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Stefania GATTI

Professore Associato
Dipartimento di Scienze Fisiche, Informatiche e Matematiche sede ex-Matematica

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Pubblicazioni

2024 - A perturbation of the Cahn–Hilliard equation with logarithmic nonlinearity [Articolo su rivista]
Conti, Monica; Gatti, Stefania; Miranville, Alain
abstract

Our aim in this paper is to study a perturbation of the Cahn–Hilliard equation with nonlinear terms of logarithmic type. This new model is based on an unconstrained theory recently proposed in [5]. We prove the existence, regularity and uniqueness of solutions, as well as (strong) separation properties of the solutions from the pure states, also in three space dimensions. We finally prove the convergence to the Cahn–Hilliard equation, on finite time intervals.

2022 - An Oxygen driven proliferative-to-invasive transition of glioma cells: an analytical study [Articolo su rivista]
Gatti, Stefania
abstract

Our aim in this paper is to analyze a model of glioma where oxygen drives cancer diffusion and proliferation. We prove the global well-posedness of the analytical problem and that, in the longtime, the illness does not disappear. Besides, the tumor dynamics increase the oxygen levels.

2022 - Mathematical analysis of a phase-field model of brain cancers with chemotherapy and antiangiogenic therapy effects [Articolo su rivista]
Conti, Monica; Gatti, Stefania; Miranville, Alain
abstract

Our aim in this paper is to study a mathematical model for brain cancers with chemotherapy and antiangiogenic therapy effects. We prove the existence and uniqueness of biologically relevant (nonnegative) solutions. We then address the important question of optimal treatment. More precisely, we study the problem of finding the controls that provide the optimal cytotoxic and antiangiogenic effects to treat the cancer.

2022 - On a tumor growth model with brain lactate kinetics [Articolo su rivista]
Cherfils, Laurence; Gatti, Stefania; Guillevin, Carole; Miranville, Alain; Guillevin, Rémy
abstract

Our aim in this paper is to study a mathematical model for high grade gliomas, taking into account lactates kinetics, as well as chemotherapy and antiangiogenic treatment. In particular, we prove the existence and uniqueness of biologically relevant solutions. We also perform numerical simulations based on di erent therapeutical situations which can be found in the literature. These simulations are consistent with what is expected in these situations.

2021 - Analysis of a model for tumor growth and lactate exchanges in a glioma [Articolo su rivista]
Cherfils, L; Gatti, S; Miranville, A; Guillevin, R
abstract

Our aim in this paper is to study a mathematical model for tumor growth and lactate exchanges in a glioma. We prove the existence of nonnegative (i.e. biologically relevant) solutions and, under proper assumptions, the uniqueness of the solution. We also state the permanence of the tumor when necrosis is not taken into account in the model and obtain linear stability results. We end the paper with numerical simulations.

2019 - Mathematical analysis of a model for proliferative-to-invasive transition of hypoxic glioma cells [Articolo su rivista]
Conti, Monica; Gatti, Stefania; Miranville, Alain
abstract

Our aim in this paper is to study a mathematical model for the proliferative-toinvasive transition of hypoxic glioma cells. We prove the existence and uniqueness of nonnegative solutions and then address the important question of whether the positive solutions undergo extinction or permanence. More precisely, we prove that this depends on the boundary conditions: there is no extinction when considering Neumann boundary conditions, while we prove extinction when considering Dirichlet boundary conditions.

2019 - Regularity and long-time behavior for a thermodynamically consistent model for complex fluids in two space dimensions [Articolo su rivista]
Eleuteri, Michela; Gatti, Stefania; Schimperna, Giulio
abstract

We consider a thermodynamically consistent model for the evolution of thermally conducting two-phase incompressible fluids. Complementing previous results, we prove additional regularity properties of solutions in the case when the evolution takes place in the two-dimensional flat torus. Thanks to improved regularity, we can also prove uniqueness and characterize the long-time behavior of trajectories showing existence of the global attractor in a suitable phase space.

