Artículos (Ecuaciones Diferenciales y Análisis Numérico)

URI permanente para esta colecciónhttps://hdl.handle.net/11441/10834

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Estimates for the asymptotic expansion of a viscous fluid satisfying Navier's law on a rugous boundary
(Wiley, 2011) Casado Díaz, Juan; Luna Laynez, Manuel; Suárez Grau, Francisco Javier; Ecuaciones Diferenciales y Análisis Numérico; Ministerio de Ciencia e Innovación (MICIN). España; FQM309: Control y Homogeneización de Ecuaciones en Derivadas Parciales
In a previous paper, we have studied the asymptotic behavior of a viscous fluid satisfying Navier's law on a periodic rugous boundary of period ε and amplitude δε, with δε/ε tending to zero. In the critical size, δε∼ε3/2, in order to obtain a strong approximation of the velocity and the pressure it is necessary to consider a boundary layer term in the corresponding ansatz. The purpose of this paper is to estimate the approximation given by this ansatz.
Uniform Bounds with Difference Quotients for Proper Orthogonal Decomposition Reduced Order Models of the Burgers Equation
(Springer, 2023-03-20) Koc, Birgul; Rubino, Samuele; Chacón Rebollo, Tomás; Ecuaciones Diferenciales y Análisis Numérico
In this paper, we prove uniform error bounds for proper orthogonal decomposition (POD) reduced ordermodeling (ROM) of Burgers equation, considering difference quotients (DQs), introduced in Kunisch and Volkwein (Numer Math 90(1):117–148, 2001). In particular, we study the behavior of the DQ ROM error bounds by considering L2(_) and H1 0 (_) POD spaces and l∞ (L2) and natural-norm errors. We present some meaningful numerical tests checking the behavior of error bounds. Based on our numerical results, DQ ROM errors are several orders of magnitude smaller than noDQ ones (in which the POD is constructed in a standard way, i.e., without the DQ approach) in terms of the energy kept by the ROM basis. Further, noDQ ROM errors have an optimal behavior, while DQ ROM errors, where the DQ is added to the POD process, demonstrate an optimality/super-optimality behavior. It is conjectured that this possibly occurs because the DQ inner products allow the time dependency in the ROM spaces to make an impact.
Residual-based data-driven variational multiscale reduced order models for parameter-dependent problems
(Springer, 2025-06-04) Koc, Birgul; Rubino, Samuele; Chacón Rebollo, Tomás; Iliescu, Traian; Ecuaciones Diferenciales y Análisis Numérico
In this paper, we propose a novel residual-based data-driven closure strategy for reduced order models (ROMs) of under-resolved, convection-dominated problems. The new ROM closure model is constructed in a variational multiscale (VMS) framework by using the available full order model data and a model form ansatz that depends on the ROM residual. We emphasize that this closure modeling strategy is fundamentally different from the current data-driven ROM closures, which generally depend on the ROM coefficients. We investigate the new residual-based data-driven VMS ROM closure strategy in the numerical simulation of three test problems: (i) a one-dimensional parameter-dependent advection-diffusion problem; (ii) a two-dimensional time-dependent advection-diffusion-reaction problem with a small diffusion coefficient (ε=1e−4); and (iii) a two-dimensional flow past a cylinder at Reynolds number Re=1000. Our numerical investigation shows that the new residual-based data-driven VMS-ROM is more accurate than the standard coefficient-based data-driven VMS-ROM.
Pullback asymptotic behavior and statistical solutions for lattice Klein-Gordon-Schrödinger equations with varying coefficient
(American Institute of Mathematical Sciences, 2025-02-14) Zhao, Caidi; Zhuang, Rong; Caraballo Garrido, Tomás; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM314: Análisis Estocástico de Sistemas Diferenciales
In this article, the authors investigate the pullback asymptotic behavior and statistical solutions for lattice Klein-Gordon-Schrödinger equations with varying coefficient. They first prove the global well-posedness of the addressed equations and the existence of a family of time-dependent pullback attractor for the associated process acting on the time-dependent phase spaces. Then they verify that the process possesses a family of invariant Borel probability measures with support contained in the time-dependent pullback attractor. Further, they reformulate the definition of statistical solution for the evolutionary equations on time-dependent phase spaces. As a result, they prove the existence of statistical solution for the lattice Klein-Gordon-Schrödinger equations with varying coefficient and show that it satisfies the Liouville theorem.
