DSpace Collection:
http://hdl.handle.net/2117/3339
Wed, 01 Oct 2014 22:35:59 GMT2014-10-01T22:35:59Zwebmaster.bupc@upc.eduUniversitat Politècnica de Catalunya. Servei de Biblioteques i DocumentaciónoOn the backward in time problem for the thermoelasticity with two temperatures
http://hdl.handle.net/2117/23165
Title: On the backward in time problem for the thermoelasticity with two temperatures
Authors: Leseduarte Milán, María Carme; Quintanilla de Latorre, Ramón
Abstract: This paper is devoted to the study of the existence, uniqueness, continuous dependence and spatial behaviour of the solutions for the backward in time problem determined by the Type III with two temperatures thermoelastodynamic theory. We first show the existence, uniqueness and continuous dependence of the solutions. Instability of the solutions for the Type II with two temperatures theory is proved later. For the one-dimensional Type III with two temperatures theory, the exponential instability is also pointed-out. We also analyze the spatial behaviour of the solutions. By means of the exponentially weighted Poincare inequality, we are able to obtain a function that defines a measure on the solutions and, therefore, we obtain the usual exponential type alternative for the solutions of the problem defined in a semi-infinite cylinder.
Description: Electronic version of an article published as "Discrete and continuous dynamical series", vol. 19, no 3, May 2014, p. 679-695. DOI 10.3934/dcdsb.2014.19.679.Thu, 05 Jun 2014 13:19:17 GMThttp://hdl.handle.net/2117/231652014-06-05T13:19:17ZLeseduarte Milán, María Carme; Quintanilla de Latorre, RamónnoBackward in time problem, Type III with two temperatures thermoelasticity, existence, continuous dependence, spatial growth/decay of solutions, energy arguments, WELL-POSED PROBLEM, ENERGY-DISSIPATION, CONTINUOUS DEPENDENCE, HEAT-CONDUCTION, SPATIAL DECAY, TEMPERATURE THERMOELASTICITY, UNIQUENESS, STABILITY, PROPAGATION, BOUNDARYThis paper is devoted to the study of the existence, uniqueness, continuous dependence and spatial behaviour of the solutions for the backward in time problem determined by the Type III with two temperatures thermoelastodynamic theory. We first show the existence, uniqueness and continuous dependence of the solutions. Instability of the solutions for the Type II with two temperatures theory is proved later. For the one-dimensional Type III with two temperatures theory, the exponential instability is also pointed-out. We also analyze the spatial behaviour of the solutions. By means of the exponentially weighted Poincare inequality, we are able to obtain a function that defines a measure on the solutions and, therefore, we obtain the usual exponential type alternative for the solutions of the problem defined in a semi-infinite cylinder.On the uniqueness and analyticity of solutions in micropolar thermoviscoelasticity
http://hdl.handle.net/2117/21495
Title: On the uniqueness and analyticity of solutions in micropolar thermoviscoelasticity
Authors: Magaña Nieto, Antonio; Quintanilla de Latorre, Ramón
Abstract: This paper deals with the linear theory of isotropic micropolar thermoviscoelastic materials. When the dissipation is positive definite, we present two uniqueness theorems. The first one requires the extra assumption that some coupling terms vanish; in this case, the instability of solutions is also proved. When the internal energy and the dissipation are both positive definite, we prove the well-posedness of the problem and the analyticity of the solutions. Exponential decay and impossibility of localization are corollaries of the analyticity.Mon, 10 Feb 2014 14:06:59 GMThttp://hdl.handle.net/2117/214952014-02-10T14:06:59ZMagaña Nieto, Antonio; Quintanilla de Latorre, RamónnoMicropolar thermoviscoelasticity, Uniqueness, Analyticity, Exponential decayThis paper deals with the linear theory of isotropic micropolar thermoviscoelastic materials. When the dissipation is positive definite, we present two uniqueness theorems. The first one requires the extra assumption that some coupling terms vanish; in this case, the instability of solutions is also proved. When the internal energy and the dissipation are both positive definite, we prove the well-posedness of the problem and the analyticity of the solutions. Exponential decay and impossibility of localization are corollaries of the analyticity.On the spatial behavior in type III thermoelastodynamics
http://hdl.handle.net/2117/21453
Title: On the spatial behavior in type III thermoelastodynamics
Authors: Leseduarte Milán, María Carme; Quintanilla de Latorre, Ramón