2018 - A singular Cahn--Hilliard--Oono phase-field system with hereditary memory [Articolo su rivista]
Conti, Monica; Gatti, Stefania; Miranville, Alain
abstract

We consider a phase-field system modeling phase transition phenomena, where the Cahn--Hilliard--Oono equation for the order parameter is coupled with the Coleman--Gurtin heat law for the temperature. The former suitably describes both local and nonlocal (long-ranged) interactions in the material undergoing phase-separation, while the latter takes into account thermal memory effects. We study the well-posedness and longtime behavior of the corresponding dynamical system in the history space setting, for a class of physically relevant and singular potentials. Besides, we investigate the regularization properties of the solutions and, for sufficiently smooth data, we establish the strict separation property from the pure phases.

2018 - Asymptotic behavior of higher‐order Navier‐Stokes‐Cahn‐Hilliard systems [Articolo su rivista]
Cherfils, Laurence; Gatti, Stefania; Miranville, Alain
abstract

Our aim in this paper is to study the asymptotic behavior, in terms of finite‐dimensional attractors, for higher‐order Navier‐Stokes‐Cahn‐Hilliard systems. Such equations describe the evolution of a mixture of 2 immiscible incompressible fluids. We also give several numerical simulations.

2017 - A phase-field system with two temperatures and memory [Articolo su rivista]
Conti, Monica; Gatti, Stefania; Miranville, Alain
abstract

Abstract. Our aim, in this paper, is to study a generalization of the Caginalp phase-field system based on the Gurtin-Pipkin law with two temperatures for heat conduction with memory. In particular, we obtain well-posedness results and study the dissipativity, in terms of the global attractor with optimal regularity, of the associated solution operators. We also study the stability of the system as the memory kernel collapses to a Dirac mass.

2017 - On a Caginalp Phase-Field System with Two Temperatures and Memory [Articolo su rivista]
Conti, M.; Gatti, Stefania; Miranville, A.; Quintanilla,
abstract

The Caginalp phase-field system has been proposed in [4] as a simple mathematical model for phase transition phenomena. In this paper, we are concerned with a generalization of this system based on the Gurtin-Pipkin law with two temperatures for heat conduction with memory, apt to describe transition phenomena in nonsimple materials. The model consists of a parabolic equation governing the order parameter which is linearly coupled with a nonclassical integrodifferential equation ruling the evolution of the thermodynamic temperature of the material. Our aim is to construct a robust family of exponential attractors for the associated semigroup, showing the stability of the system with respect to the collapse of the memory kernel. We also study the spatial behavior of the solutions in a semi-infinite cylinder, when such solutions exist.

2016 - Robust family of exponential attractors for isotropic crystal models [Articolo su rivista]
Cherfils, Laurence; Gatti, Stefania
abstract

Our aim in this paper is to study, in term of finite dimensional exponential attractors, the Willmore regularization, (depending on a small regularization parameter epsilon >0), of two phase-field equations, namely, the Allen–Cahn and the Cahn–Hilliard equations. In both cases, we construct robust families of exponential attractors, that is, attractors that are continuous with respect to the perturbation parameter.

2015 - Multi-component Cahn–Hilliard systems with dynamic boundary conditions [Articolo su rivista]
Conti, Monica; Gatti, Stefania; Miranville, Alain
abstract

Our aim in this paper is to study the well-posedness and the asymptotic behavior, in terms of finite-dimensional attractors, of Cahn–Hilliard systems describing phase separation processes in multi-component alloys, endowed with dynamic boundary conditions. Such boundary conditions take into account the interactions with the walls when considering confined systems.

2014 - Pullback exponential attractor for a Cahn-Hilliard-Navier-Stokes system in 2D [Articolo su rivista]
Stefano, Bosia; Gatti, Stefania
abstract

We consider a model for the evolution of a mixture of two incompressible and partially immiscible Newtonian fluids in two dimensional bounded domain. More precisely, we address the well-known model H consisting of the Navier-Stokes equation with non-autonomous external forcing term for the (average) fluid velocity, coupled with a convective Cahn-Hilliard equation with polynomial double-well potential describing the evolution of the relative density of atoms of one of the fluids. We study the long term behavior of solutions and prove that the system possesses a pullback exponential attractor. In particular the regularity estimates we obtain depend on the initial data only through fixed powers of their norms and these powers are independent of the growth of the polynomial potential considered in the Cahn-Hilliard equation.