Strong Convergence of Solutions and Attractors for Reaction-Diffusion Equations Governed by a Fractional Laplacian
(Springer, 2025-02-20) Xu, Jiaohui; Caraballo Garrido, Tomás; Valero, José; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM314: Análisis Estocástico de Sistemas Diferenciales
A nonlocal reaction-diffusion equation governed by a fractional Laplace operator on a bounded domain is studied in this paper. First, the strong convergence of solutions of the equations governed by fractional Laplacian to the solutions of the classical equations governed by a standard Laplace operator is proved, when the fractional parameter grows to 1. Second, for the autonomous case, the upper semicontinuity of global attractors with respect to the attractors of the limit problem is established. Apparently, these are the first results for this kind of problems on bounded domains.
A Delay Nonlocal Quasilinear Chafee–Infante Problem: An Approach via Semigroup Theory
(Springer, 2025-02-19) Caraballo Garrido, Tomás; Carvalho, A.N.; Julio, Yessica; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM314: Análisis Estocástico de Sistemas Diferenciales
In this work we study a dissipative one dimensional scalar parabolic problem with non-local nonlinear diffusion with delay. We consider the general situation in which the functions involved are only continuous and solutions may not be unique. We establish conditions for global existence and prove the existence of global attractors. All results are presented only in the autonomous since the non-autonomous case follows in the same way, including the existence of pullback attractors. A particularly interesting feature is that there is a semilinear problem (nonlocal in space and in time) from which one can obtain all solutions of the associated quasilinear problem and that for this semilinear problem the delay depends on the initial function making its study more involved.
Exponential attractors with explicit fractal dimensions for functional differential equations in Banach spaces
(Springer, 2025-04-17) Hu, Wenjie; Caraballo Garrido, Tomás; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM314: Análisis Estocástico de Sistemas Diferenciales
The aim of this paper is to propose a new method to construct exponential attractors for infinite dimensional dynamical systems in Banach spaces with explicit fractal dimensions. The approach is established by combining the squeezing properties and the covering of finite subspace of Banach spaces, which generalizes the method established in Hilbert spaces. The method is especially effective for functional differential equations in Banach spaces for which state decomposition of the linear part can be adopted to prove squeezing property. The theoretical results are applied to retarded functional differential equations and retarded reaction-diffusion equations for which the constructed exponential attractors possess explicit fractal dimensions that do not depend on the entropy number but only depend on the spectrum of the linear parts and Lipschitz constants of the nonlinear parts.
Asymptotical behavior of the 2D stochastic partial dissipative Boussinesq system with memory
(Elsevier, 2025-04-25) Dai, Haoran; You, Bo; Caraballo Garrido, Tomás; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM314: Análisis Estocástico de Sistemas Diferenciales
The objective of this paper is to consider the asymptotical behavior of solutions for the two-dimensional partial dissipative Boussinesq system with memory and additive noise. We first establish the existence of a random absorbing set in the phase space. However, due to the presence of the memory term, we cannot obtain some kind of compactness of the corresponding cocycle through Sobolev compactness embedding theorem or by verifying the pullback flattening property. To overcome this difficulty, we first prove the asymptotical compactness of the velocity component of weak solutions, and then we prove the asymptotical compactness of other components based on some energy estimates and the Aubin–Lions compactness lemma, which implies the asymptotical compactness of the corresponding cocycle. Thus, the existence of a random attractor is obtained. Finally, we establish an abstract result about some kind of upper semi-continuity of the random attractor, which is applied to the two-dimensional partial dissipative Boussinesq system.
Higher-Order Continuity of Pullback Random Attractors for Random Quasilinear Equations with Nonlinear Colored Noise
(Springer, 2023-12-03) Li, Yangrong; Wang, Fengling; Caraballo Garrido, Tomás; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM314: Análisis Estocástico de Sistemas Diferenciales
For a nonautonomous random dynamical system, we introduce a concept of a pullback random bi-spatial attractor (PRBA). We prove an existence theorem of a PRBA, which includes its measurability, compactness and attraction in the regular space. We then establish the residual dense continuity of a family of PRBAs from a parameter space into the space of all compact subsets of the regular space equipped by Hausdorff metric. The abstract results are illustrated in the nonautonomous random quasilinear equation driven by nonlinear colored noise, where the size of noise belongs to (0,∞] and the infinite size corresponds to the deterministic equation. The application results are the existence and residual dense continuity of PRBAs on (0,∞] in both square and p-order Lebesgue spaces, where p>2. The lower semi-continuity of attractors in the regular space seems to be a new subject even for an autonomous deterministic system.
Asymptotic behavior of stochastic delay Navier-Stokes equations on unbounded domains
(International Press, 2025-04-11) Zhang, Qiangheng; Caraballo Garrido, Tomás; Yang, Shuang; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM314: Análisis Estocástico de Sistemas Diferenciales
In this paper, the random dynamics of non-autonomous stochastic Navier-Stokes equations with variable delays on unbounded Poincaré domains is analysed. First, we establish the existence, uniqueness and backward compactness of pullback random attractors. Second, we study the upper semicontinuity of pullback random attractors as the delay time tends to zero. Finally, we investigate the backward asymptotic autonomy of pullback random attractors. Due to the non-compactness of Sobolev embeddings on unbounded domains, we introduce a stream function to prove backward uniform tailends smallest of solutions, and then establish the backward asymptotic compactness of the solution operators.
