My research is grounded in the study of differential equations, in their partial and fractional versions.

In particular, I'm intrigued by the analysis and control of non-linear models describing the motion of fluids. I mainly study the Cauchy-problem, controllability issues and qualitative behaviour of solutions related to Navier-Stokes (type) equations and problems arising in fluid-structure interactions. As a complement of these topics, I am equally fascinated by the discretization and numerical simulation of these type of models. In this link you can find one of our simulations of a fluid-structure interaction model based on a finite volume scheme.

Please have a look at my results. I'm always happy to discuss about any of these topics for projects and collaborations, so please feel free to get in touch!

Publications and preprints

1. On variable Lebesgue spaces and generalized nonlinear heat equations
   
   In this work  we address some questions concerning  the Cauchy problem for a generalized nonlinear heat equations considering as functional framework the variable Lebesgue spaces  $L^{p(\cdot)}(\R^n)$. More precisely, by mixing some structural properties of these spaces with decay estimates of the fractional heat kernel, we were able to prove two well-posedness results for these equations.  In a first theorem, we prove the existence and uniqueness of global-in-time  mild solutions in the mixed-space $\mathcal{L}^{p(\cdot)}_{ \frac{nb}{2\alpha - \langle 1 \rangle_\gamma}} (\mathbb{R}^n,L^\infty([0,T[ ))$. On the other hand,  by introducing a new class of variable exponents, we demonstrate the existence of an unique local-in-time mild solution  in the space $L^{p(\cdot)} \left( [0,T], L^{q} (\R^n) \right)$.
   
   Journal of Dynamics and Differential Equations (2025) - doi, arXiv
   
   

2. Local exact Lagrangian controllability for 1D barotropic compressible Navier-Stokes equations
   with F. Sueur and K. Koike
   
   We consider a viscous compressible barotropic flow in the interval $[0,\pi]$ with homogeneous Dirichlet boundary conditions for the flow velocity and a constant rest state as initial data. Given two sufficiently close subintervals $I=[\alpha_1,\alpha_2]$ and $J=[\beta_1,\beta_2]$ of $(0,1)$, a nonempty open set $\omega \subset (1,\pi)$, and $T>0$, we construct an external force $f$ supported in $\omega$ acting on the momentum equation such that the corresponding flow map moves the fluid particles initially occupying $I$ exactly onto $J$ in time $T$.
   
   SIAM Journal on Control and Optimization, Vol. 63, iss. 4 (2025) - doi, arXiv
   
   

3. Global well-posedness of the fractional dissipative system in the framework of variable Fourier-Besov spaces
   with J. Zhao
   
   In this paper, we are concerned with the well-posed issues of the fractional dissipative system in the framework of the Fourier--Besov spaces with variable regularity and integrability indices. By fully using some basic properties of these variable function spaces, we establish the linear estimates in variable Fourier--Besov spaces for the fractional heat equation. Such estimates are fundamental for solving certain dissipative PDE's of fractional type. As an applications, we prove global well-posedness in variable Fourier--Besov spaces for the 3D generalized incompressible  Navier--Stokes equations and the 3D fractional Keller--Segel system.
   
   Mediterranean Journal of Mathematics,  Vol. 22, 160 (2025) - doi, arXiv
   
   

4. Well-posedness of the 2D surface quasi-geostrophic equation in variable Lebesgue spaces
   with H. Chen and J. Zhao
   
   In this paper, we are mainly concerned with the well-posedness of the dissipative surface quasi-geostrophic equation in the framework of  variable Lebesgue spaces. Based on some analytical results developed in the variable Lebesgue spaces and the $L^{p}$-$L^{q}$ decay estimates of the fractional heat kernel, we establish the local existence and regularity of solutions to the 2D dissipative surface quasi-geostrophic equation in the variable Lebesgue space.
   
   Rendiconti del Circolo Matematico di Palermo, Vol. 74, 134 (2025) - doi, arXiv
   
   

5. Liouville-type theorems for stationary Navier-Stokes equations with Lebesgue spaces of variable exponent
   with D. Chamorro
   
   In this article we study some Liouville-type theorems for the stationary 3D Navier-Stokes equations. These results are related to the uniqueness of weak solutions for this system under some additional information over the velocity field, which is usually stated in the literature in terms of Lebesgue, Morrey or BMO^{-1} spaces. Here we will consider Lebesgue spaces of variable exponent which will provide us with some interesting flexibility.
   
