Luis Friz - Academia.edu (original) (raw)

Papers by Luis Friz

Research paper thumbnail of A result on the existence and uniqueness of stationary solutions for a bioconvective flow model

arXiv (Cornell University), Dec 10, 2017

Research paper thumbnail of On the existence and uniqueness of an inverse problem in epidemiology

arXiv (Cornell University), Nov 7, 2018

Research paper thumbnail of Exponential stability for magneto-micropolar fluids

Nonlinear Analysis: Theory, Methods & Applications, 2016

We prove existence, uniqueness and exponential stability of mildly decaying global strong solutio... more We prove existence, uniqueness and exponential stability of mildly decaying global strong solutions for the magneto-micropolar fluids system in three space dimensions. Our main objective is to study the convergence of nonstationary solutions to stationary solutions for the system fluids when t → ∞. We show that under mild suitable conditions the convergence is indeed exponential.

Research paper thumbnail of Pointwise Error Estimate for Spectral Galerkin Approximations of Micropolar Equations

Numerical Functional Analysis and Optimization, 2015

ABSTRACT The goal of this article is to present pointwise time error estimates in suitable Hilber... more ABSTRACT The goal of this article is to present pointwise time error estimates in suitable Hilbert spaces by considering spectral Galerkin approximations of the micropolar fluid model for strong solutions. In fact, we use the properties of the Stokes and Lamé operators for prove the pointwise convergence rate in the H2-norm for the ordinary velocity and microrotational velocity and the pointwise convergence rate in the L2-norm for the time-derivative of both velocities. The novelty of our method is that we do not impose any compatibility conditions in the initial data.

Research paper thumbnail of Global well-posedness for second grade fluids

Analysis and applications, May 23, 2024

Research paper thumbnail of On Bloch waves for the Stokes equations

In this work, we study the Bloch wave decomposition for the Stokes equations in a periodic media ... more In this work, we study the Bloch wave decomposition for the Stokes equations in a periodic media in R d. We prove that, because of the incompressibility constraint, the Bloch eigenvalues, as functions of the Bloch frequency ξ, are not continuous at the origin. Nevertheless, when ξ goes to zero in a fixed direction, we exhibit a new limit spectral problem for which the eigenvalues are directionally differentiable. Finally, we present an analogous study for the Bloch wave decomposition for a periodic perforated domain.

Research paper thumbnail of A Result on the Existence and Uniqueness of Stationary Solutions for a Bioconvective Flow Model

Journal of Function Spaces, 2018

In this note we prove the existence and uniqueness of weak solutions for the boundary value probl... more In this note we prove the existence and uniqueness of weak solutions for the boundary value problem modelling the stationary case of the bioconvective flow problem introduced by Tuval et. al. (2005, PNAS 102, 2277-2282). We derive some appropriate a priori estimates for the weak solution, which implies the existence, by application of Gossez theorem, and the uniqueness by standard methodology of comparison of two arbitrary solutions.

Research paper thumbnail of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow></mrow><annotation encoding="application/x-tex"></annotation></semantics></math></span><span class="katex-html" aria-hidden="true"></span></span>L^p$$Lp-solutions of a nonlinear third order differential equation and the Poincaré–Perron problem

Journal of Fixed Point Theory and Applications

In this paper we prove the well-posedness and we study the asymptotic behavior of nonoscillatory ... more In this paper we prove the well-posedness and we study the asymptotic behavior of nonoscillatory L^p$$Lp-solutions for a third order nonlinear scalar differential equation. The equation consists of two parts: a linear third order with constant coefficients part and a nonlinear part represented by a polynomial of fourth order in three variables with variable coefficients. The results are obtained assuming three hypotheses: (1) the characteristic polynomial associated with the linear part has simple and real roots, (2) the coefficients of the polynomial satisfy asymptotic integral smallness conditions, and (3) the polynomial coefficients are in L^p([t_0,\infty [)$$Lp([t0,∞[). These results are applied to study a fourth order linear differential equation of Poincaré type and a fourth order linear differential equation with unbounded coefficients. Moreover, we give some examples where the classical theorems cannot be applied.

