Riccarda Rossi - Academia.edu (original) (raw)
Papers by Riccarda Rossi
Journal of Physics: Conference Series, 2016
Journal of the European Mathematical Society, 2016
Balanced Viscosity solutions to rate-independent systems arise as limits of regularized rate-inde... more Balanced Viscosity solutions to rate-independent systems arise as limits of regularized rate-independent flows by adding a superlinear vanishing-viscosity dissipation.
Journal of Physics: Conference Series, 2016
In this paper we analyze a PDE system modelling (non-isothermal) phase transitions and damage phe... more In this paper we analyze a PDE system modelling (non-isothermal) phase transitions and damage phenomena in thermoviscoelastic materials. The model is thermodynamically consistent: in particular, no {\em small perturbation assumption} is adopted, which results in the presence of quadratic terms on the right-hand side of the temperature equation, only estimated in L1L^1L1. The whole system has a highly nonlinear character. We address the existence of a weak notion of solution, referred to as "entropic", where the temperature equation is formulated with the aid of an entropy inequality, and of a total energy inequality. This solvability concept reflects the basic principles of thermomechanics as well as the thermodynamical consistency of the model. It allows us to obtain \emph{global-in-time} existence theorems without imposing any restriction on the size of the initial data. We prove our results by passing to the limit in a time discretization scheme, carefully tailored to the nonlinear features of the PDE system (with its "entropic" formulation), and of the a priori estimates performed on it. Our time-discrete analysis could be useful towards the numerical study of this model.
Annali Della Scuola Normale Superiore Di Pisa Classe Di Scienze, 2008
Esaim Control Optimisation and Calculus of Variations, Jun 30, 2006
This paper addresses the Cauchy problem for the gradient flow equation in a Hilbert space H
Eprint Arxiv 0911 1590, Nov 1, 2009
We present applications of our abstract results by proving the existence of the global attractor ... more We present applications of our abstract results by proving the existence of the global attractor for the energy solutions both of abstract doubly nonlinear evolution equations in reflexive Banach spaces, and of a class of evolution equations in Wasserstein spaces, as well as for the generalized solutions of some phase-change evolutions driven by mean curvature.
Indiana University Mathematics Journal, 2007
This paper addresses the analysis of a model, proposed by M. Frémond, for the phenomenon of conta... more This paper addresses the analysis of a model, proposed by M. Frémond, for the phenomenon of contact with reversible adhesion between a viscoelastic body and a rigid support. First of all, we prove existence and uniqueness of global-in-time solutions of (the initial-boundary value problem for) the related PDE system by means of a fixed point technique. Hence, we investigate the long-time behaviour of such solutions and obtain some results on the structure of the associated ω-limit set.
This paper focuses on rate-independent damage in elastic bodies. Since the driving energy is nonc... more This paper focuses on rate-independent damage in elastic bodies. Since the driving energy is nonconvex, solutions may have jumps as a function of time, and in this situation it is known that the classical concept of energetic solutions for rate-independent systems may fail to accurately describe the behavior of the system at jumps. Therefore we resort to the (by now well-established) vanishing viscosity approach to rate-independent modeling, and approximate the model by its viscous regularization. In fact, the analysis of the latter PDE system presents remarkable difficulties, due to its highly nonlinear character. We tackle it by combining a variational approach to a class of abstract doubly nonlinear evolution equations, with careful regularity estimates tailored to this specific system, relying on a q-Laplacian type gradient regularization of the damage variable. Hence for the viscous problem we conclude the existence of weak solutions, satisfying a suitable energy-dissipation inequality that is the starting point for the vanishing viscosity analysis. The latter leads to the notion of (weak) parameterized solution to our rate-independent system, which encompasses the influence of viscosity in the description of the jump regime.
Compactness in the space L p (0, T ; B), B being a separable Banach space, has been deeply invest... more Compactness in the space L p (0, T ; B), B being a separable Banach space, has been deeply investigated by more recently, by J.M. , who have provided various criteria for relative compactness, which turn out to be crucial tools in the existence proof of solutions to several abstract time dependent problems related to evolutionary PDEs. In the present paper, the problem is examined in view of Young measure theory: exploiting the underlying principles of "tightness" and "integral equicontinuity", new necessary and sufficient conditions for compactness are given, unifying some of the previous contributions and showing that the Aubin -Lions condition is not only sufficient but also necessary for compactness. Furthermore, the related issue of compactness with respect to convergence in measure is studied and a general criterion is proved. : 28A20, 46E30, 46N20.