   Documenta Mathematica (2025) - doi, HAL
   
   

6. On the blow-up for a Kuramoto-Velarde type equation
   with O. Jarrin
   
   It is known that  the  Kuramoto-Velarde equation is globally well-posed on Sobolev spaces in the case when the parameters $\gamma_1$ and $\gamma_2$ involved in the non-linear terms verify $ \gamma_1=\frac{\gamma_1}{2}$ or $\gamma_2=0$. In the complementary case of these parameters, the global existence or blow-up of solutions is a completely open (and hard)  problem. Motivated by this fact,  in this work we consider a non-local version of the  Kuramoto-Velarde equation.  This  equation allows us to apply a Fourier-based method  and, within  the framework $\gamma_2\neq \frac{\gamma_1}{2}$ and $\gamma_2\neq 0$, we show that large values of these parameters yield a blow-up in finite time of solutions in the Sobolev norm.
   
   Physica D: Nonlinear Phenomena 470 (2024) 134407 - doi, arXiv
   
   

7. Remarks on variable Lebesgue spaces and fractional Navier-Stokes equations
   
   In this work we study the 3D Navier-Stokes equations, under the action of an external force and with the fractional Laplacian operator $(-\Delta)^{\alpha}$ in the diffusion term, from the point of view of variable Lebesgue spaces. Based on decay estimates of the fractional heat kernel we prove the existence and uniqueness of mild solutions on this functional setting. Thus, in a first theorem we obtain an unique local-in-time solution in the space $L^{p(\cdot)}_t L^{q}_x$. As a bi-product, in a second theorem we prove the existence of an unique global-in-time solution in the mixed-space $L^{p(\cdot)}_{\frac{3}{2\alpha -1}}_x L^\infty_t$.
   
   To appear in ESAIM: Proceedings and Surveys - Preprint: arXiv
   
   

8. An Lp-theory for fractional stationary Navier-Stokes equations
   with O. Jarrin
   
   We consider the stationary (time-independent) Navier-Stokes equations in the whole three-dimensional space, under the action of a source term and with the fractional Laplacian operator 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  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 \alpha. 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 (with i.e.  \alpha = 2) stationary Navier-Stokes equations.
   
   Journal of Elliptic and Parabolic Equations (2024) 10:859-898 - doi, arXiv
   
   

9. Some remarks about the stationary Micropolar fluid-equations: existence, regularity and uniqueness
   with D. Chamorro and D. Llerena
   
   We consider here the stationary Micropolar fluid equations which are a particular generalization of the usual Navier-Stokes system where the microrotations of the fluid particles must be taken into account. We thus obtain two coupled equations: one based mainly in the velocity field u and the other one based in the microrotation field \omega. We will study in this work some problems related to the existence of weak solutions as well as some regularity and uniqueness properties. Our main result establish, under some suitable decay at infinity conditions for the velocity field only, the uniqueness of the trivial solution.
   
   Journal of Mathematical Analysis and Applications 536 (2), 128201, 2024 - doi, arXiv
   
   

10. Lebesgue spaces with variable exponent: some applications to the Navier-Stokes equations
    with D. Chamorro
    
    In this article we study some problems related to the incompressible 3D Navier-Stokes equations from the point of view of Lebesgue spaces of variable exponent. These functional spaces present some particularities that make them quite different from the usual Lebesgue spaces: indeed, some of the most classical tools in analysis are not available in this framework. We will give here some ideas to overcome some of the difficulties that arise in this context in order to obtain different results related to the existence of mild solutions for this evolution problem.
    
    Positivity 28, 24, 2024 - doi, arXiv 
    
    

11. Well-posedness of an oscillating water column in shallow water with time-dependent air pressure
    with E. Bocchi and J. He
    
    We propose in this paper a new nonlinear mathematical model of an oscillating water column (OWC). The one-dimensional shallow water equations in the presence of this device is reformulated as a transmission problem related to the interaction between waves and a fixed partially-immersed structure. By imposing the conservation of the total fluid- OWC energy in the non-damped scenario, we are able to derive a transmission condition that involves a time-dependent air pressure inside the chamber of the device, instead of a constant atmospheric pressure as in our previous paper. We then show that the transmission problem can be reduced to a quasilinear hyperbolic initial boundary value problem with a semi-linear boundary condition determined by an ODE depending on the trace of the solution to the PDE at the boundary. Local well-posedness for general problems of this type is established via an iterative scheme by using linear estimates for the PDE and nonlinear estimates for the ODE
    
    Journal of Nonlinear Science 33, 103, 2023 - doi, arXiv
    
    

