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Diego Chamorro

Publications and source records attributed to Diego Chamorro.

At least 37 records · Page 2Linked to original sources
arXiv2023

Partial suitable solutions for the micropolar equations and regularity properties

The incompressible Micropolar system is given by two coupled equations: the first equation gives the evolution of the velocity field u while the second equation gives the evolution of the microrotation field $ω$. In this article we will consider regularity problems for weak solutions of this system. For this we will introduce the new notion of partial suitable solutions, which imposes a specific behavior for the velocity field u only, and under some classical hypotheses over the pressure, we will obtain a h{ö}lderian gain for both variables u and $ω$.

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arXiv2023

On an almost sharp Liouville type theorem for fractional Navier-Stokes equations

We investigate existence, Liouville type theorems and regularity results for the 3D stationary and incompressible fractional Navier-Stokes equations: in this setting the usual Laplacian is replaced by its fractional power $(-Δ)^{\fracα{2}}$ with $0<α<2$. By applying a fixed point argument, weak solutions can be obtained in the Sobolev space $\dot{H}^{\fracα{2}}(\mathbb{R})$ and if we add an extra integrability condition, stated in terms of Lebesgue spaces, then we can prove for some values of $α$ that the zero function is the unique smooth solution. The additional integrability condition is almost sharp for $3/5<α<5/3$. Moreover, in the case $1<α<2$ a gain of regularity is established under some conditions, however the study of regularity in the regime $0<α\leq 1$ seems for the moment to be an open problem.

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arXiv2022

Non Linear Singular Drifts and Fractional Operators

We consider parabolic PDEs associated with fractional type operators drifted by non-linear singular first order terms. When the drift enjoys some boundedness properties in appropriate Lebesgue and Besov spaces, we establish by exploiting a priori Besov-type estimates, the H{ö}lder continuity of the solutions. In particular, we handle the almost critical case in whole generality.

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arXiv2022

A crypto-regularity result for the micropolar fluids equations

In the analysis of PDEs, regularity of often measured in terms of Sobolev, H{ö}lder, Besov or Lipschitz spaces, etc. However, sometimes a gain of regularity can also be expressed just in terms of Lebesgue spaces, by passing from a singular setting to a less singular one. In this article we will obtain a gain of integrability for weak solutions of the micropolar fluid equations using as general framework Morrey spaces, which is a very useful language to study regularity in PDEs. An interesting point is that the two variables of the micropolar fluid equations can be studied separately.

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arXiv2022

Some existence and regularity results for a non-local transport-diffusion equation with fractional derivatives in time and space

We study the existence of global weak solutions of a nonlinear transport-diffusion equation with a fractional derivative in the time variable and under some extra hypotheses, we also study some regularity properties for this type of solutions. In the system considered here, the diffusion operator is given by a fractional Laplacian and the nonlinear drift is assumed to be divergence free and it is assumed to satisfy some general stability and boundedness properties in Lebesgue spaces.

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arXiv2021

Interior epsilon-regularity theory for the solutions of the magneto-micropolar equations with a perturbation term

We develop here a particular version of the partial regularity theory for the Magneto-Micropolar equations (MMP) where a perturbation term is added. These equations are used in some special cases, such as in the study of the evolution of liquid cristals or polymers, where the classical Navier-Stokes equations are not an accurate enough model. The incompressible Magneto-Micropolar system is composed of three coupled equations: the first one is based in the Navier-Stokes system, the second one considers mainly the magnetic field while the last equation introduces the microrotation field representing the angular velocity of the rotation of the fluid particles. External forces are considered and a specific perturbation term is added as it is quite useful in some applications.

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arXiv2020

Regularity theory for the dissipative solutions of the MHD equations

We study here a new generalization of Caffarelli, Kohn and Nirenberg's partial regularity theory for weak solutions of the MHD equations. Indeed, in this framework some hypotheses on the pressure P are usually asked (for example P $\in$ L q t L 1 x with q > 1) and then local H{ö}lder regularity, in time and space variables, for weak solutions can be obtained over small neighborhoods. By introducing the notion of dissipative solutions, we weaken the hypothesis on the pressure (we will only assume that P $\in$ D) and we will obtain H{ö}lder regularity in the space variable for weak solutions.

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arXiv2020

Blow-up for a nonlinear PDE with fractional Laplacian and singular quadratic nonlinearity

We consider a parabolic-type PDE with a diffusion given by a fractional Laplacian operator and with a quadratic nonlinearity of the 'gradient' of the solution, convoluted with a singular term b. Our first result is the well-posedness for this problem: We show existence and uniqueness of a (local in time) mild solution. The main result is about blow-up of said solution, and in particular we find sufficient conditions on the initial datum and on the term b to ensure blow-up of the solution in finite time.

