Research

Submitted

J.W. Bueno and G. Peralta, Long-term dynamics of Stuart–Landau model and Kuramoto model with non-local coupling, 2023.

G. Peralta, Partially dissipative viscous system of balance laws and application to Kuznetsov–Westervelt equation, 2023.
abstract We provide the well-posedness for partially dissipative viscous system of balance laws in smooth Sobolev spaces under the same assumptions as in the case of inviscid balance laws. A priori estimates for coupled hyperbolic-parabolic linear systems with coefficients having limited regularity are derived using Friedrichs regularization and Moser-type estimates. Local existence for nonlinear systems will be established using the results of the linear theory and a suitable iteration scheme. The local existence theory is then applied to the Kuznetsov–Westervelt equation with damping for nonlinear wave acoustic propagation. Existence of global solutions for small data and their asymptotic stability are established.

G. Peralta, Optimal control for non-isothermal binary viscoelastic and incompressible flows with stress diffusion, 2023.
abstract Distributed optimal control problems for a binary mixture of non-isothermal, incompressible, and non-Newtonian fluids under the framework of diffusive Johnson–Segalman models will be discussed. The flow is governed by a coupling of the two-dimensional Cahn–Hilliard equation for the order parameter and chemical potential, the biharmonic heat equation with Voigt-type damping for the temperature, the incompressible Navier–Stokes equation for the mean velocity, and a Jeffreys-type differential constitutive equation for the extra stress tensor. The total Cauchy stress tensor for the model is given by the sum of the viscous stress, the contribution due to surface tension, and a quadratic function of the elastic stress. The latter is based on a recent non-standard constitutive law for the Helmholtz free energy. The coefficients pertaining to diffusion processes depend on the concentration and temperature. We provide regularity results for the optimal control with various objective cost functionals. Such results rely on careful analysis of the corresponding linearized and adjoint problems. In particular, we study the strong, weak, and very weak solutions of the linearized and adjoint systems, and present the function spaces for the distributional time-derivatives.

Published Articles

[25] G. Peralta, Stability properties of multidimensional symmetric hyperbolic systems with damping, differential constraints and delay, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, pp. 1-43, 2023.
abstract Multidimensional linear hyperbolic systems with constraints and delay are considered. The existence and uniqueness of solutions for rough data are established using Friedrichs method. With additional regularity and compatibility on the initial data and initial history, the stability of such systems are discussed. Under suitable assumptions on the coefficient matrices, we establish standard or regularity-loss type decay estimates. For data that are integrable, better decay rates are provided. The results are applied to the wave, Timoshenko, and linearized Euler–Maxwell systems with delay.   pdf   link

[24] G. Peralta, Optimal Borel measure-valued controls to the viscous Cahn–Hilliard–Oberbeck–Boussinesq phase-field system on two-dimensional bounded domains, ESAIM Control, Optimisation and Calculus of Variations 29, Paper No. 32, 43 p., 2023.
abstract We consider an optimal control problem for the two-dimensional viscous Cahn–Hilliard–Oberbeck–Boussinesq system with controls that take values in the space of regular Borel measures. The state equation models the interaction between two incompressible non-isothermal viscous fluids. Local distributed controls with constraints are applied in either of the equations governing the dynamics for the concentration, mean velocity, and temperature. Necessary and sufficient conditions characterizing local optimality in terms of the Lagrangian will be demonstrated. These conditions will be obtained through regularity results for the associated adjoint system, a priori estimates for the solutions of the linearized system in a weaker norm compared to that of the state space, and the Lebesgue decomposition of Borel measures.   pdf   link   Zbl 1514.35446