2013 - A generalization of the Caginalp phase-field system with Neumann boundary conditions [Articolo su rivista]
Conti, M.; Gatti, Stefania; Miranville, A.
abstract

We study a generalized Caginalp phase-field system based on the theory of type III heat conduction proposed by Green and Naghdi and supplemented with Neumann boundary conditions. In contrast to the Dirichlet case, the system exhibits a lack of dissipation on the thermal displacement variable α. However, α minus its spatial average is dissipative and we are able to prove the existence of the global attractor with optimal regularity for the associated semigroup.

2013 - A variational approach to a Cahn-Hillliard model in a domain with nonpermeable walls. [Articolo su rivista]
Cherfils, L.; Gatti, Stefania; Miranville, A.
abstract

This paper deals with the well-posedness and the long time behavior of a Cahn-Hilliard model with a singular bulk potential and suitable dynamic boundary conditions. We assume here that the system is confined in a vessel with non-permeable walls and that the total mass, in the bulk and on the boundary, is conserved. As a result, the well-posedness in the sense of distributions may not hold and new notions of solutions are required. The same problem has been analyzed in \cite{Gold}, relying on duality techniques, under weak assumptions on the nonlinearities. However, the regularity of solutions and the study of the asymptotic behavior of the system required growth restrictions on the bulk nonlinearity which exclude the thermodynamically relevant logarithmic potentials. Our aim in this paper is to improve these results by introducing instead, in the spirit of [23], a variational formulation of the problem, based on a proper variational inequality.

2013 - Attractors for a Caginalp model with a logarithmic potential and coupled dynamic boundary conditions [Articolo su rivista]
M., Conti; Gatti, Stefania; A., Miranville
abstract

We study the longtime behavior of the Caginalp phase-field model with a logarithmic potential and dynamic boundary conditions both for the order parameter and the temperature. Due to the possible lack of distributional solutions, we deal with a suitable definition of solutions based on variational inequalities, for which we prove well-posedness and the existence of global and exponential attractors with finite fractal dimension.

2012 - Asymptotic behavior of the Caginalp phase-field system with coupled dynamic boundary conditions [Articolo su rivista]
M., Conti; Gatti, Stefania; A., Miranville
abstract

This paper deals with the longtime behavior of the Caginalp phase-field system with coupled dynamic boundary conditions on both state variables.We prove that the system generates a dissipative semigroup in a suitable phase-spaceand possesses the finite-dimensional smooth global attractor and an exponential attractor.

2012 - Long time behavior of the Caginalp system with singular potentials and dynamic boundary conditions [Articolo su rivista]
L., Cherfils; Gatti, Stefania; A., Miranville
abstract

This paper is devoted to the study of the well-posedness and the long time behavior of the Caginalpphase-field model with singular potentials and dynamic boundary conditions.Thanks to a suitable definition of solutions, coinciding with the strong ones underproper assumptions on the bulk and surfacepotentials, we are able to get dissipative estimates, leading tothe existence of the global attractor with finite fractal dimension,as well as of an exponential attractor.

2011 - A doubly nonlinear parabolic equation with a singular potential [Articolo su rivista]
L., Cherfils; Gatti, Stefania; A., Miranville
abstract

Our aim in this paper is to study the long time behavior, in termsof finite dimensional attractors, of doubly nonlinear Allen-Cahntype equations with singular potentials.

2010 - Continuous families of exponential attractors for singularly perturbed equations with memory [Articolo su rivista]
Gatti, Stefania; A., Miranville; V., Pata; S., Zelik
abstract

We establish, for a family of semigroups S_(ε:) H_ε→H_ε depending on aperturbation parameter ε∈[0,1], where the perturbation is allowed tobecome singular at ε=0, a general theorem on the existenceof exponential attractors E_ε satisfying a suitableHölder continuity property with respect tothe symmetric Hausdorff distanceat every ε∈[0,1]. The result is appliedto the abstract evolution equations with memoryx_t (t)+∫_0^∞▒█(k_ε (s) B_0 (x(t-s) )ds+B_1 (x(t) )=0, ε∈(0,1] )where k_ε (s)=1/ε k(s/ε)is the rescaling of a convex summable kernel k with unit mass.Such a family can be viewed as a memory perturbationof the equation x_t (t)+B_0 (x(t) )+B_1 (x(t) )=0formally obtained in the singular limit ε→0.

