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Manuel Fernando Cortez

Publications and source records attributed to Manuel Fernando Cortez.

12 recordsLinked to original sources
arXiv2026

On Stationary Gevrey Solutions to the Gravitational Boussinesq System and Applications to Uniqueness

The stationary version of the Boussinesq system with a general gravitational acceleration term is considered. Under suitable assumptions on this term, as well as on the external forces acting on each equation of this coupled system, we first establish the existence of weak solutions in the natural energy space $\dot{H}^1(\mathbb{R}^3)$. The uniqueness of these solutions is a challenging open problem. Within this framework, our first main contribution is to show that \emph{any} weak $\dot{H}^1$-solution exhibits an analytic smoothing effect in the Gevrey class. Our second main contribution is to show that the Gevrey class regularity can also be used to study the uniqueness problem, provided that these solutions satisfy a suitable low-frequency control. As a by-product, we also obtain new regularity results and a \emph{new Liouville-type result} for weak $\dot{H}^1$-solutions of the classical Navier--Stokes equations.

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arXiv2025

Remarks on the Spatial Asymptotic Behavior of Solutions to a 1D Model of Equatorial Oceanic Flows

We consider a new nonlocal and nonlinear one-dimensional evolution model arising in the study of oceanic flows in equatorial regions, recently derived in [A. Constantin and L. Molinet, Global Existence and Finite-Time Blow-Up for a Nonlinear Nonlocal Evolution Equation, Commun. Math. Phys. 402 (2023), 3233-3252]. We investigate the spatial asymptotic behavior of its solutions. In particular, we observe the influence of the Coriolis effect, which, even for rapidly decaying initial data, yields solutions that decay at the rate $1 / |x|$. Thereafter, we shed light on the optimality of this decay rate.

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arXiv2025

Sharp Well-Posedness and Parameter Asymptotics for a Nonlocal Model of Thin Film Flows

This work focuses on the mathematical analysis of the Cauchy problem associated with a two-dimensional equation describing the dynamics of a thin fluid film flowing down an inclined flat plate under the influence of gravity and an electric field. As a first objective, we study the sharp global well-posedness of solutions within the framework of Sobolev spaces. Specifically, we show that the equation is well-posed in $H^s(\mathbb{R}^2)$ for $s>-2$, and ill-posed for $s<-2$. Our main contribution is to investigate the asymptotic behavior of solutions with respect to the physical parameters of the model. This behavior is not only of mathematical interest but also of physical relevance. We rigorously show that, as the effects of the electric field vanish and the inclination angle increases, the model converges to a related one describing thin film flow down a vertical plane.

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arXiv2025

Mathematical study of a new Navier-Stokes-alpha model with nonlinear filter equation -- Part I

This article is devoted to the mathematical study of a new Navier-Stokes-alpha model with a nonlinear filter equation. For a given indicator function, this filter equation was first considered by W. Layton, G. Rebholz, and C. Trenchea to select eddies for damping based on the understanding of how nonlinearity acts in real flow problems. Numerically, this nonlinear filter equation was applied to the nonlinear term in the Navier-Stokes equations to provide a precise analysis of numerical diffusion and error estimates. Mathematically, the resulting alpha-model is described by a doubly nonlinear parabolic-elliptic coupled system. We therefore undertake the first theoretical study of this system by considering periodic boundary conditions in the spatial variable. Specifically, we address the existence and uniqueness of weak Leray-type solutions, their rigorous convergence to weak Leray solutions of the classical Navier-Stokes equations, and their long-time dynamics through the concept of the global attractor and some upper bounds for its fractal dimension. Handling the nonlinear filter equation together with the well-known nonlinear transport term makes certain estimates delicate, particularly when deriving upper bounds on the fractal dimension. For the latter, we adapt techniques developed for hyperbolic-type equations.

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arXiv2024

The role of the dimension in uniqueness results for the stationary quasi-geostrophic system

In this paper, we study a Liouville-type theorem for the stationary fractional quasi-geostrophic equation in various dimensions. Indeed, our analysis focuses on dimensions n = 2, 3, 4 and we explore the uniqueness of weak solutions for this fractional system. We demonstrate here that, under some specific Lebesgue integrability information, the only admissible solution to the stationary fractional quasi-geostrophic system is the trivial one and this result provides a comprehensive understanding of how the dimension in connection to the fractional power of the Laplacian influences the uniqueness properties of weak solutions.

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arXiv2023

Sharp well-posedness and spatial decaying for a generalized dispersive-dissipative Kuramoto-type equation and applications to related models

We introduce a fairly general dispersive-dissipative nonlinear equation, which is characterized by fractional Laplacian operators in both the dispersive and dissipative terms. This equation includes some physically relevant models of fluid dynamics as particular cases. Among them are the \emph{dispersive Kuramoto-Velarde}, the \emph{Kuramoto-Sivashinsky} equation, and some nonlocal perturbations of the \emph{KdV} and the \emph{Benjamin-Ono} equations. We thoroughly study the effects of the fractional Laplacian operators in the qualitative study of solutions: on the one hand, we prove a sharp well-posedness result in the framework of Sobolev spaces of negative order, and on the other hand, we investigate the pointwise decaying properties of solutions in the spatial variable, which are optimal in some cases. These last results are of particular interest for the corresponding physical models. Precisely, they align with previous numerical works on the spatial decay of a particular kind of solutions, commonly referred to as solitary waves.