We address the long-time behaviour of a class of viscous Cahn-Hilliard equations, modelling phase... more We address the long-time behaviour of a class of viscous Cahn-Hilliard equations, modelling phase separation in mixtures and alloys. Specifically, we prove the existence of (a suitable notion of) the global attractor for the weak solutions of the so-called generalized viscous Cahn-Hilliard equation. November 4, 2005 17:43 Dissipative Phase Transitions P. Colli, N. Kenmochi, J. Sprekels Attractor for generalized viscous Cahn-Hilliard equations 5
Compactness in the space Lp(0;T ;B), B being a separable Banach space, has been deeply investigat... more Compactness in the space Lp(0;T ;B), B being a separable Banach space, has been deeply investigated by J.P. Aubin (1963), J.L. Lions (1961,1969), J. Simon (1987), and, more recently, by J.M. Rakotoson and R. Temam (2001), who have provided various criteria for relative compactness, which turn out to be crucial tools in the existence proof of solutions to many abstract
SIAM Journal on Mathematical Analysis, 2015
In this paper we analyze a PDE system modeling (non-isothermal) phase transitions and damage phen... more In this paper we analyze a PDE system modeling (non-isothermal) phase transitions and damage phenomena in thermoviscoelastic materials. The model is thermodynamically consistent: in particular, no small perturbation assumption is adopted, which results in the presence of quadratic terms on the right-hand side of the temperature equation, only estimated in L 1 . The whole system has a highly nonlinear character.
Series on Advances in Mathematics for Applied Sciences, 2006
We address the long-time behaviour of a class of viscous Cahn-Hilliard equations, modelling phase... more We address the long-time behaviour of a class of viscous Cahn-Hilliard equations, modelling phase separation in mixtures and alloys. Specifically, we prove the existence of (a suitable notion of) the global attractor for the weak solutions of the so-called generalized viscous Cahn-Hilliard equation. November 4, 2005 17:43 Dissipative Phase Transitions P. Colli, N. Kenmochi, J. Sprekels Attractor for generalized viscous Cahn-Hilliard equations 5
Nonlinear Analysis: Real World Applications, 2015
ABSTRACT In this paper we introduce and investigate a model for adhesive contact with friction be... more ABSTRACT In this paper we introduce and investigate a model for adhesive contact with friction between a thermoviscoelastic body and a rigid support.A PDE system, consisting of the evolution equations for the temperatures in the bulk domain and on the contact surface, of the momentum balance, and of the equation for the internal variable describing the state of the adhesion, is derived on the basis of a surface damage theory by M. Frémond.The existence of global-in-time solutions to the associated initial–boundary value problem is proved by passing to the limit in a carefully tailored time-discretization scheme.
Physica D: Nonlinear Phenomena, 2014
We propose a model for (unilateral) contact with adhesion between a viscoelastic body and a rigid... more We propose a model for (unilateral) contact with adhesion between a viscoelastic body and a rigid support, encompassing thermal and frictional effects. Following Frémond's approach, adhesion is described in terms of a surface damage parameter χ . The related equations are the momentum balance for the vector of small displacements, and parabolic-type evolution equations for χ and for the absolute temperatures of the body and of the adhesive substance on the contact surface. All of the constraints on the internal variables, as well as the contact and the friction conditions, are rendered by means of subdifferential operators. Furthermore, the temperature equations, derived from an entropy balance law, feature singular functions. Therefore, the resulting PDE system has a highly nonlinear character.
Nonlinear Analysis: Real World Applications, 2015
This paper focuses on rate-independent damage in elastic bodies. Since the driving energy is nonc... more This paper focuses on rate-independent damage in elastic bodies. Since the driving energy is nonconvex, solutions may have jumps as a function of time, and in this situation it is known that the classical concept of energetic solutions for rate-independent systems may fail to accurately describe the behavior of the system at jumps.