Research paper thumbnail of On the existence and uniqueness of an inverse problem in epidemiology

Applicable Analysis

In this paper we introduce the functional framework and the necessary conditions for the well-pos... more In this paper we introduce the functional framework and the necessary conditions for the well-posedness of an inverse problem arising in the mathematical modeling of disease transmission. The direct problem is given by an initial boundary value problem for a reaction diffusion system. The inverse problem consists in the determination of the disease and recovery transmission rates from observed measurement of the direct problem solution at the end time. The unknowns of the inverse problem are coefficients of the reaction term. We formulate the inverse problem as an optimization problem for an appropriate cost functional. Then, the existence of solutions of the inverse problem is deduced by proving the existence of a minimizer for the cost functional. Moreover, we establish the uniqueness up an additive constant of identification problem. The uniqueness is a consequence of the first order necessary optimality condition and a stability of the inverse problem unknowns with respect to the observations.

Research paper thumbnail of Solution reproductive d'un système de fluide de grade deux

Comptes Rendus Mathematique, 2010

Research paper thumbnail of Allaire-Conca-Friz-Ortega

Research paper thumbnail of On Bloch waves for the Stokes equations

Discrete and Continuous Dynamical Systems - Series B, 2006

Research paper thumbnail of Time-reproductive solutions for a bioconvective flow

We consider the existence and uniqueness of periodic solutions for the generalized bioconvective ... more We consider the existence and uniqueness of periodic solutions for the generalized bioconvective flow, which is a well known model to describe the convection caused by the concentration of upward swimming microorganism in a fluid.

Research paper thumbnail of Reproductive solution of a second-grade fluid system

Comptes Rendus Mathematique, 2010

... solution of a second-grade fluid system Solutionreproductived unsyst mede fluidedegradedeux L... more ... solution of a second-grade fluid system Solutionreproductived unsyst mede fluidedegradedeux Luis Friz a , Francisco Guill n-Gonz lez b , MA Rojas-Medar aa ... Version fran aise abr ge On consid re l existence et l unicit de la solution reproductive d un syst me de fluide de grade ...

Research paper thumbnail of Identification of immersed obstacles via boundary measurements

Inverse Problems

We study the following inverse problem: an inaccessible rigid body D is immersed in a viscous flu... more We study the following inverse problem: an inaccessible rigid body D is immersed in a viscous fluid, in such a way that D plays the role of an obstacle around which the fluid is flowing in a greater bounded domain Ω, and we wish to determine D (i.e., its form and location) via boundary measurement on the boundary ∂Ω. Both for the stationary and the evolution problem, we show that under reasonable smoothness assumptions on Ω and D, one can identify D via the measurement of the velocity of the fluid and the Cauchy forces on some part of the boundary ∂Ω. We also show that the dependence of the Cauchy forces on deformations of D is analytic, and give some stability results for the inverse problem.

Research paper thumbnail of Localization of high-frequency waves propagating in a locally periodic medium

Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2010

We study the homogenization and localization of high-frequency waves in a locally periodic media ... more We study the homogenization and localization of high-frequency waves in a locally periodic media with period ε. We consider initial data that are localized Bloch-wave packets, i.e. that are the product of a fast oscillating Bloch wave at a given frequency ξ and of a smooth envelope function whose support is concentrated at a pointxwith length scale$\sqrt\varepsilon$. We assume that (ξ,x) is a stationary point in the phase space of the Hamiltonian λ(ξ,x), i.e. of the corresponding Bloch eigenvalue. Upon rescaling at size$\sqrt\varepsilon$we prove that the solution of the wave equation is approximately the sum of two terms with opposite phases which are the product of the oscillating Bloch wave and of two limit envelope functions which are the solution of two Schrödinger type equations with quadratic potential. Furthermore, if the full Hessian of the Hamiltonian λ(ξ,x) is positive definite, then localization takes place in the sense that the spectrum of each homogenized Schrödinger eq...

Research paper thumbnail of Direct Integral Decomposition for Periodic Function Spaces and Application to Bloch Waves

aimsciences.org

... Blanco Encalada 2120, 7mo. ... the sets O and Y ∗ are defined in (3) and (2) respectively, in... more ... Blanco Encalada 2120, 7mo. ... the sets O and Y ∗ are defined in (3) and (2) respectively, in the sense given by J. Dixmier [8] (see also the book of GW Mackey [12]), and thus, a decomposition based on (ξ, Y ∗ )−periodic functions of the Green&#x27;s operator for the Stokes system is ...

Research paper thumbnail of Reproductive solution for grade-two fluid model in two dimensions

We treat the existence of reproductive solution (weak periodic solution) of a second-grade fluid ... more We treat the existence of reproductive solution (weak periodic solution) of a second-grade fluid system in two dimensions, by using the Galerkin approximation method and compactness arguments.