2010 - Two-dimensional reaction-diffusion equations with memory [Articolo su rivista]
M., Conti; Gatti, Stefania; M., Grasselli; V., Pata
abstract

In a two-dimensional space domain, we consider a reaction-diffusion equation whose diffusion term is a time convolution of the Laplace operator against a nonincreasing summable memory kernel k. This equation models several phenomena arising from many different areas. After rescaling k by a relaxation time ε>0, we formulate the corresponding Cauchy-Dirichlet problem, which is rigorously translated into a similar problem for a semilinear hyperbolic integro-differential equation with nonlinear damping,for a particular choice of the initial data. Using the past history approach, we show that the hyperbolic equation generates a dynamical system, which is dissipative provided that ε is small enough, namely, when the equation is sufficiently ``close" to thestandard reaction-diffusion equation formally obtained by replacing k with the Dirac mass at 0. Then, we provide an estimate of the difference between ε-trajectories and 0-trajectories, and we construct a family of regular exponential attractors which is robust with respect to the singular limit ε→0.In particular, this yields the existence of a regular global attractor of finite fractal dimension. Convergence to equilibria is also examined. Finally, all the results are reinterpreted within the original framework.

2009 - A Gronwall-type lemma with parameter and dissipative estimates for PDEs [Articolo su rivista]
Gatti, Stefania; Pata, Vittorino; Zelik, Sergey
abstract

We discuss the problem of establishing dissipative estimates for certain differential equations for which the usual methods apparently do not work. The main tool is a new Gronwall-type lemma with parameter. As an application, we consider a semilinear equation of viscoelasticity with low dissipation.

2009 - FINITE DIMENSIONAL ATTRACTORS FOR THE CAGINALPSYSTEM WITH SINGULAR POTENTIALS ANDDYNAMIC BOUNDARY CONDITIONS [Relazione in Atti di Convegno]
L., Cherfils; Gatti, Stefania; A., Miranville
abstract

Our aim in this paper is to prove the existence of finite dimensional attractors forthe Caginalp system with dynamic boundary conditions and singular potentials.

2008 - Attractors for semilinear equations of viscoelasticity with very low dissipation [Articolo su rivista]
Gatti, Stefania; Miranville, A; Pata, V; Zelik, S.
abstract

We analyze a differential system arising in the theory of isothermal viscoelasticity. This system is equivalent to an integrodifferential equation of hyperbolic type with a cubic nonlinearity, where the dissipation mechanism is contained only in the convolution integral, accounting for the past history of the displacement. In particular, we consider here a convolution kernelwhich entails an extremely weak dissipation. In spite of that,we show that the related dynamical system possesses a globalattractor of optimal regularity.

2008 - Corrigendum to "Existence of global solutions to the Caginalp phase-field system with dynamic boundary conditions and singular potentials" [J. Math. Anal. Appl. 343 (2008) 557-566] (DOI:10.1016/j.jmaa.2008.01.077) [Articolo su rivista]
L., Cherfils; Gatti, Stefania; A., Miranville
abstract

We correct the proof of Theorem 2.3 of the previous paper, assuming in advance a suitable growth of the potential approaching the singularities.

2008 - Existence of global solutions to the Caginalp phase-field system with dynamic boundary conditions and singular potentials [Articolo su rivista]
Cherfils, Laurence; Gatti, Stefania; Miranville, Alain
abstract

Our aim in this article is to prove the global (in time) existence ofsolutions to a Caginalp phase-field system with dynamic boundaryconditions and a singular potential. The main difficulty is to provethat the solutions are strictly separated from the singular values ofthe potential. This is achieved by studying an auxiliary elliptic problem.

2008 - Uniform decay properties of linear Volterra integro-differential equations [Articolo su rivista]
Conti, Monica; Gatti, Stefania; Pata, Vittorino
abstract

We establish some new results concerning the exponential decay and the polynomial decay of the energy associated with a linear Volterra integro-differential equation of hyperbolic type in a Hilbert space, which is an abstract version of the equation∂_tt u(t)-α∆u(t)+β∂_t u(t)+∫_0^t μ(s)∆u(t-s)ds=0describing the motion of linearly viscoelastic solids. We provide sufficient conditions for the decay to hold, without invoking differential inequalities involving the convolution kernel μ.