Journal of Physics: Conference Series, 2016
Journal of the European Mathematical Society, 2016
Balanced Viscosity solutions to rate-independent systems arise as limits of regularized rate-inde... more Balanced Viscosity solutions to rate-independent systems arise as limits of regularized rate-independent flows by adding a superlinear vanishing-viscosity dissipation.
Journal of Physics: Conference Series, 2016
In this paper we analyze a PDE system modelling (non-isothermal) phase transitions and damage phe... more In this paper we analyze a PDE system modelling (non-isothermal) phase transitions and damage phenomena in thermoviscoelastic materials. The model is thermodynamically consistent: in particular, no {\em small perturbation assumption} is adopted, which results in the presence of quadratic terms on the right-hand side of the temperature equation, only estimated in L1L^1L1. The whole system has a highly nonlinear character. We address the existence of a weak notion of solution, referred to as "entropic", where the temperature equation is formulated with the aid of an entropy inequality, and of a total energy inequality. This solvability concept reflects the basic principles of thermomechanics as well as the thermodynamical consistency of the model. It allows us to obtain \emph{global-in-time} existence theorems without imposing any restriction on the size of the initial data. We prove our results by passing to the limit in a time discretization scheme, carefully tailored to the nonlinear features of the PDE system (with its "entropic" formulation), and of the a priori estimates performed on it. Our time-discrete analysis could be useful towards the numerical study of this model.
Annali Della Scuola Normale Superiore Di Pisa Classe Di Scienze, 2008
Esaim Control Optimisation and Calculus of Variations, Jun 30, 2006
This paper addresses the Cauchy problem for the gradient flow equation in a Hilbert space H
Eprint Arxiv 0911 1590, Nov 1, 2009
We present applications of our abstract results by proving the existence of the global attractor ... more We present applications of our abstract results by proving the existence of the global attractor for the energy solutions both of abstract doubly nonlinear evolution equations in reflexive Banach spaces, and of a class of evolution equations in Wasserstein spaces, as well as for the generalized solutions of some phase-change evolutions driven by mean curvature.
Indiana University Mathematics Journal, 2007
This paper addresses the analysis of a model, proposed by M. Frémond, for the phenomenon of conta... more This paper addresses the analysis of a model, proposed by M. Frémond, for the phenomenon of contact with reversible adhesion between a viscoelastic body and a rigid support. First of all, we prove existence and uniqueness of global-in-time solutions of (the initial-boundary value problem for) the related PDE system by means of a fixed point technique. Hence, we investigate the long-time behaviour of such solutions and obtain some results on the structure of the associated ω-limit set.
This paper focuses on rate-independent damage in elastic bodies. Since the driving energy is nonc... more This paper focuses on rate-independent damage in elastic bodies. Since the driving energy is nonconvex, solutions may have jumps as a function of time, and in this situation it is known that the classical concept of energetic solutions for rate-independent systems may fail to accurately describe the behavior of the system at jumps. Therefore we resort to the (by now well-established) vanishing viscosity approach to rate-independent modeling, and approximate the model by its viscous regularization. In fact, the analysis of the latter PDE system presents remarkable difficulties, due to its highly nonlinear character. We tackle it by combining a variational approach to a class of abstract doubly nonlinear evolution equations, with careful regularity estimates tailored to this specific system, relying on a q-Laplacian type gradient regularization of the damage variable. Hence for the viscous problem we conclude the existence of weak solutions, satisfying a suitable energy-dissipation inequality that is the starting point for the vanishing viscosity analysis. The latter leads to the notion of (weak) parameterized solution to our rate-independent system, which encompasses the influence of viscosity in the description of the jump regime.
Compactness in the space L p (0, T ; B), B being a separable Banach space, has been deeply invest... more Compactness in the space L p (0, T ; B), B being a separable Banach space, has been deeply investigated by more recently, by J.M. , who have provided various criteria for relative compactness, which turn out to be crucial tools in the existence proof of solutions to several abstract time dependent problems related to evolutionary PDEs. In the present paper, the problem is examined in view of Young measure theory: exploiting the underlying principles of "tightness" and "integral equicontinuity", new necessary and sufficient conditions for compactness are given, unifying some of the previous contributions and showing that the Aubin -Lions condition is not only sufficient but also necessary for compactness. Furthermore, the related issue of compactness with respect to convergence in measure is studied and a general criterion is proved. : 28A20, 46E30, 46N20.