Research paper thumbnail of A result on the existence and uniqueness of stationary solutions for a bioconvective flow model

arXiv (Cornell University), Dec 10, 2017

Research paper thumbnail of On the existence and uniqueness of an inverse problem in epidemiology

arXiv (Cornell University), Nov 7, 2018

Research paper thumbnail of Exponential stability for magneto-micropolar fluids

Nonlinear Analysis: Theory, Methods & Applications, 2016

We prove existence, uniqueness and exponential stability of mildly decaying global strong solutio... more We prove existence, uniqueness and exponential stability of mildly decaying global strong solutions for the magneto-micropolar fluids system in three space dimensions. Our main objective is to study the convergence of nonstationary solutions to stationary solutions for the system fluids when t → ∞. We show that under mild suitable conditions the convergence is indeed exponential.

Research paper thumbnail of Pointwise Error Estimate for Spectral Galerkin Approximations of Micropolar Equations

Numerical Functional Analysis and Optimization, 2015

ABSTRACT The goal of this article is to present pointwise time error estimates in suitable Hilber... more ABSTRACT The goal of this article is to present pointwise time error estimates in suitable Hilbert spaces by considering spectral Galerkin approximations of the micropolar fluid model for strong solutions. In fact, we use the properties of the Stokes and Lamé operators for prove the pointwise convergence rate in the H2-norm for the ordinary velocity and microrotational velocity and the pointwise convergence rate in the L2-norm for the time-derivative of both velocities. The novelty of our method is that we do not impose any compatibility conditions in the initial data.

Research paper thumbnail of Global well-posedness for second grade fluids

Analysis and applications, May 23, 2024

Research paper thumbnail of On Bloch waves for the Stokes equations

In this work, we study the Bloch wave decomposition for the Stokes equations in a periodic media ... more In this work, we study the Bloch wave decomposition for the Stokes equations in a periodic media in R d. We prove that, because of the incompressibility constraint, the Bloch eigenvalues, as functions of the Bloch frequency ξ, are not continuous at the origin. Nevertheless, when ξ goes to zero in a fixed direction, we exhibit a new limit spectral problem for which the eigenvalues are directionally differentiable. Finally, we present an analogous study for the Bloch wave decomposition for a periodic perforated domain.

Research paper thumbnail of A Result on the Existence and Uniqueness of Stationary Solutions for a Bioconvective Flow Model

Journal of Function Spaces, 2018

In this note we prove the existence and uniqueness of weak solutions for the boundary value probl... more In this note we prove the existence and uniqueness of weak solutions for the boundary value problem modelling the stationary case of the bioconvective flow problem introduced by Tuval et. al. (2005, PNAS 102, 2277-2282). We derive some appropriate a priori estimates for the weak solution, which implies the existence, by application of Gossez theorem, and the uniqueness by standard methodology of comparison of two arbitrary solutions.

Research paper thumbnail of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow></mrow><annotation encoding="application/x-tex"></annotation></semantics></math></span><span class="katex-html" aria-hidden="true"></span></span>L^p$$Lp-solutions of a nonlinear third order differential equation and the Poincaré–Perron problem

Journal of Fixed Point Theory and Applications

In this paper we prove the well-posedness and we study the asymptotic behavior of nonoscillatory ... more In this paper we prove the well-posedness and we study the asymptotic behavior of nonoscillatory L^p$$Lp-solutions for a third order nonlinear scalar differential equation. The equation consists of two parts: a linear third order with constant coefficients part and a nonlinear part represented by a polynomial of fourth order in three variables with variable coefficients. The results are obtained assuming three hypotheses: (1) the characteristic polynomial associated with the linear part has simple and real roots, (2) the coefficients of the polynomial satisfy asymptotic integral smallness conditions, and (3) the polynomial coefficients are in L^p([t_0,\infty [)$$Lp([t0,∞[). These results are applied to study a fourth order linear differential equation of Poincaré type and a fourth order linear differential equation with unbounded coefficients. Moreover, we give some examples where the classical theorems cannot be applied.

