Mauro Fabrizio | Università di Bologna (original) (raw)
Papers by Mauro Fabrizio
Springer eBooks, 2012
, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection w... more , except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.
Acta Applicandae Mathematicae
Progress in Fractional Differentiation & Applications, 2019
Thermodynamics of Materials with Memory, 2021
This chapter deals with the following two topics [17, 165]. Using a standard representation of a ... more This chapter deals with the following two topics [17, 165]. Using a standard representation of a free energy associated with a linear memory constitutive relation, a new condition, involving linear functionals, is derived which, if satisfied, ensures that the free energy is a functional of the minimal state. Using this condition and results on constructing free energy functionals in Chap. 17, it is shown that if the kernel of the rate of dissipation functional is given by sums of products, the associated free energy functional is a FMS. Because this condition is linear rather than a quadratic, it is easier to explore and to apply in new contexts. Also, it is argued that for a free energy functionals to be physically realistic, it should exhibit strict periodic behavior for histories that have been periodic for all past times. The work function does not have this property, which causes difficulties in ascribing free energies to materials with singleton minimal states. A method of avoiding this difficulty is outlined.
arXiv: Fluid Dynamics, 2018
Through Ginzburg-Landau and Navier-Stokes equations, we study turbulence phenomena for viscous in... more Through Ginzburg-Landau and Navier-Stokes equations, we study turbulence phenomena for viscous incompresible and compressible fluids by a second order phase transition. For this model, the velocity is defined by the sum of classical and whirling components. Moreover, the laminar-turbulent transition is controlled by rotational effects of the fluid. Hence, the thermodynamic compatibility of the differential system is proved. This model can explain the turbulence by instability effects motivated by a double well potential of the Ginzburg-Landau equation. The same model is used to understand the origins of tornadoe and the birth of the vortices resulting from the fall of water in a vertical tube. Finally, we demonstrate how the weak Coriolis force is able to change the direction of rotation of the vortices by modifying the minima of the phase field potential.
In this paper we introduce a new model describing the behavior of auxetic materials in terms of a... more In this paper we introduce a new model describing the behavior of auxetic materials in terms of a phase-field PDE system. More precisely, the evolution equations are recovered by a generalization of the principle of virtual power in which microscopic motions and forces, responsible for the phase transitions, are included. The momentum balance is written in the setting of a second gradient theory, and it presents nonlinear contributions depending on the phases. The evolution of the phases is governed by variational inclusions with non-linear coupling terms. By use of a fixed point theorem and monotonicity arguments, we are able to show that the resulting initial and boundary value problem admits a weak solution.
Thermodynamics of Materials with Memory, 2021
Thermodynamics of Materials with Memory, 2021
Springer INdAM Series, 2017
We are concerned with two different inverse problems for degenerate integro-differential equation... more We are concerned with two different inverse problems for degenerate integro-differential equations in Banach spaces. In the first, we handle a strongly degenerate problem on a finite interval, while in the second we consider a related inverse problem for integro-differential equations studied by G. Da Prato and A. Lunardi in the regular case. All these results can be applied to inverse problems for equations from mathematical physics.
Mathematics, 2020
This paper deals with inverse problems related to degenerate fractional integro-differential equa... more This paper deals with inverse problems related to degenerate fractional integro-differential equations in Banach spaces. We study existence, uniqueness and regularity of solutions to the problem, claiming to extend well known studies for the case of non-fractional equations. Our method is based on transforming the inverse problem to a direct problem and identifying the conditions under which this direct problem has a unique solution. The conditions under which the unique strict solution can be compared with the case of a mild solution, obtained in previous studies under quite restrictive requirements, are on the underlying functions. Applications from partial differential equations are given to illustrate our abstract results.