We address the long-time behaviour of a class of viscous Cahn-Hilliard equations, modelling phase... more We address the long-time behaviour of a class of viscous Cahn-Hilliard equations, modelling phase separation in mixtures and alloys. Specifically, we prove the existence of (a suitable notion of) the global attractor for the weak solutions of the so-called generalized viscous Cahn-Hilliard equation. November 4, 2005 17:43 Dissipative Phase Transitions P. Colli, N. Kenmochi, J. Sprekels Attractor for generalized viscous Cahn-Hilliard equations 5
Compactness in the space Lp(0;T ;B), B being a separable Banach space, has been deeply investigat... more Compactness in the space Lp(0;T ;B), B being a separable Banach space, has been deeply investigated by J.P. Aubin (1963), J.L. Lions (1961,1969), J. Simon (1987), and, more recently, by J.M. Rakotoson and R. Temam (2001), who have provided various criteria for relative compactness, which turn out to be crucial tools in the existence proof of solutions to many abstract
SIAM Journal on Mathematical Analysis, 2015
In this paper we analyze a PDE system modeling (non-isothermal) phase transitions and damage phen... more In this paper we analyze a PDE system modeling (non-isothermal) phase transitions and damage phenomena in thermoviscoelastic materials. The model is thermodynamically consistent: in particular, no small perturbation assumption is adopted, which results in the presence of quadratic terms on the right-hand side of the temperature equation, only estimated in L 1 . The whole system has a highly nonlinear character.
Series on Advances in Mathematics for Applied Sciences, 2006
We address the long-time behaviour of a class of viscous Cahn-Hilliard equations, modelling phase... more We address the long-time behaviour of a class of viscous Cahn-Hilliard equations, modelling phase separation in mixtures and alloys. Specifically, we prove the existence of (a suitable notion of) the global attractor for the weak solutions of the so-called generalized viscous Cahn-Hilliard equation. November 4, 2005 17:43 Dissipative Phase Transitions P. Colli, N. Kenmochi, J. Sprekels Attractor for generalized viscous Cahn-Hilliard equations 5
Nonlinear Analysis: Real World Applications, 2015
ABSTRACT In this paper we introduce and investigate a model for adhesive contact with friction be... more ABSTRACT In this paper we introduce and investigate a model for adhesive contact with friction between a thermoviscoelastic body and a rigid support.A PDE system, consisting of the evolution equations for the temperatures in the bulk domain and on the contact surface, of the momentum balance, and of the equation for the internal variable describing the state of the adhesion, is derived on the basis of a surface damage theory by M. Frémond.The existence of global-in-time solutions to the associated initial–boundary value problem is proved by passing to the limit in a carefully tailored time-discretization scheme.
Physica D: Nonlinear Phenomena, 2014
We propose a model for (unilateral) contact with adhesion between a viscoelastic body and a rigid... more We propose a model for (unilateral) contact with adhesion between a viscoelastic body and a rigid support, encompassing thermal and frictional effects. Following Frémond's approach, adhesion is described in terms of a surface damage parameter χ . The related equations are the momentum balance for the vector of small displacements, and parabolic-type evolution equations for χ and for the absolute temperatures of the body and of the adhesive substance on the contact surface. All of the constraints on the internal variables, as well as the contact and the friction conditions, are rendered by means of subdifferential operators. Furthermore, the temperature equations, derived from an entropy balance law, feature singular functions. Therefore, the resulting PDE system has a highly nonlinear character.
Nonlinear Analysis: Real World Applications, 2015
This paper focuses on rate-independent damage in elastic bodies. Since the driving energy is nonc... more This paper focuses on rate-independent damage in elastic bodies. Since the driving energy is nonconvex, solutions may have jumps as a function of time, and in this situation it is known that the classical concept of energetic solutions for rate-independent systems may fail to accurately describe the behavior of the system at jumps.