Capítulos (Ecuaciones Diferenciales y Análisis Numérico)

URI permanente para esta colecciónhttps://hdl.handle.net/11441/10835

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Remarks on the Control of Family of b–Equations
(Elsevier, 2019) Fernández Cara, Enrique; Araujo de Souza, Diego; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM131: Ec.diferenciales,Simulacion Num.y Desarrollo Software
Presents cutting-edge results in the areas of control theory and PDEs, giving a broad picture of recent advances Contains contributions from leading experts Includes theoretical studies and models for applications
Nonlocal elliptic system arising from the growth of cancer stem cells
(AIMS, 2018-04-18) Delgado Delgado, Manuel; Mendes Duarte, Ítalo Bruno; Suárez Fernández, Antonio; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM131: Ec.diferenciales,Simulacion Num.y Desarrollo Software
In this work we show the existence of coexistence states for a nonlocal elliptic system arising from the growth of cancer stem cells. For this, we use the bifurcation method and the theory of the fixed point index in cones. Moreover, in some cases we study the behaviour of the coexistence region, depending on the parameters of the problem.
Multi-valued random dynamics og stochastic wave equations with infinite delays
(AIMS, 2022-10-01) Wang, Jingyu; Wang, Yejuan; Caraballo Garrido, Tomás; Universidad de Sevilla. Departamento de Ecuaciones diferenciales y Análisis numérico; Universidad de Sevilla. FQM314: Análisis Estocástico de Sistemas Diferenciales
This paper is devoted to the asymptotic behavior of solutions to a non-autonomous stochastic wave equation with infinite delays. The nonlinear terms of the equation are not expected to be Lipschitz continuous, but only satisfy continuity assumptions along with growth conditions, under which the uniqueness of the solutions may not hold. Using the theory of multi-valued non-autonomous random dynamical systems, we prove the existence and measurability of a compact global pullback attractor.
Low rank approximation of multidimensional data
(Springer, 2019) Azaïez, Mejdi; Lestandi, Lucas; Chacón Rebollo, Tomás; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Universidad de Sevilla. FQM120: Modelado Matemático y Simulación de Sistemas Medioambientales
In the last decades, numerical simulation has experienced tremendous improvements driven by massive growth of computing power. Exascale computing has been achieved this year and will allow solving ever more complex problems. But such large systems produce colossal amounts of data which leads to its own difficulties. Moreover, many engineering problems such as multiphysics or optimisation and control, require far more power that any computer architecture could achieve within the current scientific computing paradigm. In this chapter, we propose to shift the paradigm in order to break the curse of dimensionality by introducing decomposition to reduced data. We present an extended review of data reduction techniques and intends to bridge between applied mathematics community and the computational mechanics one. The chapter is organized into two parts. In the first one bivariate separation is studied, including discussions on the equivalence of proper orthogonal decomposition (POD, continuous framework) and singular value decomposition (SVD, discrete matrices). Then, in the second part, a wide review of tensor formats and their approximation is proposed. Such work has already been provided in the literature but either on separate papers or into a pure applied mathematics framework. Here, we offer to the data enthusiast scientist a description of Canonical, Tucker, Hierarchical and Tensor train formats including their approximation algorithms. When it is possible, a careful analysis of the link between continuous and discrete methods will be performed.
A random model for immune response to virus in fluctuating environments
(Springer, 2016) Asai, Yusuke; Caraballo Garrido, Tomás; Han, Xiaoying; Kloeden, Peter E.; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Sadovnichiy, Victor A.; Zgurovsky, Mikhail Z.; Universidad de Sevilla. FQM314: Análisis Estocástico de Sistemas Diferenciales
In this work we study a model for virus dynamics with a random immune response and a random production rate of susceptible cells from cell proliferation. In traditional models for virus dynamics, the rate at which the viruses are cleared by the immune system is constant, and the rate at which susceptible cells are provided is constant or a function depending on the population of all cells. However, the human body in general is never stationary, and thus these rates can barely be constant. Here we assume that the human body is a random environment and models the rates by random processes, which result in a system of random differential equations. We then analyze the long term behavior of the random system, in particular the existence and geometric structure of the random attractor, by using the theory of random dynamical systems. Numerical simulations are provided to illustrate the theoretical result.
Some aspects concerning the dynamics of stochastic chemostats
(Springer, 2016) Caraballo Garrido, Tomás; Garrido Atienza, María José; López de la Cruz, Javier; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Sadovnichiy, Victor A.; Zgurovsky, Mikhail Z.; Universidad de Sevilla. FQM314: Análisis Estocástico de Sistemas Diferenciales
In this paper we study a simple chemostat model influenced by white noise which makes this kind of models more realistic. We use the theory of random attractors and, to that end, we first perform a change of variable using the OrnsteinUhlenbeck process, transforming our stochastic model into a system of differential equations with random coefficients. After proving that this random system possesses a unique solution for any initial value, we analyze the existence of random attractors. Finally we illustrate our results with some numerical simulations.
