arXiv subjects

Diego Alonso-Orán

Publications and source records attributed to Diego Alonso-Orán.

27 records · Page 2Linked to original sources

arXiv2022

Global existence and blow-up for a stochastic transport equation with non-local velocity

In this paper we investigate a non-linear and non-local one dimensional transport equation under random perturbations on the real line. We first establish a local-in-time theory, i.e., existence, uniqueness and blow-up criterion for pathwise solutions in Sobolev spaces $H^{s}$ with $s>3$. Thereafter, we give a complete picture of the long time behavior of the solutions based on the type of noise we consider. On one hand, we identify a family of noises such that blow-up can be prevented with probability $1$, guaranteeing the existence and uniqueness of global solutions almost surely. On the other hand, in the particular linear noise case, we show that singularities occur in finite time with positive probability, and we derive lower bounds of these probabilities. To conclude, we introduce the notion of stability of exiting times and show that one cannot improve the stability of the exiting time and simultaneously improve the continuity of the dependence on initial data.

math.AP↗

arXiv2022

On the Grad-Rubin boundary value problem for the two-dimensional magneto-hydrostatic equations

In this work, we study the solvability of a boundary value problem for the magneto-hydrostatic equations originally proposed by Grad and Rubin in the late 50's. The proof relies on a fixed point argument which combines the so-called current transport method together with Hölder estimates for a class of non-convolution singular integral operators. The same method allows to solve an analogous boundary value problem for the steady incompressible Euler equations.

math.AP↗

arXiv2021

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

In this work we study the inhomogeneous Muskat problem, \emph{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.

math.AP↗

arXiv2021

Boundary value problems for two dimensional steady incompressible fluids

In this paper we study the solvability of different boundary value problems for the two dimensional steady incompressible Euler equation. Two main methods are currently available to study those problems, namely the Grad-Shafranov method and the vorticity transport method. We describe for which boundary value problems these methods can be applied. The obtained solutions have non-vanishing vorticity.

math.AP↗

arXiv2020

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 Córdoba-Córdoba-Fontelos (CCF) model.

math.PR↗

arXiv2020

Asymptotic shallow models arising in magnetohydrodynamics

In this paper, we derive a new shallow asymptotic model for the free boundary plasma-vacuum problem governed by the magnetohydrodynamic equation, vital in describing large-scale processes in flows of astrophysical plasma. More precisely, we present the magnetic analogue of the 2D Green-Naghdi equations for water waves in the presence of weakly shared vorticity and magnetic currents. The method is inspired by developed ideas for hydrodynamics flows in by Castro and Lannes (2014) to reduce the $(d+1)$-dimensional dynamics of the problem to a finite cascade of equations which can be closed at the precision of the model.

math.AP↗

arXiv2019

Pointwise monotonicity of heat kernels

In this paper the authors present a proof of a pointwise radial monotonicity property of heat kernels that is shared by the euclidean spaces, spheres and hyperbolic spaces. The main result deals with monotonicity from special points on revolution hypersurfaces from which the aforementioned are deduced. The proof relies on a non straightforward but elementary application of the parabolic maximum principle.

math.CA↗

arXiv2018

Stability, well-posedness and blow-up criterion for the Incompressible Slice Model

In atmospheric science, slice models are frequently used to study the behaviour of weather, and specifically the formation of atmospheric fronts, whose prediction is fundamental in meteorology. In 2013, Cotter and Holm introduced a new slice model, which they formulated using Hamilton's variational principle, modified for this purpose. In this paper, we show the local existence and uniqueness of strong solutions of the related ISM (Incompressible Slice Model). The ISM is a modified version of the Cotter-Holm Slice Model (CHSM) that we have obtained by adapting the Lagrangian function in Hamilton's principle for CHSM to the Euler-Boussinesq Eady incompressible case. Besides proving local existence and uniqueness, in this paper we also construct a blow-up criterion for the ISM, and study Arnold's stability around a restricted class of equilibrium solutions. These results establish the potential applicability of the ISM equations in physically meaningful situations.

math.AP↗

arXiv2018

On the well-posedness of stochastic Boussinesq equations with cylindrical multiplicative noise

The Boussinesq equations are fundamental in meteorology. Among other aspects, they aim to model the process of front formation. We use the approach presented in [Hol15] to introduce stochasticity into the incompressible Boussinesq equations. This is, we introduce cylindrical transport noise in a way that the geometric properties in the Euler-Poincaré formulation are preserved. One of our main results establishes the local well-posedness of regular solutions for these new stochastic Boussinesq equations. We also construct a blow-up criterion and derive some general estimates, which are crucial for showing well-posedness of a wide range of similar SPDEs.

math.AP↗