Existence and dimensions of global attractors for a delayed reaction-diffusion equation on an unbounded domain
(IOP Publishing, 2025-06-05) Hu, Wenjie; Caraballo Garrido, Tomás; Miranville, Alain; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM314: Análisis Estocástico de Sistemas Diferenciales
The purpose of this paper is to investigate the existence and Hausdorff dimension as well as fractal dimension of global attractors for a delayed reaction-diffusion equation on an unbounded domain. The noncompactness of the domain causes the Laplace operator to have a continuous spectrum, the semigroup generated by the linear part and the Sobolev embeddings are no longer compact, making the problem more difficult compared with the bounded domain case. We first obtain the existence of an absorbing set for the infinite dimensional dynamical system generated by the equation thanks to a priori estimates of the solution. Then, we show the asymptotic compactness of the solution semiflow by uniform a priori estimates for far-field values of solutions together with the Arzelà–Ascoli theorem, which facilitates us to show the existence of global attractors. By decomposing the solution into three parts and establishing squeezing properties of each part, we obtain the explicit upper bounds of both Hausdorff dimension and fractal dimension of the global attractors, which only depend on the inner characteristics of the equation, while not related to the entropy number compared with the existing literature.
Ecuaciones diferenciales como jeroglíficos
(2025-01-15) Langa Rosado, José Antonio; Lukaszewicz, Grzegorz; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM314: Análisis Estocástico de Sistemas Diferenciales
La matemática supone un lenguaje que se adapta a lo real de manera sorprendente. En particular, las ecuaciones diferenciales utilizan una serie de símbolos propios, con un significado muy rico, no solo por su contenido abstracto, sino por la manera en que describen fenómenos del mundo físico o natural. En este sentido, es apropiada la metáfora de que el estudio de un sistema de ecuaciones diferenciales se asemeja a enfrentrarse a un jeroglífico. Este no solo informa de un conjunto de propiedades matemáticas del modelo, sino que es capaz de aprehender lo íntimo de lo real, con la materia, la energía y la información como sus constitu- yentes básicos, los cuales encuentran una expresión en este lenguaje tan impresionante que supone la matemática.
Stability and convergence for a complete model of mass diffusion
(Elsevier, 2011) Cabrales, R. C.; Guillén González, Francisco Manuel; Gutiérrez Santacreu, Juan Vicente; Universidad de Sevilla. Departamento de Matemática Aplicada I (ETSII); Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico
We propose a fully discrete scheme for approximating a three-dimensional, strongly nonlinear model of mass diffusion, also called the complete Kazhikhov–Smagulov model. The scheme uses a C0 finite-element approximation for all unknowns (density, velocity and pressure), even though the density limit, solution of the continuous problem, belongs to H2. A first-order time discretization is used such that, at each time step, one only needs to solve two decoupled linear problems for the discrete density and the velocity–pressure, separately. We extend to the complete model, some stability and convergence results already obtained by the last two authors for a simplified model where λ2-terms are not considered, λ being the mass diffusion coefficient. Now, different arguments must be introduced, based mainly on an induction process with respect to the time step, obtaining at the same time the three main properties of the scheme: an approximate discrete maximum principle for the density, weak estimates for the velocity and strong ones for the density. Furthermore, the convergence towards a weak solution of the density-dependent Navier–Stokes problem is also obtained as λ→0 (jointly with the space and time parameters). Finally, some numerical computations prove the practical usefulness of the scheme.
Theoretical and numerical results for some bi-objective optimal control problems
(AIMS, 2019-04-14) Fernández Cara, Enrique; Marín Gayte, Irene; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM131: Ec.diferenciales,Simulacion Num.y Desarrollo Software
This article deals with the solution of some multi-objective optimal control problems for several PDEs: linear and semilinear elliptic equations and stationary Navier-Stokes systems. More precisely, we look for Pareto equilibria associated to standard cost functionals. First, we study the linear and semilinear cases. We prove the existence of equilibria, we deduce appropriate optimality systems, we present some iterative algorithms and we establish convergence results. Then, we analyze the existence and characterization of Pareto equilibria for the Navier-Stokes equations. Here, we use the formalism of Dubovitskii and Milyutin. In this framework, we also present a finite element approximation of the bi-objective problem and we illustrate the techniques with several numerical experiments.