[23] G. Peralta, Weak and very weak solutions to the viscous Cahn–Hilliard–Oberbeck–Boussinesq phase-field system on two-dimensional bounded domains, Journal of Evolution Equations 22 (1), Paper No. 12, 71 p., 2022.
abstract In this paper, we consider weak and very weak solutions to the viscous Cahn–Hilliard–Oberbeck–Boussinesq system for non-isothermal, viscous and incompressible binary fluid flows in two-dimensional bounded domains. The source functions have low spatial regularities, and the initial data belong to some interpolation spaces. The essential tools employed in the analysis are the extended maximal parabolic regularity for the associated linearized system and the well-posedness of the nonlinear part with the solution of the linearized dynamics as the frozen coefficients. We resolve the linear system by decomposition into the viscous biharmonic heat, Stokes, and heat equations. A spectral Faedo–Galerkin framework shall be pursued for the nonlinear part. Higher integrability with respect to time will be established using interpolation and compactness methods.   pdf   link   Zbl 1490.35346

[22] G. Peralta, Error estimates for mixed and hybrid FEM for elliptic optimal control problems with penalizations, Advances in Computational Mathematics 48 (6), Paper No. 70, 35 p., 2022.
abstract Mixed and hybrid finite element discretizations for distributed optimal control problems governed by an elliptic equation are analyzed. A cost functional keeping track of both the state and its gradient is studied. A priori error estimates and super-convergence properties for the continuous and discrete optimal states, adjoint states, and controls will be given. The approximating finite-dimensional systems will be solved by adding penalization terms for the state and the associated Lagrange multipliers. In general, performing optimization, discretization, hybridization, and penalization in any order lead to the same optimality system. Numerical examples based on the Raviart–Thomas finite elements will be presented.   pdf   link   python   Zbl 1505.35108

[21] G. Peralta, Optimal Borel measure controls for the two-dimensional stationary Boussinesq system, ESAIM Control, Optimisation and Calculus of Variations 28, Paper No. 22, 33 p., 2022.
abstract We analyze an optimal control problem for the stationary two-dimensional Boussinesq system with controls taken in the space of regular Borel measures. Such measure-valued controls are known to produce sparse solutions. First-order and second-order necessary and sufficient optimality conditions are established. Following an optimize-then-discretize strategy, the corresponding finite element approximation will be solved by a semi-smooth Newton method initialized by a continuation strategy. The controls are discretized by finite linear combinations of Dirac measures concentrated at the nodes associated with the degrees of freedom for the mini-finite element.   pdf   link   python   Zbl 1485.49011

[20] G. Peralta and K. Kunisch, Mixed and hybrid Petrov–Galerkin finite element discretization for optimal control of the wave equation, Numerische Mathematik 150 (2), pp. 591-627, 2022.
abstract A mixed finite element discretization of an optimal control problem for the linear wave equation with homogeneous Dirichlet boundary condition is considered. For the temporal discretization, a Petrov–Galerkin scheme is utilized and the Raviart–Thomas finite elements for spatial discretization is used. A priori error analysis is proved for this numerical scheme. A hybridized formulation is proposed and if the Arnold–Brezzi post-processing method is applied, better convergence rates with respect to space are observed. The interchangeability of discretization and optimization holds both for mixed and hybrid formulations. Numerical experiments illustrating the theoretical results are presented using the lowest-order Raviart–Thomas elements.   pdf   link   python   Zbl 1487.49007

[19] G.M. Angeles and G. Peralta, Energy method for exponential stability of coupled one-dimensional hyperbolic PDE-ODE systems, Evolution Equations and Control Theory 11 (1), pp. 199-224, 2022.
abstract We consider a hyperbolic system of partial differential equations on a bounded interval coupled with ordinary differential equations on both ends. The evolution is governed by linear balance laws, which we treat with semigroup and time-space methods. Our goal is to establish the exponential stability in the natural state space by utilizing the stability with respect to the first-order energy of the system. Derivation of a priori estimates plays a crucial role in obtaining energy and dissipation functionals. The theory is then applied to specific physical models.   pdf   link   Zbl 1486.35044

