arXiv subjects

Oscar Jarrín

Publications and source records attributed to Oscar Jarrín.

14 recordsLinked to original sources

arXiv2026

Mathematical study of a new Navier-Stokes-alpha model with nonlinear filter equation - Regularity theory and refined time asymptotics

This article is devoted to the mathematical analysis of a new Navier--Stokes-$α$ model involving a nonlinear filter equation. The resulting model is governed by a doubly nonlinear parabolic--elliptic coupled system. In our previous work [M.~F.~Cortez and O.~Jarrín, \emph{Mathematical study of a new Navier--Stokes-alpha model with nonlinear filter equation -- Part I}, J. Math. Fluid Mech. 28, no.~12 (2026)], we established the global well-posedness of weak Leray-type solutions and the existence of a global attractor. In the present work, under natural assumptions on $A(\cdot)$, we investigate several additional properties of this model, with particular emphasis on higher-order regularity and its consequences for the long-time dynamics. The regularity analysis constitutes a central and delicate issue, since the nonlinear structure of the elliptic filter and its coupling with the evolution equation preclude a direct application of the standard regularity arguments available for linearly filtered Navier--Stokes-$α$ models. Overcoming this difficulty requires new higher-order estimates for the nonlinear elliptic filter equation, which may also be of independent interest. Among the results obtained, two constitute the main contributions of the article. First, we establish uniform higher-order regularity for the global attractor. This result is then used as a fundamental ingredient in proving an exact determining-modes property for complete trajectories on the attractor: if the projections of two complete trajectories onto a sufficiently large finite-dimensional space of Stokes modes coincide at every time, then the two trajectories coincide identically.

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arXiv2026

On a pointwise estimate for a weighted rough singular integral operator in stratified Lie groups and applications

In this article, we present a new pointwise estimate for a weighted rough singular integral operator in the setting of stratified Lie groups. This operator, $T_{Ω, \varpi}$, is based on a kernel $Ω$ and a weight $\varpi$, where the kernel satisfies a natural size condition and a cancellation property with respect to the weight $\varpi$. Moreover, we do not assume any kind of regularity on these objects. This weighted rough singular integral operator, applied to a function $f$, is estimated through a combination of information involving a weighted maximal function of the gradient of $f$ and a weighted Morrey space. We also deduce from this pointwise estimate some new weighted functional inequalities and, as an application, we obtain a uniqueness result for a rough version of the stationary Navier-Stokes equation over the Heisenberg group.

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arXiv2026

A remark on very weak suitable solutions and Leray solutions of the Navier-Stokes equations

We introduce the notion of very weak suitable solutions for the Navier--Stokes equations. Here, the velocity and the pressure satisfy minimal conditions that make sense of the local energy inequality in the distributional setting. A well-known but still relevant question is to find sufficient conditions ensuring that very-weak solutions are in fact Leray solutions. Exploiting the \emph{local} energy inequality within the general framework of \emph{local} Morrey spaces, we establish such conditions. Local Morrey spaces provide a general framework that contains other useful functional settings in the theoretical analysis of the Navier--Stokes equations, such as Lebesgue, Lorentz, homogeneous Morrey, and parabolic Morrey spaces. As a by-product, we also derive some sufficient conditions to study the uniqueness and regularity of the resulting Leray solutions.

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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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arXiv2024

An Lp-theory for fractional stationary Navier-Stokes equations

We consider the stationary (time-independent) Navier-Stokes equations in the whole threedimensional space, under the action of a source term and with the fractional Laplacian operator (--$Δ$) $α$/2 in the diffusion term. In the framework of Lebesgue and Lorentz spaces, we find some natural sufficient conditions on the external force and on the parameter $α$ to prove the existence and in some cases nonexistence of solutions. Secondly, we obtain sharp pointwise decaying rates and asymptotic profiles of solutions, which strongly depend on $α$. Finally, we also prove the global regularity of solutions. As a bi-product, we obtain some uniqueness theorems so-called Liouville-type results. On the other hand, our regularity result yields a new regularity criterion for the classical ( i.e. with $α$ = 2) stationary Navier-Stokes equations. Contents

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arXiv2024

Mild solutions to the 3D-Boussinesq system with weakened initial temperature

In this research, the Cauchy problem of the 3D viscous Boussinesq system is studied considering an initial temperature with negative Sobolev regularity. Precisely, we construct local in time mild solutions to this system where the temperature term belongs to Sobolev spaces of negative order. Our main contribution is to show how the coupled structure of the Boussinesq system allows us to considerably weaken the regularity in the temperature term.