Theoretical and numerical local null controllability of a quasi-linear parabolic equation in dimensions 2 and 3
(Elsevier, 2021-01-27) Fernández Cara, Enrique; Límaco, Juan; Marín Gayte, Irene; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM131: Ec.diferenciales,Simulacion Num.y Desarrollo Software
This paper is devoted to the theoretical and numerical analysis of the null controllability of a quasi-linear parabolic equation. First, we establish a local controllability result. The proof relies on an appropriate inverse function argument. Then, we formulate an iterative algorithm for the computation of the null control and we prove a convergence result. Finally, we illustrate the analysis with some numerical experiments.
Multiobjective optimal control problems. Stationary Navier-Stokes equations
(Taylor & Francis, 2024-07-31) Gayte Delgado, Inmaculada; Marín Gayte, Irene; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM131: Ec.diferenciales,Simulacion Num.y Desarrollo Software
This paper deals with the solution of some multi-objective optimal control problems for stationary Navier-Stokes equations. More precisely, we look for Pareto and Nash equilibria associated to standard cost functionals. First, we prove the existence of equilibria and we deduce appropriate optimality systems. Then, we analyze the existence and characterization of Pareto and Nash equilibria for the Navier-Stokes equations. Here, we use the formalism of Dubovitskii and Milyoutin., see [13]. Finally, we also present a finite element approximation of the bi-objective problem, we illustrate the techniques with several numerical experiments and we compare the Pareto and Nash equilibria.
Mathematical modelling and computational reduction of molten glass fluid flow in a furnace melting basin
(Springer, 2024-08-25) Ballarin, Francesco; Delgado Ávila, Enrique; Mola, Andrea; Rozza, Gianluigi; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM120: Modelado Matemático y Simulación de Sistemas Medioambientales
In this work, we present the modelling and numerical simulation of a molten glass fluid flow in a furnace melting basin. We first derive a model for a molten glass fluid flow and present numerical simulations based on the finite element method (FEM). We further discuss and validate the results obtained from the simulations by comparing them with experimental results. Finally, we also present a non-intrusive proper orthogonal decomposition (POD) based on artificial neural networks (ANN) to efficiently handle scenarios which require multiple simulations of the fluid flow upon changing parameters of relevant industrial interest. This approach lets us obtain solutions of a complex 3D model, with good accuracy with respect to the FEM solution, yet with negligible associated computational times.
A logistic type equation in Rᴺ with a nonlocal reaction term via bifurcation method
(Elsevier, 2021-01-01) Delgado Delgado, Manuel; Molina Becerra, Mónica; Suárez Fernández, Antonio; Universidad de Sevilla. Departamento de Matemática Aplicada II (ETSI); Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; European Commission (EC). Fondo Europeo de Desarrollo Regional (FEDER); Universidad de Sevilla. TIC130: Investigación en Sistemas Dinámicos en Ingeniería; Universidad de Sevilla. FQM131: Ecuaciones Diferenciales, Simulación Numérica y Desarrollo Software
We study the existence of positive solutions of a logistic equation in the entire space with a nonlocal reaction term. Mainly, we apply a bifurcation method and singular boundary equations to obtain a priori bounds of the solutions. Our results show a drastic change of behaviour of the set of positive solutions depending on the sign of the nonlocal term.
Some control results for simplified one-dimensional models of fluid-solid interaction
(World Scientific, 2005-05-10) Doubova Krasotchenko, Anna; Fernández Cara, Enrique; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM131: Ec.diferenciales,Simulacion Num.y Desarrollo Software
We analyze the null controllability of a one-dimensional nonlinear system which models the interaction of a fluid and a particle. This can be viewed as a first step in the control analysis of fluid-solid systems. The fluid is governed by the Burgers equation and the control is exerted at the boundary points. We present two main results: the global null controllability of a linearized system and the local null controllability of the nonlinear original model. The proofs rely on appropriate global Carleman inequalities, observability estimates and fixed point arguments.
Reconstruction of degenerate conductivity region for parabolic equations
(IOP Publishing, 2024-03-15) Cannarsa, Piermarco; Doubova Krasotchenko, Anna; Yamamoto, Masahiro; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM131: Ec.diferenciales,Simulacion Num.y Desarrollo Software
We consider an inverse problem of reconstructing a degeneracy point in the diffusion coefficient in a one-dimensional parabolic equation by measuring the normal derivative on one side of the domain boundary. We analyze the sensitivity of the inverse problem to the initial data. We give sufficient conditions on the initial data for uniqueness and stability for the one-point measurement and show some examples of positive and negative results. On the other hand, we present more general uniqueness results, also for the identification of an initial data by measurements distributed over time. The proofs are based on an explicit form of the solution by means of Bessel functions of the first type. Finally, the theoretical results are supported by numerical experiments.

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