Control of weakly blowing up semilinear heat equations
(Springer, 2002) Fernández Cara, Enrique; Zuazua Iriondo, Enrique; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Berestycki, Henri; Pomeau, Yves; Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software
In these notes we consider a semilinear heat equation in a bounded domain of IRd , with control on a subdomain and homogeneous Dirichlet boundary conditions. We consider nonlinearities for which, in the absence of control, blow up arises. We prove that when the nonlinearity grows at infinity fast enough, due to the local (in space) nature of the blow up phenomena, the control may not avoid the blow up to occur for suitable initial data. This is done by means of localized energy estimates. However, we also show that when the nonlinearity is weak enough, and provided the system admits a globally defined solution (for some initial data and control), the choice of a suitable control guarantees the global existence of solutions and moreover that the solution may be driven in any finite time to the globally defined solution. In order for this to be true we require the nonlinearity f to satisfy at infinity the growth condition f(s) |s| log3/2 (1 + |s|) → 0 as |s| → ∞. This is done by means of a fixed point argument and a careful analysis of the control of linearized heat equations relying on global Carleman estimates. The problem of controlling the blow up in this sense remains open for nonlinearities growing at infinity like f(s) ∼ |s|logp (1 + |s|) with 3/2 ≤ p ≤ 2.
The control of PDEs: some basic concepts, recent results and open problems
(Jindřich Nečas Center for Mathematical Modeling, 2012) Fernández Cara, Enrique; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Kaplický, Petr; Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software
These Notes deal with the control of systems governed by some PDEs. I will mainly consider time-dependent problems. The aim is to present some fundamental results, some applications and some open problems related to the optimal control and the controllability properties of these systems. In Chapter 1, I will review part of the existing theory for the optimal control of partial differential systems. This is a very broad subject and there have been so many contributions in this field over the last years that we will have to limit considerably the scope. In fact, I will only analyze a few questions concerning some very particular PDEs. We shall focus on the Laplace, the stationary Navier-Stokes and the heat equations. Of course, the existing theory allows to handle much more complex situations. Chapter 2 is devoted to the controllability of some systems governed by linear time-dependent PDEs. I will consider the heat and the wave equations. I will try to explain which is the meaning of controllability and which kind of controllability properties can be expected to be satisfied by each of these PDEs. The main related results, together with the main ideas in their proofs, will be recalled. Finally, Chapter 3 is devoted to present some controllability results for other time-dependent, mainly nonlinear, parabolic systems of PDEs. First, we will revisit the heat equation and some extensions. Then, some controllability results will be presented for systems governed by stochastic PDEs. Finally, I will consider several nonlinear systems from fluid mechanics: Burgers, NavierStokes, Boussinesq, micropolar, etc. Along these Notes, a set of questions (some of them easy, some of them more intrincate or even difficult) will be stated. Also, several open problems will be mentioned. I hope that all this will help to understand the underlying basic concepts and results and to motivate research on the subject.
Some indefinite nonlinear eigenvalue problems
(World Scientific, 2004) Suárez Fernández, Antonio; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Delgado Delgado, Manuel; López Gómez, Julián; Ortega Ríos, Rafael; Suárez Fernández, Antonio; Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software
In this work we study the structure of the set of positive solutions of a nonlinear eigenvalue problem with a weight changing sign. Specifically, the reaction term arises from a population dynamic model. We use mainly bifurcation methods to obtain our results.
Some elliptic problems with nonlinear boundary conditions
(World Scientific, 2005) Morales Rodrigo, Cristian; Suárez Fernández, Antonio; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Cano Casanova, Santiago; López Gómez, Julián; Mora Corral, Carlos; Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software
This paper concerns with some elliptic equations with non-linear boundary conditions. Sub-supersolution and bifurcation methods are used in order to obtain existence, uniqueness or multiplicity of positive solutions.
Attracting complex networks
(Springer, 2016) Guerrero Suárez, Giovanny Fabián; Langa Rosado, José Antonio; Suárez Fernández, Antonio; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Commendatore, Pasquale; Matilla García, Mariano; Varela Cabo, Luis Miguel; Cánovas Peña, José Salvador; Universidad de Sevilla. FQM314: Análisis Estocástico de Sistemas Diferenciales
Real phenomena from different areas of Life Sciences can be described by complex networks, whose structure is usually determining their intrinsic dynamics. On the other hand, Dynamical Systems Theory is a powerful tool for the study of evolution processes in real situations. The concept of global attractor is the central one in this theory. In the last decades there has been an intensive research in the geometrical characterization of global attractors. However, there still exists a weak connection between the asymptotic dynamics of a complex network and the structure of associated global attractors. In this paper we show that, in order to analyze the long-time behavior of the dynamics on a complex network, it is the topological and geometrical structure of the attractor the subject to take into account. In fact, given a complex network, a global attractor can be understood as the new attracting complex network which is really describing and determining the forwards dynamics of the phenomena. We illustrate our discussion with models of differential equations related to mutualistic complex networks in Economy and Ecology.