[18] G. Peralta and J. S. Simon, Optimal control for the Navier–Stokes with time delay in the convection: analysis and finite element approximations, Journal of Mathematical Fluid Mechanics 23 (3), Paper No. 56, 49 p., 2021.
abstract A distributed optimal control problem for the 2D incompressible Navier–Stokes equation with delay in the convection term is studied. The delay corresponds to the non-instantaneous effect of the motion of a fluid parcel on the mass transfer, and can be realized as a regularization or stabilization to the Navier–Stokes equation. The existence of optimal controls is established, and the corresponding first-order necessary optimality system is determined. A semi-implicit discontinuous Galerkin scheme with respect to time and conforming finite elements for space is considered. Error analysis for this numerical scheme is discussed and optimal convergence rates are proved. The fully discrete problem is solved by the Barzilai–Borwein gradient method. Numerical examples for the velocity-tracking and vorticity minimization problems based on the Taylor-Hood elements are presented.   pdf   link   python   Zbl 1465.49005

[17] G. Peralta, Distributed optimal control of the 2D Cahn–Hilliard–Oberbeck–Boussinesq system for nonisothermal viscous two-phase flows, Applied Mathematics and Optimization 84, Suppl. 2, pp. 1219-1279, 2021.
abstract We analyze a distributed optimal control problem where the state equation is governed by the coupling of the two-dimensional Cahn–Hilliard and Oberbeck–Boussinesq systems modelling incompressible viscous two-phase flows with convective heat transfer. Pointwise constraints are imposed on the controls that act as external sources in the fluid and convection-diffusion equations. The objective functional is of tracking-type that consists of a weighted energy of the difference between the state and a desired target. We establish the existence of optimal controls, the differentiability of the control-to-state operator, and the necessary and sufficient optimality conditions. For initial and target data with finite energy norms, limited space–time regularity of the adjoint states arises due to convection and surface tension.   pdf   link   Zbl 1507.35196

[16] G. Peralta and K. Kunisch, Analysis and finite element discretization for optimal control of a linear fluid-structure interaction problem with delay, IMA Journal of Numerical Analysis 40 (1), pp. 140-206, 2020.
abstract An optimal control problem for a linearized fluid–structure interaction model with a delay term in the structural damping is analyzed. A distributed control acting on the fluid domain, structure domain or both is considered. The necessary optimality conditions are derived both for rough and smooth initial data. A parabolic regularization of the problem and its convergence are investigated. Finite element discretization for the regularized problem and error estimates are provided. Piecewise linear elements with bubble functions for the fluid and a discontinuous Galerkin scheme for the spatial and temporal discretizations are utilized respectively. Numerical experiments illustrating the theoretical results are given.   pdf   link   python   Zbl 1464.76059

[15] G. Peralta, Uniform exponential stability of a fluid-plate interaction model due to thermal effects, Evolution Equations and Control Theory 9 (1), pp. 39-60, 2020.
abstract We consider a coupled fluid-thermoelastic plate interaction model. The fluid velocity is modeled by the linearized 3D Navier-Stokes equation while the plate dynamics is described by a thermoelastic Kirchoff system. By eliminating the pressure term, the system is reformulated as an abstract evolution problem and its well-posedness is proved by semigroup methods. The dissipation in the system is due to the diffusion of the fluid and heat components. Uniform stability of the coupled system is established through multipliers and the energy method. The multipliers used for thermoelastic plate models in the literature are modified in accordance to the applicability of a certain Stokes map   pdf   link   Zbl 1435.35283

[14] G. Peralta and K. Kunisch, Analysis of a nonlinear fluid-structure interaction model with mechanical dissipation and delay, Nonlinearity 32 (12), pp. 5110-5149, 2019.
abstract A fluid-structure interaction model with discrete and distributed delays in the structural damping is studied. The fluid and structure dynamics are governed by the Navier–Stokes and linear elasticity equations, respectively. Due to the presence of delay, a crucial ingredient of the weak formulation is the use of hidden boundary regularity for transport equations. In two space dimension, it is shown that weak solutions are unique. For smooth and compatible data, we establish the existence of the pressure and by applying micro-local analysis, further regularity of the solutions are available. Finally, the exponential stability of the system is obtained through an appropriate Lyapunov functional.   pdf   link   Zbl 1425.74163