2007 - Decay rates of Volterra equations on R^n [Articolo su rivista]
Conti, Monica; Gatti, Stefania; Pata, Vittorino
abstract

This note is concerned with the linear Volterra equation of hyperbolic type ∂_tt u(t)-α∆u(t)+∫_0^t μ(s)∆u(t-s)ds=0 on the whole space R^n. New results concerning the decay of the associated energy as time goes to infinity were established.

2006 - A construction of a robust family of exponential attractors [Articolo su rivista]
Gatti, Stefania; Grasselli, M; Miranville, A; Pata, V.
abstract

Given a dissipative strongly continuous semigroup depending onsome parameters, we construct a family of exponential attractorswhich is robust, in the sense of the symmetric Hausdorff distance,with respect to (even singular) perturbations.

2006 - A one-dimensional wave equation with nonlinear damping [Articolo su rivista]
Gatti, Stefania; Pata, V.
abstract

We consider a one-dimensional weakly damped wave equation, with a damping coefficient depending on the displacement. We prove the existence of a regular connected global attractor of finite fractal dimension for the associated dynamical system, as well as the existence of an exponential attractor.

2006 - Asymptotic behavior of a phase-field system with dynamic boundary conditions [Capitolo/Saggio]
Gatti, Stefania; Miranville, A.
abstract

This article is devoted to the study of the asymptotic behavior of aCaginalp phase-field system with nonlinear dynamic boundary conditions. As a proper parameter ε goes to zero, this problem converges to the viscous Cahn-Hilliard equation. We firstprove the existence and uniqueness of the solution to the system and then provide an upper semicontinuous family of globalattractors A_ε . Furthermore, we prove the existence of anexponential attractor for each problem, which yields, since it contains the aforementioned global attractor, the finite fractal dimensionality of A_ε.

2006 - Convergence to stationary states of solutions to the semilinear equation of viscoelasticity [Capitolo/Saggio]
Gatti, Stefania; Grasselli, M.
abstract

We consider the equation of viscoelasticity characterized by a nonlinear elastic force φ depending on the displacement u and subject to a time dependent external load F. The dissipativity of the corresponding evolution system is only due to the presence of the relaxation kernel k. Rescaling k(s)-k(∞) with a relaxation time ε>0, we can find a sufficiently small ε_0>0, such that, if φ is real analytic and ε∈ (0,ε_0], then any sufficiently smooth u converges to a single stationary state with an algebraic decay rate, provided that F suitably converges to a time independent load.The proof relies on the well-known Łojasiewicz-Simon inequality.

2006 - Memory relaxation of the one-dimensional Cahn-Hilliard equation [Capitolo/Saggio]
Gatti, Stefania; Grasselli, M; Pata, V; Miranville, A.
abstract

We consider the memory relaxation of the one-dimensionalCahn-Hilliard equation endowed with the no-flux boundaryconditions. The resulting integrodifferential equation ischaracterized by a memory kernel which is the rescaling of a given positive decreasing function. The Cahn-Hilliard equation is then viewed as the formal limit of the relaxed equation, when thescaling parameter (or relaxation time) ε tends to zero. Inparticular, if the memory kernel is the decreasing exponential,then the relaxed equation is equivalent to the standard hyperbolicrelaxation. The main result of this note is the existence of afamily of robust exponential attractors for the one-parameterdissipative dynamical system generated by the relaxed equation,which is stable with respect to the singular limit ε→0.This theorem is obtained as a nontrivial application of a recentabstract result.

2006 - Parabolic-hyperbolic time-dependent Ginzburg-Landau-Maxwell equations [Articolo su rivista]
Berti, V; Gatti, Stefania
abstract

This article is devoted to the long-term dynamics of a parabolic-hyperbolic system arising in superconductivity. In the literature, the existence and uniqueness of the solution have been investigated but, to our knowledge, no asymptotic result is available. For the bi-dimensional model we prove that the system generates a dissipative semigroup in a proper phase-space where it possesses a (regular) global attractor. Then, we show the existence of an exponential attractor whose basin of attraction coincides with the whole phase-space. Thus, in particular, this exponential attractor contains the global attractor which, as a consequence, is of finite fractal dimension.