On the regularity of the Q-Tensor depending on the data
(Universidad de Zaragoza, 2015) Guillén González, Francisco Manuel; Rodríguez Bellido, María Ángeles; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Ahusborde, Étienne; Amrouche, Chérif; López de Silanes Busto, María Cruz; Palacios Latasa, Manuel Pedro; Sanz Sáiz, Gerardo; Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software
The coupled Navier-Stokes and Q-Tensor system is one of the models used to describe the behavior of the nematic liquid crystals, an intermediate phase between crystalline solids and isotropic fluids. These equations model the dynamics of the fluid via velocity and pressure (u, p) and the orientation of the molecules via a tensor Q. A review on the existence of weak solutions, maximum principle and a uniqueness criteria can be seen in [Guillen-González, F., and Rodríguez-Bellido, M. Á. Some properties on the Q-tensor system. Monogr. Mat. García Galdeano 39 (2014), 133–145] (the corresponding Cauchy problem in the whole R3 is analyzed by Zarnescu. However, the regularity of such solutions is only analyzed under some restrictive conditions: large viscosity or periodic boundary conditions. In this work, we study two different types of regularity for the Q-Tensor model: one inherited from the usual strong solution for the Navier-Stokes equations, and another one where (u, Q) and (∂tu, ∂tQ) have weak regularity (weak-t). This latter regularity is introduced due to the impossibility of obtaining local in time strong estimates for nonperiodic boundary conditions, where only the existence (and uniqueness) of local weak-t solution is obtained. Some regularity criteria for (u, Q) will also be given. In the particular case of Neumann boundary conditions for Q, the regularity criteria only must be imposed for the velocity u).
Some properties on the Q-Tensor system
(Universidad de Zaragoza, 2014) Guillén González, Francisco Manuel; Rodríguez Bellido, María Ángeles; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; López de Silanes Busto, María Cruz; Palacios Latasa, Manuel Pedro; Sanz Sáiz, Gerardo; Amrouche, Chérif; Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software
We study the coupled Navier-Stokes and Q-Tensor system (analyzed in cf. [Paicu, M., and Zarnescu, A. Energy dissipation and regularity for a coupled Navier-Stokes and Q-tensor system. Arch. Ration. Mech. Anal. 203 (2012), 45–67] in the whole R3) in a bounded three-dimensional domain for several boundary conditions, rewriting the system in a way that properties as symmetry and null-trace for the tensor Q can be proved. We show some analytical results such as: the existence of global in time weak solution, a maximum principle for the Q-tensor, local in time strong solution (which is global assuming an additional regularity criterion for the velocity in the space-periodic boundary condition case), global in time strong solution imposing dominant viscosity (for the space-periodic or homogeneous Neumann boundary condition cases) and regularity criteria for uniqueness of weak solutions.
Lévy-areas of Ornstein-Uhlenbeck processes in Hilbert-spaces
(Springer, 2015) Garrido Atienza, María José; Lu, Kening; Schmalfuss, Björn; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Sadovnichiy, Victor A.; Zgurovsky, Mikhail Z.; Universidad de Sevilla. FQM314: Análisis Estocástico de Sistemas Diferenciales
In this paper we investigate the existence and some useful properties of the Lévy areas of Ornstein-Uhlenbeck processes associated to Hilbert-space-valued fractional Brownian-motions with Hurst parameter H ∈ (1/3, 1/2]. We prove that this stochastic area has a Hölder-continuous version with sufficiently large Hölder-exponent and that can be approximated by smooth areas. In addition, we prove the stationarity of this area.
The sub-supersolution method for Kirchhoff systems: applications
(Springer, 2015) Malcher Figueiredo, Giovany de Jesus; Suárez Fernández, Antonio; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Carvalho, Alexandre Nolasco; Ruf, Bernhard; Moreira dos Santos, Ederson; Gossez, Jean-Pierre; Monari Soares, Sergio Henrique; Cazenave, Thierry; Universidad de Sevilla. FQM131: Ec.diferenciales,Simulación Num.y Desarrollo Software
In this paper we prove that the sub-supersolution method works for general Kirchhoff systems. We apply the cited method to prove the existence of positive solutions for some specific models.
Well-balanced Finite Volume schemes: some stability and convergence results
(Universidad de Zaragoza, 2003) Chacón Rebollo, Tomás; Domínguez Delgado, Antonio; Fernández Nieto, Enrique Domingo; Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico; Palacios Latasa, Manuel Pedro; Trujillo, David; Torrens Iñigo, Juan José; Madaune-Tort, Monique; López de Silanes Busto, María Cruz; Sanz Sáiz, Gerardo
We report a stability and convergence analysis for some simplified well-balanced Finite Volume solvers of Hyperbolic Systems of Conservation Laws. These are specific solvers, recently introduced, that balance all steady solutions up to second order of accuracy by means of an additional numerical source term. We prove the stability and convergence of some of these solvers for scalar hyperbolic equations under reasonable conditions on the additional term.

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