Archivio della ricerca dell'Università di Modena e Reggio Emiliahttps://iris.unimore.itIl sistema di repository digitale IRIS acquisisce, archivia, indicizza, conserva e rende accessibili prodotti digitali della ricerca.Fri, 17 Sep 2021 13:37:28 GMT2021-09-17T13:37:28Z10481Convergence to stationary states of solutions to the semilinear equation of viscoelasticityhttp://hdl.handle.net/11380/461846Titolo: Convergence to stationary states of solutions to the semilinear equation of viscoelasticity
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.
Sun, 01 Jan 2006 00:00:00 GMThttp://hdl.handle.net/11380/4618462006-01-01T00:00:00ZAsymptotic behavior of a phase-field system with dynamic boundary conditionshttp://hdl.handle.net/11380/461845Titolo: Asymptotic behavior of a phase-field system with dynamic boundary conditions
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.
Sun, 01 Jan 2006 00:00:00 GMThttp://hdl.handle.net/11380/4618452006-01-01T00:00:00ZMemory relaxation of the one-dimensional Cahn-Hilliard equationhttp://hdl.handle.net/11380/461847Titolo: Memory relaxation of the one-dimensional Cahn-Hilliard equation
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.
Sun, 01 Jan 2006 00:00:00 GMThttp://hdl.handle.net/11380/4618472006-01-01T00:00:00ZAutomatic control of the temperature in phase change problems with memoryhttp://hdl.handle.net/11380/453681Titolo: Automatic control of the temperature in phase change problems with memory
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.
Mon, 01 Jan 2001 00:00:00 GMThttp://hdl.handle.net/11380/4536812001-01-01T00:00:00ZParabolic equation in theory of combustion: Direct and inverse problemshttp://hdl.handle.net/11380/453682Titolo: Parabolic equation in theory of combustion: Direct and inverse problems
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)
Thu, 01 Jan 1998 00:00:00 GMThttp://hdl.handle.net/11380/4536821998-01-01T00:00:00ZA generalization of the Caginalp phase-field system with Neumann boundary conditionshttp://hdl.handle.net/11380/935889Titolo: A generalization of the Caginalp phase-field system with Neumann boundary conditions
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.
Tue, 01 Jan 2013 00:00:00 GMThttp://hdl.handle.net/11380/9358892013-01-01T00:00:00ZA singular Cahn--Hilliard--Oono phase-field system with hereditary memoryhttp://hdl.handle.net/11380/1160566Titolo: A singular Cahn--Hilliard--Oono phase-field system with hereditary memory
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.
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/11380/11605662018-01-01T00:00:00ZPullback exponential attractor for a Cahn-Hilliard-Navier-Stokes system in 2Dhttp://hdl.handle.net/11380/1007513Titolo: Pullback exponential attractor for a Cahn-Hilliard-Navier-Stokes system in 2D
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.
Wed, 01 Jan 2014 00:00:00 GMThttp://hdl.handle.net/11380/10075132014-01-01T00:00:00ZA phase-field system with two temperatures and memoryhttp://hdl.handle.net/11380/1136670Titolo: A phase-field system with two temperatures and memory
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/11380/11366702017-01-01T00:00:00ZAsymptotic behavior of the Caginalp phase-field system with coupled dynamic boundary conditionshttp://hdl.handle.net/11380/668845Titolo: Asymptotic behavior of the Caginalp phase-field system with coupled dynamic boundary conditions
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.
Sun, 01 Jan 2012 00:00:00 GMThttp://hdl.handle.net/11380/6688452012-01-01T00:00:00Z