[13] G. Peralta and K. Kunisch, Interface stabilization of a parabolic-hyperbolic PDE system with delay in the interaction, Discrete and Continuous Dynamical Systems - A 38 (6), pp. 3055-3083, 2018.
abstract A coupled parabolic-hyperbolic system of partial differential equations modeling the interaction of a structure submerged in a fluid is studied. The system being considered incorporates delays in the interaction on the interface between the fluid and the solid. We study the stability properties of the interaction model under suitable assumptions between the competing strengths of the delays and the feedback controls.   pdf   link   Zbl 1397.35032

[12] G. Peralta, Stabilization of the wave equation with acoustic and delay boundary conditions, Semigroup Forum 96 (2), pp. 357-376, 2018.
abstract In this paper, we consider the wave equation on a bounded domain with mixed Dirichlet-impedance type boundary conditions coupled with oscillators on the Neumann boundary. The system has either a delay in the pressure term of the wave component or the velocity of the oscillator component. Using the velocity as a bound- ary feedback it is shown that if the delay factor is less than that of the damping factor then the energy of the solutions decays to zero exponentially. The results are based on the energy method, a compactness-uniqueness argument and an appropriate weighted trace estimate. In the critical case where the damping and delay factors are equal, it is shown using variational methods that the energy decays to zero asymptotically.   pdf   link   Zbl 1403.35151

[11] G. Peralta, A fluid-structure interaction model with interior damping and delay in the structure, Zeitschrift fuer Angewandte Mathematik und Physik 67 (1), Article ID 10, 20 p., 2016.
abstract A coupled system of partial differential equations modeling the interaction of a fluid and a structure with delay in the feedback is studied. The model describes the dynamics of an elastic body immersed in a fluid that is contained in a vessel, whose boundary is made of a solid wall. The fluid component is modeled by the linearized Navier-Stokes equation, while the solid component is given by the wave equation neglecting transverse elastic force. Spectral properties and exponential or strong stability of the interaction model under appropriate conditions on the damping factor, delay factor and the delay parameter are established using a generalized Lax-Milgram method.   pdf   link   Zbl 1341.74054

[10] G. Peralta, Stabilization of viscoelastic wave equations with distributed or boundary delay, Zeitschrift fuer Analysis und Ihre Anwendungen 35 (3), pp. 359-381, 2016.
abstract The wave equation with viscoelastic boundary damping and internal or boundary delay is considered. The memory kernel is assumed to be integrable and completely monotonic. Under certain conditions on the damping factor, delay factor and the memory kernel it is shown that the energy of the solutions decay to zero either asymptotically or exponentially. In the case of internal delay, the result is obtained through spectral analysis and the Gearhart-Prüss Theorem, whereas in the case of boundary delay, it is obtained using the energy method.   pdf   link   Zbl 1379.35030

[09] G. Peralta and G. Propst, Existence of local-in-time classical solutions of a model of flow in a bounded elastic tube, Mathematical Methods in the Applied Sciences 39 (18), pp. 5315-5329, 2016.
abstract This paper studies the local-in-time existence of classical solutions to a hyperbolic system with differential boundary conditions modelling a flow in an elastic tube. The well-known Lax transformations used for obtaining a priori estimates for conservation laws are difficult to apply here because of the inhomogeneity of the partial differential equations (PDE). Rather, our method relies on a suitable splitting of the original system into the PDE part and the ODE part, the characteristics for the PDE part, appropriate modulus of continuity estimates and a compactness argument.   pdf   link   Zbl 1352.35077