12. Boundary controllability of a system modelling a partially immersed obstacle
    with G. Leugering and Y. Wang
    
    In this paper, we address the problem of boundary controllability for the one-dimensional nonlinear shallow water system, describing the free surface flow of water as well as the flow under a fixed gate structure. The system of differential equations considered can be interpreted as a simplified model of a particular type of wave energy device converter called oscillating water column. The physical requirements naturally lead to the problem of exact controllability in a prescribed region. In particular, we use the concept of nodal profile controllability in which at a given point (the node) time-dependent profiles for the states are required to be reachable by boundary controls. By rewriting the system into a hyperbolic system with nonlocal boundary conditions, we at first establish the semi-global classical solutions of the system, then get the local controllability and nodal profile using a constructive method. In addition, based on this constructive process, we provide an algorithmic concept to calculate the required boundary control function for generating a solution for solving these control problem.
    
    ESAIM: Control, Optimisation and Calculus of Variations 27, 80, 2021  - doi, HAL
    
    

13. Asymptotic behaviour of a system modelling rigid structures floating in a viscous fluid
    with D. Matignon and M. Tucsnak
    
    The PDE system introduced in Maity et al. (2019) describes the interaction of surface water waves with a floating solid, and takes into account the viscosity µ of the fluid. In this work, we study the Cummins type integro-differential equation for unbounded domains, that arises when the system is linearized around equilibrium conditions. A proof of the input-output stability of the system is given, thanks to a diffusive representation of the generalized fractional operator. Moreover, relying on Matignon (1996) stability result for fractional systems, explicit solutions are established both in the frequency and the time domains, leading to an explicit knowledge of the decay rate of the solution. Finally, numerical evidence is provided of the transition between different decay rates as a function of the viscosity µ.
    
    IFAC-PapersOnLine 54 (9), 205-212, 2021 - doi, HAL
    
    

14. Extended SIRU model with dynamic transmission rate and its application in the forecasting of COVID-19 under temporally varying public intervention
    with Y. Jiang
    
    By considering the recently introduced SIRU model, in this paper we study the dynamic of COVID-19 pandemic under the temporally varying public intervention in the Chilean context. More precisely, we propose a method to forecast cumulative daily reported cases $CR(t)$, and a systematic way to  identify the unreported daily cases given $CR(t)$ data. We firstly base on the recently introduced epidemic model SIRU (Susceptible, Asymptomatic Infected, Reported infected, Unreported infected), and focus on the transmission rate parameter $\tau$. To understand the dynamic of the data, we extend the scalar $\tau$ to an unknown function $\tau(t)$ in the SIRU system, which is then inferred directly from the historical $CR(t)$ data, based on nonparametric estimation. The estimation of $\tau(t)$ leads to the estimation of other unobserved functions in the system, including the daily unreported cases. Furthermore, the estimation of $\tau(t)$ allows us to build links between the pandemic evolution and the public intervention, which is modeled by logistic regression. We then employ polynomial approximation to construct a predicted curve which evolves with the latest trend of $CR(t)$. In addition, we regularize the evolution of the forecast in such a way that it corresponds to the future intervention plan based on the previously obtained link knowledge. We test the proposed predictor on different time windows. The promising results show the effectiveness of the proposed methods.
    
    BioMath 13 (2024), 2412176.
    
    

15. Well-posedness and input-output stability for a system modelling rigid structures floating in a viscous fluid
    with D. Matignon and M. Tucsnak
    
    We study a PDE based linearized model for the vertical motion of a solid floating at the free surface of a shallow viscous fluid. The solid is controlled by a vertical force exerted via an actuator. This force is the input of the system, whereas the output is the distance from the solid to the bottom. The first novelty we bring in is that we prove that the governing equations define a well-posed linear system. This is done by considering adequate function spaces and convenient operators between them. Another contribution of this work is establishing that the system is input-output stable. To this aim, we give an explicit form of the transfer function and we show that it lies in the Hardy space H∞ of the right-half plane.
    
    IFAC-PapersOnLine 53 (2), 7491-7496, 2021 - doi, HAL
    
    

16. Modelling and simulation of a wave energy converter
    with E. Bocchi and J. He
    
    In this work we present the mathematical model and simulations of a particular wave energy converter, the so-called oscillating water column. In this device, waves governed by the one-dimensional nonlinear shallow water equations arrive from offshore, encounter a step in the bottom and then arrive into a chamber to change the volume of the air to activate the turbine. The system is reformulated as two transmission problems: one is related to the wave motion over the stepped topography and the other one is related to the wave-structure interaction at the entrance of the chamber. We finally use the characteristic equations of Riemann invariants to obtain the discretized transmission conditions and we implement the Lax-Friedrichs scheme to get numerical solutions.
    