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arXiv2023

A turbulent study for a damped Navier-Stokes equation: turbulence and problems

In this article we consider a damped version of the incompressible Navier-Stokes equations in the whole three-dimensional space with a divergence-free and time-independent external force. Within the framework of a well-prepared force and with a particular choice of the damping parameter, when the Grashof numbers are large enough, we are able to prove some estimates from below and from above between the fluid characteristic velocity and the energy dissipation rate according to the Kolmogorov dissipation law. Precisely, our main contribution concerns the estimate from below which is not often studied in the existing literature. Moreover, we address some remarks which open the door to a deep discussion on the validity of this theory of turbulence.

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arXiv2023

A short note on the Liouville problem for the steady-state Navier-Stokes equations

Uniqueness of the trivial solution (the zero solution) for the steady-state Navier-Stokes equations is an interesting problem who has known several recent contributions. These results are also known as the Liouville type problem for the steady-state Navier-Stokes equations. In the setting of the $L^p-$ spaces, when $3\leq p \leq 9/2$ it is known that the trivial solution of these equations is the unique one. In this note, we extend this previous result to other values of the parameter $p$. More precisely, we prove that the velocity field must be zero provided that it belongs to the $L^p -$ space with $3/2<p<3$. Moreover, for the large interval of values $9/2<p<+\infty$, we also obtain a partial result on the vanishing of the velocity under an additional hypothesis in terms of the Sobolev space of negative order $\dot{H}^{-1}$. This last result has an interesting corollary when studying the Liouville problem in the natural energy space of these solutions $\dot{H}^{1}$.

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arXiv2022

Some remarks on the regularity of weak solutions for the stationary Ericksen-Leslie and MHD systems

We consider two elliptic coupled systems of relevance in the fluid dynamics. These systems are posed on the whole three-dimensional space and they consider the action of external forces. The first system deals with the simplified Ericksen-Leslie (SEL) system, which describes the dynamics of liquid crystal flows. The second system is the time-independent magneto-hydrodynamic (MHD) equations. For the (SEL) system, we obtain a new criterion to improve the regularity of weak solutions, provided that they belong to some homogeneous Morrey space. As a bi-product, we also obtain some new regularity criterion for the stationary Navier-Stokes equations and for a nonlinear harmonic map flow. This new regularity criterion also holds true for the (MHD) equations. Furthermore, for this last system we are able to use the Gevrey class to prove that all finite energy weak solutions are analytic functions, provided the external forces belong to some Gevrey class.

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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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arXiv2020

Weak-strong uniqueness in weighted $L^2$ spaces and weak suitable solutions in local Morrey spaces for the MHD equations

We consider here the magneto-hydrodynamics (MHD) equations on the whole space. For the 3D case, in the setting of the weighted $L^2$ spaces we obtain a weak-strong uniqueness criterion provided that the velocity field and the magnetic field belong to a fairly general multipliers space. On the other hand, we study the local and global existence of weak suitable solutions for intermittent initial data, which is characterized through a local Morrey space. This large initial data space was also exhibit in a contemporary work [4] in the context of 3D Navier-Stokes equations. Finally, we make a discussion on the local and global existence problem in the 2D case.

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arXiv2019

Existence of infinite-energy and discretely self-similar global weak solutions for 3D MHD equations

This paper deals with the existence of global weak solutions for 3D MHD equations when the initial data belong to the weighted spaces $L^2_{w_γ}$, with $w_γ(x)=(1+| x|)^{-γ}$ and $0 \leq γ\leq 2$. Moreover, we prove the existence of discretely self-similar solutions for 3D MHD equations for discretely self-similar initial data which are locally square integrable. Our methods are inspired of a recent work of P. Fernández-Dalgo and P.G. Lemarié-Riseusset for the 3D Navier-Stokes equations.

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arXiv2019

On the Kolmogorov dissipation law in a damped Navier-Stokes equation

We consider here the Navier-Stokes equations in $\mathbb{R}^{3}$ with a stationary, divergence-free external force and with an additional damping term that depends on two parameters. We first study the well-posedness of weak solutions for these equations and then, for a particular set of the damping parameters, we will obtain an upper and lower control for the energy dissipation rate $\varepsilon$ according to the Kolmogorov K41 theory. However, although the behavior of weak solutions corresponds to the K41 theory, we will show that in some specific cases the damping term introduced in the Navier-Stokes equations could annihilate the turbulence even though the Grashof number (which are equivalent to the Reynolds number) are large.

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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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