[08] G. Peralta and G. Propst, Nonlinear and linear hyperbolic systems with dynamic boundary conditions, Bulletin of the Brazilian Mathematical Society, New Series 47 (2), pp. 671-683, 2016.
abstract We consider first order hyperbolic systems on an interval with dynamic boundary conditions. The well-posedness for linear systems is established by using a variational method. The linear theory is used to analyze the local-in-time well-posedness for nonlinear systems. The results are applied to a model describing the flow of an incompressible fluid inside an elastic tube whose ends are attached to tanks. Global existence and stability for data that are smooth enough and close to the steady state are obtained by using energy and entropy methods.   pdf   link   Zbl 1356.35129

[07] G. Peralta and G. Propst, Well-posedness and regularity of linear hyperbolic systems with dynamic boundary conditions, Proceedings of the Royal Society of Edinburgh Section A: Mathematics 146 (5), pp. 1047-1080, 2016.
abstract We consider first-order hyperbolic systems on an interval with dynamic boundary conditions. These systems occur when the ordinary differential equation dynamics on the boundary interact with the waves in the interior. The well-posedness for linear systems is established using an abstract Friedrichs theorem. Due to the limited regularity of the coefficients, we need to introduce the appropriate space of test functions for the weak formulation. It is shown that the weak solutions exhibit a hidden regularity at the boundary as well as at interior points. As a consequence, the dynamics of the boundary components satisfy an additional regularity. Neither result can be achieved from standard semigroup methods. Nevertheless, we show that our weak solutions and the semigroup solutions coincide. For illustration, we give three particular physical examples that fit into our framework.   pdf   link   Zbl 1432.35137

[06] G. Peralta and G. Propst, Global smooth solution to a hyperbolic system on an interval with dynamic boundary conditions, Quarterly of Applied Mathematics 74 (3), pp. 539-570, 2016.
abstract We consider a hyperbolic two component system of partial differential equations in one space dimension with ODE boundary conditions describing the flow of an incompressible fluid in an elastic tube that is connected to a tank at each end. Using the local-existence theory together with entropy methods, the existence and uniqueness of a global-in-time smooth solution is established for smooth initial data sufficiently close to the equilibrium state. Energy estimates are derived using the relative entropy method for zero order estimates while constructing entropy-entropy flux pairs for the corresponding diagonal system of the shifted Riemann invariants to deal with higher order estimates. Finally, using the linear theory and interpolation estimates, we show that the solution converges exponentially to the equilibrium state.   pdf   link   Zbl 1347.35152

[05] G. Peralta and G. Propst, Stability and boundary controllability of a linearized model of flow in an elastic tube, ESAIM Control, Optimisation and Calculus of Variations 21 (2), pp. 583-601, 2015.
abstract We consider a model describing the flow of a fluid inside an elastic tube that is connected to two tanks. We study the linearized system through semigroup theory. Controlling the pressures in the tanks renders a hyperbolic PDE with boundary control. The linearization induces a one-dimensional linear manifold of equilibria; when those are factored out, the corresponding semigroup is exponentially stable. The location of the eigenvalues in dependence on the viscosity is discussed. Exact boundary controllability of the system is achieved by the Riesz basis approach including generalized eigenvectors. A minimal time for controllability is given. The corresponding result for internal distributed control is stated.   pdf   link   Zbl 1330.35033

[04] G. Peralta and G. Propst, Local well-posedness of a class of hyperbolic PDE-ODE systems on a bounded interval, Journal of Hyperbolic Differential Equations 11 (4), pp. 705-747, 2014.
abstract The well-posedness theory for hyperbolic systems of first-order quasilinear PDE's with ODE's boundary conditions (on a bounded interval) is discussed. Such systems occur in multi-scale blood flow models, as well as valveless pumping and fluid mechanics. The theory is presented in the setting of Sobolev spaces Hm (m ≥ 3 being an integer), which is an appropriate set-up when it comes to proving existence of smooth solutions using energy estimates. A blow-up criterion is also derived, stating that if the maximal time of existence is finite, then the state leaves every compact subset of the hyperbolicity region, or its first-order derivatives blow-up. Finally, we discuss physical examples which fit in the general framework presented.   pdf   link   Zbl 1316.35176