2006 - Singular limit of equations for linear viscoelastic fluids with periodic boundary conditions [Articolo su rivista]
Gatti, Stefania; Vuk, E.
abstract

We consider a one-parameter family of problems, governing, for any fixed parameter, the motion of a linear viscoelastic fluid in a two-dimensional domain with periodic boundary conditions. The asymptotic behavior of each problem is analyzed, by proving the existence of the global attractor. Moreover, letting the parameter go to zero, since the memory effect disappears, we obtain a limiting problem, given by the Navier–Stokes equations. For any fixed parameter, we construct an exponential attractor. The resulting family is robust, meaning that these exponential attractors converge, in an appropriate sense, to an exponential attractor of the limiting problem.

2006 - Trajectory and global attractors for evolution equations with memory [Articolo su rivista]
Chepyzhov, Vv; Gatti, Stefania; Grasselli, M; Miranville, A; Pata, V.
abstract

Our aim in this note is to analyze the relation between two notionsof attractors for the study of the long time behavior of equationswith memory, namely, the global attractor in the so-called pasthistory approach, and the more recently proposed notion of trajectory attractor.

2005 - Hyperbolic relaxation of the viscous Cahn-Hilliard equation in 3-D [Articolo su rivista]
Gatti, Stefania; Grasselli, M; Pata, V; Miranville, A.
abstract

We consider a modified version of the viscous Cahn-Hilliard equation governing the relative concentration u of one component in a binary system. This equation is characterized by the presence of the additional inertial term ω∂_tt u that accounts for the relaxation of the diffusion flux. Here ω≥0 is an inertial parameter which is supposed to be dominated from above by the viscosity coefficient δ. Endowing the equation with suitable boundary conditions, we show that it generates a dissipative dynamical system acting on a certain phase-space depending on ω. This system is shown to possess a global attractor that is upper-semicontinuous at ω=δ=0. Then, we construct a family of exponential attractors E_(ω,δ), which is a robust perturbation of an exponential attractor of the Cahn-Hilliard equation, namely, the symmetric Hausdorff distance between E_(ω,δ) and E_0,0 goes to 0 as (ω,δ) goes to (0,0) in an explicitly controlled way. This is done by using a general theorem which requires the construction of another dynamical system, strictly related to the original one, but acting on a different phase-space depending on both ω and δ.

2005 - Lyapunov functionals for reaction-diffusion equations with memory [Articolo su rivista]
Gatti, Stefania; Grasselli, M; Pata, V.
abstract

We consider a reaction diffusion equation in which the usual diffusion term also depends on the past history of the diffusion itself. This equation has been analysed by several authors, with an enphasis on the longtime behavior of the solutions. In this respect, the first results have been obtained by using the past history approach. They show that the equation, subject to a suitable boundary condition, defines a dissipative dynamical system which possesses a global attractor. A similar theorem has been recently proved by Chepyzhov and Miranville, using a different method based on the notion of trajectory attractors. In addition, those authors provide sufficient conditions that ensure the existence of a Lyapunov functional. Here we show that a similar result can be demonstrated within the past history approach, with less restrictive conditions.

2005 - Memory relaxation of first order evolution equations [Articolo su rivista]
Gatti, Stefania; Grasselli, M; Miranville, A; Pata, V.
abstract

A first order nonlinear evolution equation is relaxed by means of a time convolution operator, with a kernel obtained by rescaling a given positive decreasing function. This relaxation produces an integrodifferential equation, whose formal limit, as the scaling parameter (or relaxation time) ε tends to zero, is the original equation. The relaxed equation is equivalent to the widely studied hyperbolic relaxation when the memory kernel, in particular, is the decreasing exponential. In this work, we establish general conditions which ensure that the longterm dynamics of the two evolution equations are, in some appropriate sense, close, when ε is small. Namely, we prove the existence of a robust family of exponential attractors for the related dissipative dynamical systems, which is stable with respect to the singular limit ε→0. The abstract result is then applied to Allen-Cahn and Cahn-Hilliard type equations.

2005 - Navier-Stokes limit of Jeffreys type flows [Articolo su rivista]
Gatti, Stefania; Giorgi, C; Pata, V.
abstract

We analyze a Jeffreys type model ruling the motion of a viscoelastic polymeric solution with linear memory in a two-dimensional domain with nonslip boundary conditions. For fixed values of the concentrations, we describe the asymptotic dynamics and we prove that, when the scaling parameter in the memory kernel (physically, the Weissenberg number of the flow) tends to zero, the model converges in an appropriate sense to the Navier-Stokes equations.