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arXiv2020

On the partial regularity theory for the MHD equations

We generalize here the celebrated Partial Regularity Theory of Caffarelli, Kohn and Nirenberg to the MHD equations in the framework of parabolic Morrey spaces. This type of parabolic generalization using Morrey spaces appears to be crucial when studying the role of the pressure in the regularity theory for the classical Navier-Stokes equations as well as for the MHD 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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arXiv2018

Functional Inequalities in Stratified Lie groups with Sobolev, Besov, Lorentz and Morrey spaces

The study of Sobolev inequalities can be divided in two cases: p = 1 and 1 < p < +$\infty$. In the case p = 1 we study here a relaxed version of refined Sobolev inequalities. When p > 1, using as base space classical Lorentz spaces associated to a weight from the Arino-Muckenhoupt class Bp, we will study Gagliardo-Nirenberg inequalities. As a by-product we will also consider Morrey-Sobolev inequalities. These arguments can be generalized to many different frameworks, in particular the proofs are given in the setting of stratified Lie groups.

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arXiv2018

Deterministics descriptions of the turbulence in the Navier-Stokes equations

This PhD thesis is devoted to deterministic study of the turbulence in the Navier- Stokes equations. The thesis is divided in four independent chapters.The first chapter involves a rigorous discussion about the energy's dissipation law, proposed by theory of the turbulence K41, in the deterministic setting of the homogeneous and incompressible Navier-Stokes equations, with a stationary external force (the force only depends of the spatial variable) and on the whole space R3. The energy's dissipation law, also called the Kolmogorov's dissipation law, characterizes the energy's dissipation rate (in the form of heat) of a turbulent fluid and this law was developed by A.N. Kolmogorov in 1941. However, its deduction (which uses mainly tools of statistics) is not fully understood until our days and then an active research area consists in studying this law in the rigorous framework of the Navier-Stokes equations which describe in a mathematical way the fluids motion and in particular the movement of turbulent fluids. In this setting, the purpose of this chapter is to highlight the fact that if we consider the Navier-Stokes equations on R3 then certain physical quantities, necessary for the study of the Kolmogorov's dissipation law, have no a rigorous definition and then to give a sense to these quantities we suggest to consider the Navier-Stokes equations with an additional damping term. In the framework of these damped equations, we obtain some estimates for the energy's dissipation rate according to the Kolmogorov's dissipation law.In the second chapter we are interested in study the stationary solutions of the damped Navier- Stokes introduced in the previous chapter. These stationary solutions are a particular type of solutions which do not depend of the temporal variable and their study is motivated by the fact that we always consider the Navier-Stokes equations with a stationary external force. In this chapter we study two properties of the stationary solutions : the first property concerns the stability of these solutions where we prove that if we have a control on the external force then all non stationary solution (with depends of both spatial and temporal variables) converges toward a stationary solution. The second property concerns the decay in spatial variable of the stationary solutions. These properties of stationary solutions are a consequence of the damping term introduced in the Navier-Stokes equations.In the third chapter we still study the stationary solutions of Navier-Stokes equations but now we consider the classical equations (without any additional damping term). The purpose of this chapter is to study an other problem related to the deterministic description of the turbulence : the frequency decay of the stationary solutions. Indeed, according to the K41 theory, if the fluid is in a laminar setting then the stationary solutions of the Navier-Stokes equations must exhibit a exponential frequency decay which starts at lows frequencies. But, if the fluid is in a turbulent setting then this exponential frequency decay must be observed only at highs frequencies. In this chapter, using some Fourier analysis tools, we give a precise description of this exponential frequency decay in the laminar and in the turbulent setting.In the fourth and last chapter we return to the stationary solutions of the classical Navier-Stokes equations and we study the uniqueness of these solutions in the particular case without any external force. Following some ideas of G. Seregin, we study the uniqueness of these solutions first in the framework of Lebesgue spaces of and then in the a general framework of Morrey spaces.

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arXiv2018

Some Liouville theorems for stationary Navier-Stokes equations in Lebesgue and Morrey spaces

Uniqueness of Leray solutions of the 3D Navier-Stokes equations is a challenging open problem. In this article we will study this problem for the 3D stationary Navier-Stokes equations and under some additional hypotheses, stated in terms of Lebesgue and Morrey spaces, we will show that the trivial solution U = 0 is the unique solution. This type of results are known as Liouville theorems.

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arXiv2017

Frequency decay for Navier-Stokes stationary solutions

We consider stationary Navier-Stokes equations in R 3 with a regular external force and we prove exponential frequency decay of the solutions. Moreover, if the external force is small enough, we give a pointwise exponential frequency decay for such solutions according to the K41 theory. If a damping term is added to the equation, a pointwise decay is obtained without the smallness condition over the force.

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arXiv2017

Non Linear Singular Drifts and Fractional Operators: when Besov meets Morrey and Campanato

Within the global setting of singular drifts in Morrey-Campanato spaces presented in [6], we study now the H{ö}lder regularity properties of the solutions of a transport-diffusion equation with nonlinear singular drifts that satisfy a Besov stability property. We will see how this Besov information is relevant and how it allows to improve previous results. Moreover, in some particular cases we show that as the nonlinear drift becomes more regular, in the sense of Morrey-Campanato spaces, the additional Besov stability property will be less useful.

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arXiv2016

The role of the pressure in the partial regularity theory for weak solutions of the Navier--Stokes equations

We study the role of the pressure in the partial regularity theory for weak solutions of the Navier--Stokes equations. By introducing the notion of dissipative solutions, due to Duchon \& Robert, we will provide a generalization of the Caffarelli, Kohn and Nirenberg theory. Our approach gives a new enlightenment of the role of the pressure in this theory in connection to Serrin's local regularity criterion.

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