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arXiv2021

On the long-time behavior for a damped Navier-Stokes-Bardina model

In this paper, we consider a damped Navier-Stokes-Bardina model posed on the whole three-dimensional. These equations have an important physical motivation and they arise from some oceanic model. From the mathematical point of view, they write down as the well-know Navier-Stokes equations with an additional nonlocal operator in their nonlinear transport term, and moreover, with an additional damping term depending of a parameter $β>0$. We study first the existence and uniqueness of global in time weak solutions in the energy space. Thereafter, our main objective is to describe the long time behavior of these solutions. For this, we use some tools in the theory of dynamical systems to prove the existence of a global attractor, which is a compact subset in the energy space attracting all the weak solutions when the time goes to infinity. Moreover, we derive an upper bound for the fractal dimension of the global attractor associated to these equations. Finally, we find a range of values for the damping parameter $β>0$, where we are able to give an acutely description of the internal structure of the global attractor. More precisely, we prove that the global attractor only contains the stationary (time-independing) solution of the damped Navier-Stokes-Bardina equations.

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arXiv2019

On decay properties and asymptotic behavior of solutions to a non-local perturbed KdV equation

We consider the \emph{KdV} equation with an additional non-local perturbation term defined through the Hilbert transform, also known as the OST-equation. We prove that the solutions $u(t,x)$ has a pointwise decay in spatial variable: $\vert u(t,x)\vert \lesssim \frac{1}{1 + |x|^{2}}$, provided that the initial data has the same decaying and moreover we find the asymptotic profile of $u(t,x)$ when $|x| \to +\infty$. Next, we show that decay rate given above is optimal when the initial data is not a zero-mean function and in this case we derive an estimate from below $\frac{1}{\vert x\vert^2} \lesssim \vert u(t,x)\vert$ for $\vert x \vert$ large enough. In the case when the initial datum is a zero-mean function, we prove that the decay rate above is improved to $\frac{1}{1+\vert x \vert^{2+\varepsilon}}$ for $0<\varepsilon \leq 1$. Finally, we study the local-well posedness of the OST-equation in the framework of Lebesgue spaces.

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arXiv2014

On permanent and breaking waves in hyperelastic rods and rings

We prove that the only global strong solution of the periodic rod equation vanishing in at least one point $(t_0,x_0)$ is the identically zero solution. Such conclusion holds provided the physical parameter $γ$ of the model (related to the finger deformation tensor) is outside some neighborhood of the origin and applies in particular for the Camassa--Holm equation, corresponding to $γ=1$. We also establish the analogue of this unique continuation result in the case of non-periodic solutions defined on the whole real line with vanishing boundary conditions at infinity. Our analysis relies on the application of new local-in-space blowup criteria and involves the computation of several best constants in convolution estimates and weighted Poincaré inequalities.

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arXiv2014

Blowup issues for a class of nonlinear dispersive wave equations

In this paper we consider the nonlinear dispersive wave equation on the real line, $u_t-u_{txx}+[f(u)]_x-[f(u)]_{xxx}+\bigl[g(u)+\frac{f''(u)}{2}u_x^2\bigr]_x=0$, that for appropriate choices of the functions $f$ and $g$ includes well known models, such as Dai's equation for the study of vibrations inside elastic rods or the Camassa--Holm equation modelling water wave propagation in shallow water. We establish a local-in-space blowup criterion (i.e., a criterion involving only the properties of the data $u_0$ in a neighbourhood of a single point) simplifying and extending earlier blowup criteria for this equation. Our arguments apply both to the finite and infinite energy case, yielding the finite time blowup of strong solutions with possibly different behavior as $x\to+\infty$ and $x\to-\infty$.

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arXiv2014

PDEs in Moving Time Dependent Domains

In this work we study partial differential equations defined in a domain that moves in time according to the flow of a given ordinary differential equation, starting out of a given initial domain. We first derive a formulation for a particular case of partial differential equations known as balance equations. For this kind of equations we find the equivalent partial differential equations in the initial domain and later we study some particular cases with and without diffusion. We also analyze general second order differential equations, not necessarily of balance type. The equations without diffusion are solved using the characteristics method. We also prove that the diffusion equations, endowed with Dirichlet boundary conditions and initial data, are well posed in the moving domain. For this we show that the principal part of the equivalent equation in the initial domain is uniformly elliptic. We then prove a version of the weak maximum principle for an equation in a moving domain. Finally we perform suitable energy estimates in the moving domain and give sufficient conditions for the solution to converge to zero as time goes to infinity.

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