Abstract: This note is concerned with the linear (and linearized) Type III thermoelastodynamic theory proposed by Green and Naghdi. We investigate the spatial behavior of the solutions when we assume the positivity of the elasticity tensor, the thermal conductivity tensor, the mass density and the heat capacity. However, we do not assume (a priori) the positivity of the internal energy. We first obtain a Phragmén-Lindelöf alternative of exponential type for the solutions. Later, we prove that the decay can be controlled by the exponential of a second-degree polynomial. This is similar to other thermoelastic situations.Tue, 04 Feb 2014 16:20:58 GMThttp://hdl.handle.net/2117/214532014-02-04T16:20:58ZLeseduarte Milán, María Carme; Quintanilla de Latorre, RamónnoType III thermoelastodynamics, Spatial growth/decay of solutions, Energy methods, Comparison argumentsThis note is concerned with the linear (and linearized) Type III thermoelastodynamic theory proposed by Green and Naghdi. We investigate the spatial behavior of the solutions when we assume the positivity of the elasticity tensor, the thermal conductivity tensor, the mass density and the heat capacity. However, we do not assume (a priori) the positivity of the internal energy. We first obtain a Phragmén-Lindelöf alternative of exponential type for the solutions. Later, we prove that the decay can be controlled by the exponential of a second-degree polynomial. This is similar to other thermoelastic situations.Spatial decay for several phase-field models
http://hdl.handle.net/2117/20931
Title: Spatial decay for several phase-field models
Authors: Miranville, Alain; Quintanilla de Latorre, Ramón
Abstract: In this paper, we study the spatial behavior of three phase-field models. First, we consider the Cahn-Hilliard equation and
we obtain the exponential decay of solutions under suitable assumptions on the data. Then, for the classical isothermal
phase-field equation (i.e., the Allen-Cahn equation), we prove the nonexistence and the fast decay of solutions and, for
the nonisothermal case governed by the Fourier law, we obtain a Phragm ́
en-Lindel ̈
of alternative of exponential type,
respectively.Thu, 05 Dec 2013 15:30:33 GMThttp://hdl.handle.net/2117/209312013-12-05T15:30:33ZMiranville, Alain; Quintanilla de Latorre, RamónnoPhase-field models, Spatial decay, Exponential decay, Fast decay, Phragmén-Lindelöf alternativeIn this paper, we study the spatial behavior of three phase-field models. First, we consider the Cahn-Hilliard equation and
we obtain the exponential decay of solutions under suitable assumptions on the data. Then, for the classical isothermal
phase-field equation (i.e., the Allen-Cahn equation), we prove the nonexistence and the fast decay of solutions and, for
the nonisothermal case governed by the Fourier law, we obtain a Phragm ́
en-Lindel ̈
of alternative of exponential type,
respectively.Decay of solutions for a mixture of thermoelastic one dimensional solids
http://hdl.handle.net/2117/19560
Title: Decay of solutions for a mixture of thermoelastic one dimensional solids
Authors: Muñoz Rivera, Jaime E.; Naso, Maria-Grazia; Quintanilla de Latorre, Ramón
Abstract: We study a PDE system modeling thermomechanical deformations for a mixture of thermoelastic solids. In particular we investigate the asymptotic behavior of the solutions. First, we identify conditions on the constitutive coefficients to guarantee that the imaginary axis is contained in the resolvent. Subsequently, we find the necessary and sufficient conditions to guarantee the exponential decay of solutions. When the decay is not of exponential type, we prove that the solutions decay polynomially and we find the optimal polynomial decay rate.Mon, 17 Jun 2013 16:52:48 GMThttp://hdl.handle.net/2117/195602013-06-17T16:52:48ZMuñoz Rivera, Jaime E.; Naso, Maria-Grazia; Quintanilla de Latorre, RamónnoWe study a PDE system modeling thermomechanical deformations for a mixture of thermoelastic solids. In particular we investigate the asymptotic behavior of the solutions. First, we identify conditions on the constitutive coefficients to guarantee that the imaginary axis is contained in the resolvent. Subsequently, we find the necessary and sufficient conditions to guarantee the exponential decay of solutions. When the decay is not of exponential type, we prove that the solutions decay polynomially and we find the optimal polynomial decay rate.On the decay of solutions for the heat conduction with two temperatures
http://hdl.handle.net/2117/18775
Title: On the decay of solutions for the heat conduction with two temperatures
Authors: Leseduarte Milán, María Carme; Quintanilla de Latorre, Ramón
Abstract: This paper is devoted to the study of the asymptotic behavior of the solutions of the system of
equations that models the heat conduction with two temperatures. That is, we consider a mixture of isotropic
and homogeneous rigid solids. We analyze the static problem in a semi-infinite cylinder where every material