[03] G. Peralta, Convergence, isometry and quasiconvexity in the modular sequence space w_0^0(Φ), Southeast Asian Bulletin of Mathematics 35 (5), 2011, pp. 813-823, 2011.
abstract We consider a modular sequence space generated by a sequence of Orlicz functions. Sufficient conditions were imposed in this sequence to guarantee that the modular of the space generates a Luxemburg norm. The (Δ2,𝛿2) condition was used to prove the equivalence of modular convergence and Luxemburg norm convergence. Using a linear operator with certain properties, we will generate another modular sequence space and establish an isometry between this space and the previous one. Also, the quasiconvexity of the modular will be studied.   pdf   link   Zbl 1265.46012

[02] G. Peralta, On existence of solutions of a class of Volterra integral equations, Chamchuri Journal of Mathematics 1 (2), pp. 43-53, 2009.
abstract In this paper, we discuss sufficient conditions to guarantee the existence and uniqueness of a class of Volterra integral equations of the second kind. Our results are based on the resolvent and contraction mapping methods. Some classical existence and uniqueness theorems are given as corollaries.   pdf   link   Zbl 1264.45011

[01] D.J. Indong and G. Peralta, Inversions of permutations in symmetric, alternating, and dihedral groups, Journal of Integer Sequences 11 (4), Article 08.4.3, 12 p., 2008.
abstract We use two methods to obtain a formula relating the total number of inversions of all permutations and the corresponding order of symmetric, alternating, and dihedral groups. First, we define an equivalence relation on the symmetric group Sn and consider each element in each equivalence class as a permutation of a proper subset of {1,2,...,n}. Second, we look at certain properties of a backward permutation, a permutation obtained by reversing the row images of a given permutation. Lastly, we employ the first method to obtain a recursive formula corresponding to the number of permutations with k inversions.   pdf   link   Zbl 1148.05004

Conference Proceedings

[2] G. Peralta, Feedback stabilization of a linear fluid–membrane system with time-delay. InKlingenberg, Christian (ed.) et al., Theory, numerics and applications of hyperbolic problems II, Aachen, Germany, August 2016. Cham: Springer. Springer Proc. Math. Stat. 237, pp. 437-449, 2018.
abstract A coupled parabolic–hyperbolic system of partial differential equations modelling the interaction of a fluid and a membrane is considered. The model is reformulated as an abstract Cauchy problem and thereby constructing a semigroup for the evolution. This is done by eliminating the pressure. The system is stabilized through a feedback force applied to the membrane incorporating a time delay. The spectral properties and stability are considered under suitable conditions on the fluid viscosity, damping coefficient and delay coefficient.   pdf   link   Zbl 1407.74029

[1] D.J. Indong and G. Peralta, Some properties of a sequence of inversion numbers. In Kamarul Haili, Hailiza (ed.) et al., Proceedings of the 5th Asian mathematical conference (AMC), June 22–26, 2009, Kuala Lumpur, Malaysia. Vol. I: Pure Mathematics. Penang: Universiti Sains Malaysia, School of Mathematical Sciences (ISBN 978-967-5417-53-5/CD-ROM), pp. 45-52, 2009.
abstract In this paper, we consider a certain sequence of inversion numbers. We show that this sequence is a polynomial sequence and find its leading term. Using this, a characterization of the Hankel and inverse binomial transforms of these inversion numbers will be given and each of these transforms, together with an appropriate sequence, forms a basis for the space of real sequences having compact support. Also, with the aid of the generating function of the inversion numbers we will give a formula for a certain type of complex integral.   pdf   Zbl 1207.26002

Teaching

Current Teaching (AY 2023-2024)

Previous Teaching (from AY 2007-2008 to AY 2022-2023)

Education

Work Experience