Research paper thumbnail of On the existence and uniqueness of an inverse problem in epidemiology

Applicable Analysis

In this paper we introduce the functional framework and the necessary conditions for the well-pos... more In this paper we introduce the functional framework and the necessary conditions for the well-posedness of an inverse problem arising in the mathematical modeling of disease transmission. The direct problem is given by an initial boundary value problem for a reaction diffusion system. The inverse problem consists in the determination of the disease and recovery transmission rates from observed measurement of the direct problem solution at the end time. The unknowns of the inverse problem are coefficients of the reaction term. We formulate the inverse problem as an optimization problem for an appropriate cost functional. Then, the existence of solutions of the inverse problem is deduced by proving the existence of a minimizer for the cost functional. Moreover, we establish the uniqueness up an additive constant of identification problem. The uniqueness is a consequence of the first order necessary optimality condition and a stability of the inverse problem unknowns with respect to the observations.

Research paper thumbnail of Solution reproductive d'un système de fluide de grade deux

Comptes Rendus Mathematique, 2010

Research paper thumbnail of Allaire-Conca-Friz-Ortega

Research paper thumbnail of On Bloch waves for the Stokes equations

Discrete and Continuous Dynamical Systems - Series B, 2006

Research paper thumbnail of Time-reproductive solutions for a bioconvective flow

We consider the existence and uniqueness of periodic solutions for the generalized bioconvective ... more We consider the existence and uniqueness of periodic solutions for the generalized bioconvective flow, which is a well known model to describe the convection caused by the concentration of upward swimming microorganism in a fluid.

Research paper thumbnail of Reproductive solution of a second-grade fluid system

Comptes Rendus Mathematique, 2010

... solution of a second-grade fluid system Solutionreproductived unsyst mede fluidedegradedeux L... more ... solution of a second-grade fluid system Solutionreproductived unsyst mede fluidedegradedeux Luis Friz a , Francisco Guill n-Gonz lez b , MA Rojas-Medar aa ... Version fran aise abr ge On consid re l existence et l unicit de la solution reproductive d un syst me de fluide de grade ...

Research paper thumbnail of Identification of immersed obstacles via boundary measurements

Inverse Problems

We study the following inverse problem: an inaccessible rigid body D is immersed in a viscous flu... more We study the following inverse problem: an inaccessible rigid body D is immersed in a viscous fluid, in such a way that D plays the role of an obstacle around which the fluid is flowing in a greater bounded domain Ω, and we wish to determine D (i.e., its form and location) via boundary measurement on the boundary ∂Ω. Both for the stationary and the evolution problem, we show that under reasonable smoothness assumptions on Ω and D, one can identify D via the measurement of the velocity of the fluid and the Cauchy forces on some part of the boundary ∂Ω. We also show that the dependence of the Cauchy forces on deformations of D is analytic, and give some stability results for the inverse problem.

Research paper thumbnail of Localization of high-frequency waves propagating in a locally periodic medium

Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2010

We study the homogenization and localization of high-frequency waves in a locally periodic media ... more We study the homogenization and localization of high-frequency waves in a locally periodic media with period ε. We consider initial data that are localized Bloch-wave packets, i.e. that are the product of a fast oscillating Bloch wave at a given frequency ξ and of a smooth envelope function whose support is concentrated at a pointxwith length scale$\sqrt\varepsilon$. We assume that (ξ,x) is a stationary point in the phase space of the Hamiltonian λ(ξ,x), i.e. of the corresponding Bloch eigenvalue. Upon rescaling at size$\sqrt\varepsilon$we prove that the solution of the wave equation is approximately the sum of two terms with opposite phases which are the product of the oscillating Bloch wave and of two limit envelope functions which are the solution of two Schrödinger type equations with quadratic potential. Furthermore, if the full Hessian of the Hamiltonian λ(ξ,x) is positive definite, then localization takes place in the sense that the spectrum of each homogenized Schrödinger eq...

Research paper thumbnail of Direct Integral Decomposition for Periodic Function Spaces and Application to Bloch Waves

aimsciences.org

... Blanco Encalada 2120, 7mo. ... the sets O and Y ∗ are defined in (3) and (2) respectively, in... more ... Blanco Encalada 2120, 7mo. ... the sets O and Y ∗ are defined in (3) and (2) respectively, in the sense given by J. Dixmier [8] (see also the book of GW Mackey [12]), and thus, a decomposition based on (ξ, Y ∗ )−periodic functions of the Green&#x27;s operator for the Stokes system is ...

Research paper thumbnail of Reproductive solution for grade-two fluid model in two dimensions

We treat the existence of reproductive solution (weak periodic solution) of a second-grade fluid ... more We treat the existence of reproductive solution (weak periodic solution) of a second-grade fluid system in two dimensions, by using the Galerkin approximation method and compactness arguments.