point has two temperatures with nonlinear boundary conditions on the lateral side. A Phragmén–Lindelöf
alternative for the solutions is obtained by means of energy arguments. Estimates for the decay and growth of
the solutions are presented. We also prove that the only solution vanishing in the exterior of a bounded set is
the null solution for a particular subfamily of problems. Cone-like domains are considered in the last section,
and we obtain decay estimates for the solutions when the total energy is bounded.Thu, 11 Apr 2013 15:01:18 GMThttp://hdl.handle.net/2117/187752013-04-11T15:01:18ZLeseduarte Milán, María Carme; Quintanilla de Latorre, RamónnoThis paper is devoted to the study of the asymptotic behavior of the solutions of the system of
equations that models the heat conduction with two temperatures. That is, we consider a mixture of isotropic
and homogeneous rigid solids. We analyze the static problem in a semi-infinite cylinder where every material
point has two temperatures with nonlinear boundary conditions on the lateral side. A Phragmén–Lindelöf
alternative for the solutions is obtained by means of energy arguments. Estimates for the decay and growth of
the solutions are presented. We also prove that the only solution vanishing in the exterior of a bounded set is
the null solution for a particular subfamily of problems. Cone-like domains are considered in the last section,
and we obtain decay estimates for the solutions when the total energy is bounded.On the logarithmic convexity in thermoelastiscity with microtemperatures
http://hdl.handle.net/2117/18770
Title: On the logarithmic convexity in thermoelastiscity with microtemperatures
Authors: Quintanilla de Latorre, Ramón
Abstract: This article is concerned with the linear theory of thermoelasticity with
microtemperatures. In a recent article we have used the logarithmic convexity method
to investigate uniqueness, instability and structural stability. The results are restricted
to the case when the constitutive coefficients
k1 and k3 have the same signs. Here we
prove that these results also hold when the coefficients k1 and k3 have opposite signs.Thu, 11 Apr 2013 13:01:01 GMThttp://hdl.handle.net/2117/187702013-04-11T13:01:01ZQuintanilla de Latorre, RamónnoThis article is concerned with the linear theory of thermoelasticity with
microtemperatures. In a recent article we have used the logarithmic convexity method
to investigate uniqueness, instability and structural stability. The results are restricted
to the case when the constitutive coefficients
k1 and k3 have the same signs. Here we
prove that these results also hold when the coefficients k1 and k3 have opposite signs.On a strain gradient theory of thermoviscoelasticity
http://hdl.handle.net/2117/17245
Title: On a strain gradient theory of thermoviscoelasticity
Authors: Iesan, Dorin; Quintanilla de Latorre, Ramón
Abstract: This paper is concerned with a strain gradient theory of thermoviscoelasticity in which the time derivatives of the strain tensors are included in the set of independent constitutive variables. The theory is motivated by the recent interest in the study of gradient theories. First, we establish the basic equations of the linear theory and present two uniqueness results. Then, we use a semigroup approach to derive an existence result. Finally, we establish the constitutive equations for an isotropic chiral material and derive a solution of the field equations.Wed, 09 Jan 2013 18:14:58 GMThttp://hdl.handle.net/2117/172452013-01-09T18:14:58ZIesan, Dorin; Quintanilla de Latorre, RamónnoThis paper is concerned with a strain gradient theory of thermoviscoelasticity in which the time derivatives of the strain tensors are included in the set of independent constitutive variables. The theory is motivated by the recent interest in the study of gradient theories. First, we establish the basic equations of the linear theory and present two uniqueness results. Then, we use a semigroup approach to derive an existence result. Finally, we establish the constitutive equations for an isotropic chiral material and derive a solution of the field equations.A conserved phase-field system based on the Maxwell–Cattaneo law
http://hdl.handle.net/2117/17207
Title: A conserved phase-field system based on the Maxwell–Cattaneo law
Authors: Miranville, Alain; Quintanilla de Latorre, Ramón
Abstract: Our aim in this paper is to study a generalization of the conserved Caginalp phasefield
system based on the Maxwell–Cattaneo law for heat conduction and endowed with
Neumann boundary conditions. In particular, we obtain well-posedness results and study