Applied Mathematics and Computation, 2020
Through the Ginzburg-Landau and the Navier-Stokes equations, we study turbulence phenomena for vi... more Through the Ginzburg-Landau and the Navier-Stokes equations, we study turbulence phenomena for viscous incompressible and compressible fluids by a second order phase transition. For this model, the velocity is defined by the sum of classical and whirling components. Moreover, the laminar-turbulent transition is controlled by rotational effects of the fluid. Hence, the thermodynamic compatibility of the differential system is proved. The same model is used to understand the origins of tornadoes and their behavior and the birth of the vortices resulting from the fall of water in a vertical tube. Finally, we demonstrate how the weak Coriolis force is able to change the rotation direction of the vortices by modifying the minima of the Ginzburg-Landau equation. Hence, we conclude the paper with the differential system describing the water vorticity and its thermodynamic compatibility.
Fractal and Fractional, 2017
The extension of the fractional order derivative to the distributed order fractional derivative (... more The extension of the fractional order derivative to the distributed order fractional derivative (DOFD) is somewhat simple from a formal point of view, but it does not yet have a simple, obvious analytic form that allows its fast numerical calculation, which is necessary when solving differential equations with DOFD. In this paper, we supply a simple analytic kernel for the Caputo DOFD and the Caputo-Fabrizio DOFD, which may be used for numerical calculation in cases where the weight function is unity. This, in turn, could potentially allow faster solution of differential equations containing DOFD. Utilizing an analytical formulation of simple physical systems with phenomenological equations that include a DOFD, we show the relevant differences between the Caputo DOFD and the Caputo-Fabrizio DOFD. Finally, we propose a model based on DOFD for modeling composed materials that comprise different constituents, and show its compatibility with thermodynamics.
Evolution Equations and Control Theory, 2014
The free energy for most materials with memory is not unique. There is a convex set of free energ... more The free energy for most materials with memory is not unique. There is a convex set of free energy functionals with a minimum and a maximum element. Various functionals have been shown to have the properties of a free energy for materials with particular types of relaxation behaviour. Also, over the last decade or more, forms have been given for the minimum and related free energies. These are all quadratic functionals which yield linear memory terms in the constitutive equations for the stress. A difficulty in constructing free energy functionals arises in making choices that ensure a non-negative quadratic form both for the free energy and for the rate of dissipation. We propose a technique which renders this task more straightforward. Instead of constructing the free energy and determining from this the rate of dissipation, which may not have the required non-negativity, the procedure is reversed, which guarantees a satisfactory free energy functional. Certain results for quadratic functionals in the time and frequency domains are derived, providing a platform for this alternative approach, which produces new free energies, including a family of functionals that are generalizations of the minimum and related free energies. 2010 Mathematics Subject Classification. 80A17, 74A15, 74D05. Key words and phrases. Thermodynamics, memory effects, new free energy functionals, rate of dissipation, frequency domain representations, causality. 1 We consider for definiteness here isothermal mechanical problems, indeed those for solid viscoelastic materials. Also, only the scalar case is considered, which simplifies the algebra and allows us to focus on the essential structure of the arguments. It must be emphasized however that similar results can be given with little extra difficulty, for viscoelastic fluids, non-isothermal problems, electromagnetism, non-simple materials etc. as presented in the references noted above, and indeed for the general tensor cases relating to all these materials.
Journal of Non-Equilibrium Thermodynamics, 2017
In this paper, the seeming inconsistency highlighted by Fabrizio and Lazzari (Stability and secon... more In this paper, the seeming inconsistency highlighted by Fabrizio and Lazzari (Stability and second law of thermodynamics in dual-phase-lag heat conduction,
Journal of Non-Equilibrium Thermodynamics, 2016
We develop a phase-field model for the liquid–vapor phase transition. The model aims to describe ... more We develop a phase-field model for the liquid–vapor phase transition. The model aims to describe in a thermodynamically consistent way the phase change phenomenon coupled with the macroscopic motion of the fluid. The phase field
Progress in Fractional Differentiation and Applications, 2016
In the paper, we present some applications and features related with the new notions of fractiona... more In the paper, we present some applications and features related with the new notions of fractional derivatives with a time exponential kernel and with spatial Gauss kernel for gradient and Laplacian operators. Specifically, for these new models we have proved the coherence with the thermodynamic laws. Hence, we have revised the standard linear solid of Zener within continuum mechanics and the model of Cole and Cole inside electromagnetism by these new fractional operators. Moreover, by the Gaussian fractional gradient and through numerical simulations, we have studied the bell shaped filtering effects comparing the results with exponential and Caputo kernel.