2005 - On the hyperbolic relaxation of the one-dimensional Cahn-Hilliard equation [Articolo su rivista]
Gatti, Stefania; Grasselli, M; Miranville, A; Pata, V.
abstract

We consider the one-dimensional Cahn-Hilliard equation with aninertial term ε∂_tt u, for ε ≥ 0. This equation, endowed with proper boundary conditions, generates a strongly continuous semigroup S_ε(t) which acts on a suitable phase-space and possesses a global attractor. Our main result is the construction of a robust family of exponential attractors M_ε, whose common basins of attraction are the whole phase-space.

2005 - Robust exponential attractors for a family of nonconserved phase-field systems with memory [Articolo su rivista]
Gatti, Stefania; Grasselli, M; Pata, V; Squassina, M.
abstract

We consider a family of phase-field systems with memory effects in the temperature ϑ, depending on a parameter ω≥0. Setting the problems in a suitable phase-space accounting for the past history of ϑ, we prove the existence of a family of exponential attractors E_ω which is robust as ω→0.

2004 - Exponential attractors for a conserved phase-field system with memory [Articolo su rivista]
Gatti, Stefania; Grasselli, M; Pata, V.
abstract

We consider a conserved phase-field system of Caginalp type, characterized by the assumption that both the internal energy and the heat flux depend on the past history of the temperature and its gradient, respectively. The model consists of a parabolicintegrodifferential equation, coupled with a fourth-order evolutionequation for the phase-field. This system, supplemented with suitable boundary conditions, can be interpreted as a dissipative dynamical system in a proper phase-space. In a previous joint work, the last two authors have proved the existence of a global attractor of finite fractal dimension. Here we show the existence of an exponential attractor, by means of the techniques we developed for nonconserved phase-field systems with memory.

2004 - Exponential attractors for a phase-field model with memory and quadratic nonlinearities [Articolo su rivista]
Gatti, Stefania; Grasselli, M; Pata, V.
abstract

We consider a phase-field system with memory effects. This model consists of an integrodifferential equation of parabolic type describing the evolution of the (relative) temperature ϑ, and depending on its past history. This equation is nonlinearly coupled through a function λ with a semilinear parabolic equation governing the order parameter χ. The state variables ϑ and χ are subject to Neumann homogeneous boundary conditions. Themodel becomes an infinite-dimensional dynamical system in asuitable phase-space by introducing an additional variable ηaccounting for the (integrated) past history of the temperature.The evolution of η is thus ruled by a first-order hyperbolicequation. Giorgi, Grasselli, and Pata proved that the obtaineddynamical system possesses a universal attractor A, which has finite fractal dimension provided that the coupling function λ is linear. Here we prove, as main result, the existence of an exponential attractor E which entails, in particular, that A has finite fractal dimension when λ is nonlinear with quadratic growth. Since the so-called squeezing property does not work in our framework, we cannot use the standard technique to construct E. Instead, we take advantage of a recent result due to Efendiev, Miranville, and Zelik. The present paper contains, to the best of our knowledge, the first example of exponential attractor for an infinite-dimensional dynamical system with memory effects. Also, the approach introduced here can be adapted to other dynamical systems with similar features.

2003 - Phase field systems with memory effects in the order parameter dynamics: convergence to standard phase field systems [Articolo su rivista]
Gatti, Stefania; Sartori, E.
abstract

We shall deal with two models arising in phase transition dynamics. The state of the system is described by the pair (ϑ,χ), where ϑ is the (relative) temperature and χ is the order parameter or phase field. The main difference between the two models relies on whether global constraints on χ are imposed or not: the resulting models will be called conserved or nonconserved, accordingly. Memory effects influencing both the heat flux and the dynamics of χ have been considered in a number of recent papers. Here we assume the Fourier law forthe heat flux in the energy balance equation, while we considermemory effects in the order parameter dynamics. We show thatsolutions to the phase field problems with memory converge to the solution to the standard phase field model, when the memorykernels suitably converge to the Dirac mass. This is done for boththe conserved and the nonconserved cases. Some error estimates are also obtained.