the dissipativity of the associated solution operators.Tue, 08 Jan 2013 11:33:15 GMThttp://hdl.handle.net/2117/172072013-01-08T11:33:15ZMiranville, Alain; Quintanilla de Latorre, RamónnoOur aim in this paper is to study a generalization of the conserved Caginalp phasefield
system based on the Maxwell–Cattaneo law for heat conduction and endowed with
Neumann boundary conditions. In particular, we obtain well-posedness results and study
the dissipativity of the associated solution operators.Non-linear deformations of porous elastic solids
http://hdl.handle.net/2117/16847
Title: Non-linear deformations of porous elastic solids
Authors: Iesan, Dorin; Quintanilla de Latorre, Ramón
Abstract: This paper is concerned with the non-linear theory of porouselastic bodies. First, we present the basic equations in general curvilinear coordinates. The constitutive equations for porouselastic bodies with incompressible matrix material are derived. Then, the equilibrium theory is investigated. An existence result within the one-dimensional theory is presented. The theory is applied in order to study the torsion of an isotropic circular cylinder and the flexure of a cuboid made of an anisotropic material. It is shown that the equations of equilibrium reduce to a single ordinary differential equation governing an unknown function which characterizes the aforementioned deformations.Tue, 06 Nov 2012 16:00:27 GMThttp://hdl.handle.net/2117/168472012-11-06T16:00:27ZIesan, Dorin; Quintanilla de Latorre, RamónnoThis paper is concerned with the non-linear theory of porouselastic bodies. First, we present the basic equations in general curvilinear coordinates. The constitutive equations for porouselastic bodies with incompressible matrix material are derived. Then, the equilibrium theory is investigated. An existence result within the one-dimensional theory is presented. The theory is applied in order to study the torsion of an isotropic circular cylinder and the flexure of a cuboid made of an anisotropic material. It is shown that the equations of equilibrium reduce to a single ordinary differential equation governing an unknown function which characterizes the aforementioned deformations.Phragmén-Lindelöf alternative for an exact heat conduction equation with delay
http://hdl.handle.net/2117/16720
Title: Phragmén-Lindelöf alternative for an exact heat conduction equation with delay
Authors: Leseduarte Milán, María Carme; Quintanilla de Latorre, Ramón
Abstract: In this paper we investigate the spatial behavior of the solutions for
a theory for the heat conduction with one delay term. We obtain a Phragm én-
Lindelöf type alternative. That is, the solutions either decay in an exponential
way or blow-up at in nity in an exponential way. We also show how to obtain
an upper bound for the amplitude term. Later we point out how to extend
the results to a thermoelastic problem. We nish the paper by considering
the equation obtained by the Taylor approximation to the delay term. A
Phragm én-Lindelöf type alternative is obtained for the forward and backward
in time equations.Thu, 11 Oct 2012 14:06:43 GMThttp://hdl.handle.net/2117/167202012-10-11T14:06:43ZLeseduarte Milán, María Carme; Quintanilla de Latorre, RamónnoIn this paper we investigate the spatial behavior of the solutions for
a theory for the heat conduction with one delay term. We obtain a Phragm én-
Lindelöf type alternative. That is, the solutions either decay in an exponential
way or blow-up at in nity in an exponential way. We also show how to obtain
an upper bound for the amplitude term. Later we point out how to extend
the results to a thermoelastic problem. We nish the paper by considering
the equation obtained by the Taylor approximation to the delay term. A
Phragm én-Lindelöf type alternative is obtained for the forward and backward
in time equations.On uniqueness and continuous dependence in type III thermoelasticity
http://hdl.handle.net/2117/16197
Title: On uniqueness and continuous dependence in type III thermoelasticity
Authors: Leseduarte Milán, María Carme; Quintanilla de Latorre, Ramón
Abstract: This note is concerned with the linear (and linearised) type III thermoelastic theory
proposed by Green and Naghdi. First, the continuous dependence of the solutions upon
the initial data and supply terms is established for noncentrosymmetric bodies. Then a
uniqueness result for centrosymmetric materials is established.Mon, 09 Jul 2012 08:37:37 GMThttp://hdl.handle.net/2117/161972012-07-09T08:37:37ZLeseduarte Milán, María Carme; Quintanilla de Latorre, RamónnoThis note is concerned with the linear (and linearised) type III thermoelastic theory
proposed by Green and Naghdi. First, the continuous dependence of the solutions upon
the initial data and supply terms is established for noncentrosymmetric bodies. Then a
uniqueness result for centrosymmetric materials is established.Further mathematical results concerning Burgers fluids and their generalizations