Springer eBooks, 2012
, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection w... more , except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.
Acta Applicandae Mathematicae
Progress in Fractional Differentiation & Applications, 2019
Thermodynamics of Materials with Memory, 2021
This chapter deals with the following two topics [17, 165]. Using a standard representation of a ... more This chapter deals with the following two topics [17, 165]. Using a standard representation of a free energy associated with a linear memory constitutive relation, a new condition, involving linear functionals, is derived which, if satisfied, ensures that the free energy is a functional of the minimal state. Using this condition and results on constructing free energy functionals in Chap. 17, it is shown that if the kernel of the rate of dissipation functional is given by sums of products, the associated free energy functional is a FMS. Because this condition is linear rather than a quadratic, it is easier to explore and to apply in new contexts. Also, it is argued that for a free energy functionals to be physically realistic, it should exhibit strict periodic behavior for histories that have been periodic for all past times. The work function does not have this property, which causes difficulties in ascribing free energies to materials with singleton minimal states. A method of avoiding this difficulty is outlined.
arXiv: Fluid Dynamics, 2018
Through Ginzburg-Landau and Navier-Stokes equations, we study turbulence phenomena for viscous in... more Through Ginzburg-Landau and Navier-Stokes equations, we study turbulence phenomena for viscous incompresible and compressible fluids by a second order phase transition. For this model, the velocity is defined by the sum of classical and whirling components. Moreover, the laminar-turbulent transition is controlled by rotational effects of the fluid. Hence, the thermodynamic compatibility of the differential system is proved. This model can explain the turbulence by instability effects motivated by a double well potential of the Ginzburg-Landau equation. The same model is used to understand the origins of tornadoe and the birth of the vortices resulting from the fall of water in a vertical tube. Finally, we demonstrate how the weak Coriolis force is able to change the direction of rotation of the vortices by modifying the minima of the phase field potential.
In this paper we introduce a new model describing the behavior of auxetic materials in terms of a... more In this paper we introduce a new model describing the behavior of auxetic materials in terms of a phase-field PDE system. More precisely, the evolution equations are recovered by a generalization of the principle of virtual power in which microscopic motions and forces, responsible for the phase transitions, are included. The momentum balance is written in the setting of a second gradient theory, and it presents nonlinear contributions depending on the phases. The evolution of the phases is governed by variational inclusions with non-linear coupling terms. By use of a fixed point theorem and monotonicity arguments, we are able to show that the resulting initial and boundary value problem admits a weak solution.
Thermodynamics of Materials with Memory, 2021
Thermodynamics of Materials with Memory, 2021
Springer INdAM Series, 2017
We are concerned with two different inverse problems for degenerate integro-differential equation... more We are concerned with two different inverse problems for degenerate integro-differential equations in Banach spaces. In the first, we handle a strongly degenerate problem on a finite interval, while in the second we consider a related inverse problem for integro-differential equations studied by G. Da Prato and A. Lunardi in the regular case. All these results can be applied to inverse problems for equations from mathematical physics.
Mathematics, 2020
This paper deals with inverse problems related to degenerate fractional integro-differential equa... more This paper deals with inverse problems related to degenerate fractional integro-differential equations in Banach spaces. We study existence, uniqueness and regularity of solutions to the problem, claiming to extend well known studies for the case of non-fractional equations. Our method is based on transforming the inverse problem to a direct problem and identifying the conditions under which this direct problem has a unique solution. The conditions under which the unique strict solution can be compared with the case of a mild solution, obtained in previous studies under quite restrictive requirements, are on the underlying functions. Applications from partial differential equations are given to illustrate our abstract results.
Applied Mathematics and Computation, 2020
Through the Ginzburg-Landau and the Navier-Stokes equations, we study turbulence phenomena for vi... more Through the Ginzburg-Landau and the Navier-Stokes equations, we study turbulence phenomena for viscous incompressible and compressible fluids by a second order phase transition. For this model, the velocity is defined by the sum of classical and whirling components. Moreover, the laminar-turbulent transition is controlled by rotational effects of the fluid. Hence, the thermodynamic compatibility of the differential system is proved. The same model is used to understand the origins of tornadoes and their behavior and the birth of the vortices resulting from the fall of water in a vertical tube. Finally, we demonstrate how the weak Coriolis force is able to change the rotation direction of the vortices by modifying the minima of the Ginzburg-Landau equation. Hence, we conclude the paper with the differential system describing the water vorticity and its thermodynamic compatibility.