2003 - Solvability of a plane elliptic problem for the flow in a channel with a surface-piercing obstacle [Articolo su rivista]
Gatti, Stefania; Pierotti, D.
abstract

Let us consider the three-dimensional problem of the steady flow of a heavy ideal fluid past a surface-piercing obstacle in a rectangular channel of constant depth. The flow is parallel at infinity upstream, with constant velocity c. We discuss an approximate linear problem obtained in the limit of a "flat obstacle". This is a boundary value problem for the Laplace equation in a three-dimensional unbounded domain, with a second order condition on part of the boundary, the Neumann-Kelvin condition. By a Fourier expansion of the potential function, we reduce the three-dimensional problem to a sequence of plane problems for the Fourier coefficients; for every value of the velocity c, these problems can be described in terms of a two parameter elliptic problem in a strip. We discuss the two dimensional problem by a special variational approach, relying on some a priori properties of finite energy solutions; as a result, we prove unique solvability for 〖c≠c〗_(m,k) where c_(m,k) is a known sequence of values depending on the dimensions of the channel and on the limit length of the obstacle. Accordingly, we can prove the existence of a solution of the three-dimensional problem; the related flow has in general a non trivial wave pattern at infinity downstream. We also investigate the regularity of the solution in a neighborhood of the obstacle. The meaning of the singular values c_(m,k) is discussed from the point of view of the nonlinear theory.

2003 - Well-posedness results for phase field systems with memory effects in the order parameter dynamics [Articolo su rivista]
Gatti, Stefania; Sartori, E.
abstract

We study two models arising in phase transition dynamics. Thestate of the system is described by the pair (ϑ,χ), where ϑ is the (relative) temperature and χ is the order parameter or phase field. The main difference between the two models relies on whether global constraints on χ are imposed or not: accordingly, the resulting models will be called conserved or nonconserved. Memory effects influencing both the heat flux and the dynamics of χ have been considered in a number of recent papers. Here we assume the Fourier law for the heat flux in the energy balance equation, while we consider memory effects in the order parameter dynamics. We analyze the well-posedness of corresponding Cauchy-Neumann problems for both conserved and nonconserved models. Various results are derived according to properties of the memory kernel involved.

2001 - Automatic control of the temperature in phase change problems with memory [Articolo su rivista]
Gatti, Stefania
abstract

We study a parabolic two-phase system with memory occupying a bounded and smooth domain. The heat exchange at part of the boundary is controlled by a thermostat. Assuming on the phase variable either a relaxation dynamics or a Stefan condition, we prove existence and uniqueness results for feedback control problems corresponding to two different types of thermostat: the relay switch and the Preisach operator. These results are strictly related to the continuous dependence of the solution on the boundary datum, which is investigated in advance.

1998 - An existence result for an inverse problem for a quasilinear parabolic equation [Articolo su rivista]
Gatti, Stefania
abstract

In this paper we are concerned with a quasilinear parabolic equation with nonhomogeneous Cauchy and Neumann conditions arising in combustion theory: by the Schauder fixed point theorem, we give a local existence result for the solution to an inverse problem on a semi-infinite strand.

1998 - Parabolic equation in theory of combustion: Direct and inverse problems [Articolo su rivista]
Gatti, Stefania
abstract

In questo lavoro, si riassumono i risultati ottenuti nella tesi di dottorato su un sistema parabolico semi-lineare in un dominio (cilindrico) non limitato con una condizione di Neumann nonlineare. Il modello proviene da studi sulla combustione nei propellenti solidi dei razzi. In letteratura era stata considerata solo l'approssimazione mono-dimensionale del modello: nella tesi, invece, si affrontano due problemi diretti sul modello 3-D e due problemi inversi su quello mono-dimensionale, arricchito dalla considerazione della presenza di reazioni chimiche sull'interfaccia solido-gas. Questi risultati sono contenuti in tre lavori pubblicati: i problemi diretti in una nota sui Rendiconti dell'Istituto Lombardo, Sezione A: Scienze Matematichee Applicazioni (1996) e i due problemi inversi su Inverse Problems (1998) e sul Journal of Inverse and Ill-posed problems.(1997)

1997 - A stability result for an inverse problem related to a quasilinear parabolic equation [Articolo su rivista]
Gatti, Stefania
abstract

We study the stability of an unknown nonlinear term in a quasilinear parabolic equation arising in combustion theory. Choosing suitable spaces for data and nonlinear terms, we can show that the mapping data → nonlinear term is Hölder continuous.