http://hdl.handle.net/2117/16064
Title: Further mathematical results concerning Burgers fluids and their generalizations
Authors: Quintanilla de Latorre, Ramón; Rajagopal, Kumbakonam
Abstract: In this paper, we extend the earlier work by Quintanilla and Rajagopal (Math Methods Appl Sci 29: 2133–2147,
2006) and establish qualitative new results for a proper generalization of Burgers’ original work that stems form a general
thermodynamic framework. Such fluids have been used to describe the behavior of several geological materials such as
asphalt and the earth’s mantle as well as polymeric fluids. We study questions concerning stability, uniqueness and continuous
dependence on initial data for the solutions of the flows of these fluids. We show that if certain conditions are not
satisfied by the material moduli, the solutions could be unstable. The spatial behavior of the solutions is also analyzed.Fri, 15 Jun 2012 18:08:32 GMThttp://hdl.handle.net/2117/160642012-06-15T18:08:32ZQuintanilla de Latorre, Ramón; Rajagopal, KumbakonamnoIn this paper, we extend the earlier work by Quintanilla and Rajagopal (Math Methods Appl Sci 29: 2133–2147,
2006) and establish qualitative new results for a proper generalization of Burgers’ original work that stems form a general
thermodynamic framework. Such fluids have been used to describe the behavior of several geological materials such as
asphalt and the earth’s mantle as well as polymeric fluids. We study questions concerning stability, uniqueness and continuous
dependence on initial data for the solutions of the flows of these fluids. We show that if certain conditions are not
satisfied by the material moduli, the solutions could be unstable. The spatial behavior of the solutions is also analyzed.Analyticity in porous-thermoelasticity with microtemperatures
http://hdl.handle.net/2117/16054
Title: Analyticity in porous-thermoelasticity with microtemperatures
Authors: Pamplona, Paulo Xavier; Muñoz Rivera, Jaime E.; Quintanilla de Latorre, Ramón
Abstract: In this note we study the analyticity of the solutions to the one-dimensional porouselasticity
problem with temperatures and microtemperatures when viscoelasticity and
porous viscosity effects are also present. We show the lack of analyticity when the porous
dissipation is weak, and the analyticity when it is strongFri, 15 Jun 2012 16:02:55 GMThttp://hdl.handle.net/2117/160542012-06-15T16:02:55ZPamplona, Paulo Xavier; Muñoz Rivera, Jaime E.; Quintanilla de Latorre, RamónnoIn this note we study the analyticity of the solutions to the one-dimensional porouselasticity
problem with temperatures and microtemperatures when viscoelasticity and
porous viscosity effects are also present. We show the lack of analyticity when the porous
dissipation is weak, and the analyticity when it is strongMathematical results concerning a class of incompressible viscoelastic solids of differential type
http://hdl.handle.net/2117/12300
Title: Mathematical results concerning a class of incompressible viscoelastic solids of differential type
Authors: Quintanilla de Latorre, Ramón; Rajagopal, Kumbakonam
Abstract: In this paper we investigate several mathematical aspects concerning a class of incompressible viscoelastic solids of the differential type. The model that we consider can be viewed as a generalization of the Kelvin—Voigt viscoelastic solid. We obtain a uniqueness result and show that when the shear modulus of the viscoelastic solid is positive the solutions decay exponentially. We also show that if the shear modulus is negative, a physically unacceptable situation, we have exponential growth of the solutions, which is in keeping with physical expectations. The impossibility of localization of the solutions in finite time is also proved. The last section is devoted to the development of spatial decay estimates in the quasi-static case.Thu, 07 Apr 2011 14:29:35 GMThttp://hdl.handle.net/2117/123002011-04-07T14:29:35ZQuintanilla de Latorre, Ramón; Rajagopal, KumbakonamnoIn this paper we investigate several mathematical aspects concerning a class of incompressible viscoelastic solids of the differential type. The model that we consider can be viewed as a generalization of the Kelvin—Voigt viscoelastic solid. We obtain a uniqueness result and show that when the shear modulus of the viscoelastic solid is positive the solutions decay exponentially. We also show that if the shear modulus is negative, a physically unacceptable situation, we have exponential growth of the solutions, which is in keeping with physical expectations. The impossibility of localization of the solutions in finite time is also proved. The last section is devoted to the development of spatial decay estimates in the quasi-static case.