Fractal and Fractional, 2017
The extension of the fractional order derivative to the distributed order fractional derivative (... more The extension of the fractional order derivative to the distributed order fractional derivative (DOFD) is somewhat simple from a formal point of view, but it does not yet have a simple, obvious analytic form that allows its fast numerical calculation, which is necessary when solving differential equations with DOFD. In this paper, we supply a simple analytic kernel for the Caputo DOFD and the Caputo-Fabrizio DOFD, which may be used for numerical calculation in cases where the weight function is unity. This, in turn, could potentially allow faster solution of differential equations containing DOFD. Utilizing an analytical formulation of simple physical systems with phenomenological equations that include a DOFD, we show the relevant differences between the Caputo DOFD and the Caputo-Fabrizio DOFD. Finally, we propose a model based on DOFD for modeling composed materials that comprise different constituents, and show its compatibility with thermodynamics.
Evolution Equations and Control Theory, 2014
The free energy for most materials with memory is not unique. There is a convex set of free energ... more The free energy for most materials with memory is not unique. There is a convex set of free energy functionals with a minimum and a maximum element. Various functionals have been shown to have the properties of a free energy for materials with particular types of relaxation behaviour. Also, over the last decade or more, forms have been given for the minimum and related free energies. These are all quadratic functionals which yield linear memory terms in the constitutive equations for the stress. A difficulty in constructing free energy functionals arises in making choices that ensure a non-negative quadratic form both for the free energy and for the rate of dissipation. We propose a technique which renders this task more straightforward. Instead of constructing the free energy and determining from this the rate of dissipation, which may not have the required non-negativity, the procedure is reversed, which guarantees a satisfactory free energy functional. Certain results for quadratic functionals in the time and frequency domains are derived, providing a platform for this alternative approach, which produces new free energies, including a family of functionals that are generalizations of the minimum and related free energies. 2010 Mathematics Subject Classification. 80A17, 74A15, 74D05. Key words and phrases. Thermodynamics, memory effects, new free energy functionals, rate of dissipation, frequency domain representations, causality. 1 We consider for definiteness here isothermal mechanical problems, indeed those for solid viscoelastic materials. Also, only the scalar case is considered, which simplifies the algebra and allows us to focus on the essential structure of the arguments. It must be emphasized however that similar results can be given with little extra difficulty, for viscoelastic fluids, non-isothermal problems, electromagnetism, non-simple materials etc. as presented in the references noted above, and indeed for the general tensor cases relating to all these materials.
Journal of Non-Equilibrium Thermodynamics, 2017
In this paper, the seeming inconsistency highlighted by Fabrizio and Lazzari (Stability and secon... more In this paper, the seeming inconsistency highlighted by Fabrizio and Lazzari (Stability and second law of thermodynamics in dual-phase-lag heat conduction,
Journal of Non-Equilibrium Thermodynamics, 2016
We develop a phase-field model for the liquid–vapor phase transition. The model aims to describe ... more We develop a phase-field model for the liquid–vapor phase transition. The model aims to describe in a thermodynamically consistent way the phase change phenomenon coupled with the macroscopic motion of the fluid. The phase field
Progress in Fractional Differentiation and Applications, 2016
In the paper, we present some applications and features related with the new notions of fractiona... more In the paper, we present some applications and features related with the new notions of fractional derivatives with a time exponential kernel and with spatial Gauss kernel for gradient and Laplacian operators. Specifically, for these new models we have proved the coherence with the thermodynamic laws. Hence, we have revised the standard linear solid of Zener within continuum mechanics and the model of Cole and Cole inside electromagnetism by these new fractional operators. Moreover, by the Gaussian fractional gradient and through numerical simulations, we have studied the bell shaped filtering effects comparing the results with exponential and Caputo kernel.