    ESAIM: Proceedings and Surveys 70, 68-83, 2021 - doi, HAL
    
    

17. Liouville-type theorems for  the 3D stationary MHD equations in Lebesgue spaces with variable exponent
    with J. Sun and J. Zhao
    
    In this paper, we study the Liouville-type problem of the three dimensional stationary MHD equations and establish some Liouville-type results under some additional conditions imposed on the velocity field $u$ and the magnetic field $b$ in terms of the Lebesgue spaces with variable exponent $L^{p(\cdot)}$. These functional spaces present some particularities that can relax the $L^{p(\cdot)}$ norm imposed on $(u,b)$ beyond the usual range $3\leq p(\cdot)\leq\frac{9}{2}$ in some non-negligible regions of $\mathbb{R}^3$, and thus provide us with some new interesting insights to study the Liouville-type problem of the stationary MHD equations. 
    
    Preprint posted in HAL
    
    

18. Finite time blow-up for a  nonlinear parabolic equation with smooth coefficients
    with O. Jarrin
    
    In this article, we consider an $n$-dimensional parabolic partial differential equation with a smooth coefficient term in the nonlinear gradient term. This equation was first introduced and analyzed in [E. Issoglio, On a non-linear transport-diffusion equation with distributional coefficients, Journal of Differential Equations, Volume 267, Issue 10 (2019)], where one of the main open questions is the possible finite-time blow-up of solutions. Here, leveraging a virial-type estimate, we provide a positive answer to this question within the framework of smooth solutions. 
    
    Preprint posted in HAL
    
    

19. Global well-posedness of the Navier-Stokes equations and the Keller-Segel system in variable Fourier-Besov spaces
    with J. Zhao
    
    In this paper, we study the Cauchy problem of the classical incompressible Navier-Stokes equations and the parabolic-elliptic Keller-Segel system in the framework of the Fourier-Besov spaces with variable regularity and integrability indices. By fully using some basic properties of these variable function spaces, we establish the linear estimates in variable Fourier-Besov spaces for the heat equation. Such estimates are fundamental for solving certain PDE's of parabolic type. As an applications, we prove global well-posedness in variable Fourier-Besov spaces for the 3D classical incompressible Navier-Stokes equations and the 3D parabolic-elliptic Keller-Segel system.
    
    Preprint posted in arXiv
    
    

20. On the Cauchy problem for the fractional Keller-Segel system in variable Lebesgue spaces
    with J. Zhao
    
    In this paper, we are mainly concerned with the well-posed problem of the fractional Keller-Segel system in the framework of variable Lebesgue spaces. Based on carefully examining the algebraical structure of the system, we reduced the fractional Keller--Segel system into the generalized nonlinear heat equation to overcome the difficulties caused by the boundedness of the Riesz potential in a variable Lebesgue spaces, then by mixing some structural properties of the variable Lebesgue spaces with the optimal decay estimates of the fractional heat kernel, we were able to establish two well-posedness results of the fractional Keller--Segel system in this functional setting.
    
    Preprint posted in arXiv
    
    

21. On the dynamics of the Coronavirus epidemic and the unreported cases: the Chilean case
    with A. Navas
    
    We analyze the dynamics of the COVID-19 epidemic taking into account the role of the unreported cases. After a first section in which we deal with a framework of very slow test capacity, we turn to the model recently introduced/implemented by Liu, Magal, Seydi and Webb. First, we prove some basic structural results for the corresponding ODE, as for instance the convergence of S(t) to a positive limit. These are similar to those of the classical SIR model, although the maxima of the corresponding curves are not necessarily unique. Finally, we implement the model -- but with a variable transmission rate -- in the Chilean context. A key parameter adjustment (namely, the fraction of unreported cases) is done via an argument using mortality rates. We conclude with several conclusions and lines of future research.
    
    Preprint posted in arXiv
    
    

Other investigations

• Observaciones sobre la dinamica de la epidemia de Coronavirus y los casos no reportados: el caso de Chile
  with A. Navas
  Published in the webpage of VIME of the University of Santiago of Chile.
  Available here.

• Estimacion de casos no reportados de infectados de COVID-19 en Chile, el Maule y la Araucana durante marzo de 2020
  with M. Candia
  Not intended for publication in a scientific journal.
  Preprint available on the HAL website. 

• Aproximacion numérica de derivadas fraccionarias por polinomios ortogonales
  with S. Berres
  Published in the proceedings of the Mathematical Meeting of Sud Zone 2018, Chile.