decay of the inhomogeneous Muskat problem with Lipschitz initial data

In this work we study the inhomogeneous Muskat problem, i.e. the evolution of an internal wave between two different fluids in a porous medium with discontinuous permeability. In particular, under precise conditions on the initial datum and the physical quantities of the problem, our results ensure the decay of the solutions towards the equilibrium state in the Lipschitz norm. In addition, we establish the global existence and decay of Lipschitz solutions.DA-O is supported by the Alexander von Humboldt Foundation and by the Spanish MINECO through Juan de la Cierva fellowship FJC2020-046032-I. RG-B was supported by the project ‘Mathematical Analysis of Fluids and Applications’ Grant PID2019-109348GA-I00 funded by MCIN/AEI/10.13039/501100011033 and acronym ‘MAFyA’. This publication is part of the Project PID2019-109348GA-I00/AEI/10.13039/501100011033. Project supported by a 2021 Leonardo Grant for Researchers and Cultural Creators, BBVA Fundation

Regularity results for some models in geophysical fluid dynamics

Tesis Doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de lectura: 12-04-2019This thesis centers on the study of two di erent problems of partial di erential equations arising from geophysics and uid mechanics: the surface quasi-geostrophic equation and the so called, Incompressible Slice Model. The surface quasi-geostrophic equation is a two dimensional nonlo- cal partial di erential equation of geophysical importance, describing the evolution of a surface buoyancy in a rapidly rotating, strati ed potential vorticity uid. In the rst part of the talk, we will present some global regularity results for its dissipative analogue in the critical regime for the two dimensional sphere. After that, we will introduce the Incompressible Slice Model deal- ing with oceanic and atmospheric uid motions taking place in a ver- tical slice domain R2, with smooth boundary. The ISM can be understood as a toy model for the full 3D Euler-Boussinesq equa- tions. We will study the solution properties of the Incompressible Slice Model: characterizing a class of equilibrium solutions, establishing the local existence of solutions and providing a blow-up criterion.This thesis has been funded by a Severo Ochoa FPI scholarship for Centres of Excellence in R&D (SEV-2015-0554) and by the grant MTM2017-83496-P from the Spanish Ministry of Economy and Competitiveness

The Burgers' equation with stochastic transport: shock formation, local and global existence of smooth solutions

In this work, we examine the solution properties of the Burgers' equation with stochastic transport. First, we prove results on the formation of shocks in the stochastic equation and then obtain a stochastic Rankine-Hugoniot condition that the shocks satisfy. Next, we establish the local existence and uniqueness of smooth solutions in the inviscid case and construct a blow-up criterion. Finally, in the viscous case, we prove global existence and uniqueness of smooth solutions

Well-posedness for an hyperbolic-hyperbolic-elliptic system describing cold plasmas

In this short note, we provide the well-posedness for an hyperbolic-hyperbolic-elliptic system of PDEs describing the motion of collision free-plasma in magnetic fields. The proof combines a pointwise estimate together with a bootstrap type of argument for the elliptic part of the system

A local-in-time theory for singular SDEs with applications to fluid models with transport noise

In this paper, we establish a local theory, i.e., existence, uniqueness and blow-up criterion, for a general family of singular SDEs in some Hilbert space. The key requirement is an approximation property that allows us to embed the singular drift and diffusion mappings into a hierarchy of regular mappings that are invariant with respect to the Hilbert space and enjoy a cancellation property. Various nonlinear models in fluid dynamics with transport noise belong to this type of singular SDEs. With a cancellation estimate for generalized Lie derivative operators, we can construct such regular approximations for cases involving the Lie derivative operators, or more generally, differential operators of order one with suitable coefficients. In particular, we apply the abstract theory to achieve novel local-in-time results for the stochastic two-component Camassa--Holm (CH) system and for the stochastic C\'ordoba-C\'ordoba-Fontelos (CCF) model

Global well-posedness of critical surface quasigeostrophic equation on the sphere

In this paper we prove global well-posedness of the critical surface quasigeostrophic equation on the two dimensional sphere building on some earlier work of the authors. The proof relies on an improving of the previously known pointwise inequality for fractional laplacians as in the work of Constantin and Vicol for the euclidean settingThis work has been partially supported by ICMAT Severo Ochoa project SEV-2015-0554 and the MTM2011-2281 project of the MCINN (Spain