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We analyze a system of nonlinear parabolic equations which describes the evolution of an order parameter f and the relative temperature v in a superconducting material occupying a bounded domain Ω ⊂ ℝn, n ≤ 3. Here the dependent variable f is subject to the homogeneous Neumann boundary condition, while v is equal to a time-dependent Dirichlet datum ũ on the boundary. Therefore, the corresponding dynamical system is nonautonomous. Our main goal is to analyze the asymptotic behavior of its solutions. We first show that the system has a bounded absorbing set and a global attractor which are uniform with respect to a sufficiently general class of ũ. Then, we give sufficient conditions on ũ which ensure the convergence of a given trajectory to a single stationary state and we estimate the convergence rate. Finally, we demonstrate the existence of an exponential attractor of finite fractal dimension for quasi-periodic or stabilizing boundary data ũ. [ABSTRACT FROM AUTHOR]

We consider a Ginzburg-Landau-Maxwell model which describes the behavior of a two-dimensional superconducting material. The state variables are the complex-valued order parameter ψ, the magnetic potential Ã, and the electric potential Φ. Under the choice of Coulomb (i.e., London) gauge, the resulting system is a parabolic-hyperbolic coupled system of nonlinear partial differential equations subject to suitable boundary and initial conditions. Global well-posedness results were proved in [M. Tsutsumi and H. Kasai, Nonlinear Anal., 37 (1999), pp. 187-216], while the existence of global attractor and exponential attractors was proved in [V. Berti and S. Gatti, Quart. Appl. Math., 64 (2006), pp. 617-639]. In this paper we use an extended Łojasiewicz--Simon approach to show that for any initial datum in certain phase space, the corresponding global solution converges to an equilibrium as time goes to infinity. Besides, we also provide an estimate on the convergence rate with respect to the phase space metric. [ABSTRACT FROM AUTHOR]

We study a phase-field system where the energy balance equation has the standard (parabolic) form, while the kinetic equation ruling the evolution of the order parameter χ is a nonlocal and nonlinear second-order ODE. The main features of the latter equation are a space convolution term which models long-range interactions of particles and a singular configuration potential that forces χ to take values in (-1, 1). We first prove the global existence and uniqueness of a regular solution to a suitable initial and boundary value problem associated with the system. Then, we investigate its long time behavior from the point of view of ω-limits. In particular, using a nonsmooth version of the Lojasiewicz-Simon inequality, we show that the ω-limit of any trajectory contains one and only one stationary solution, provided that the configuration potential in the kinetic equation is convex and analytic. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with the well-posedness and the asymptotic behavior of solutions to the following parabolic–hyperbolic phase-field system in , subject to the Neumann boundary condition for θ the dynamical boundary condition for χ and the initial conditions where Ω is a bounded domain in with smooth boundary Γ, ν is the outward normal direction to the boundary and ϕ is a real analytic function. In this paper we first establish the existence and uniqueness of a global strong solution to (0.1)–(0.4). Then, we prove its convergence to an equilibrium as time goes to infinity and we provide an estimate of the convergence rate. [Copyright &y& Elsevier]

This paper is concerned with the asymptotic behavior of global solutions to a parabolic–hyperbolic coupled system which describes the evolution of the relative temperature θ and the order parameter χ in a material subject to phase transitions. For the system with homogeneous Neumann boundary conditions for both ¸ and χ, under the assumption that the nonlinearities λ and ϕ are real analytic functions, we prove the convergence of a global solution to an equilibrium as time goes to infinity by means of a suitable Łojasiewicz–Simon type inequality. [ABSTRACT FROM AUTHOR]

We consider a phase‐field model based on hereditary constitutive equations for the internal energy and the heat flux and on the assumption that the spatial average of the order parameter χ is conserved. This model consists of a parabolic integrodifferential equation for the (relative) temperature ϑ coupled with a nonlinear fourth‐order evolution equation for χ. We first show that the obtained system is indeed a nonautonomous dynamical system, provided that the phase space accounts for the past history of ϑ and appropriate boundary conditions are given. Then we establish the existence of an absorbing set, uniformly with respect to a certain class of heat source terms. Finally, we prove that, under suitable assumptions, our dissipative dynamical system possesses a uniform attractor of finite Hausdorff and fractal dimensions. [ABSTRACT FROM AUTHOR]

We consider the nonlocal Cahn-Hilliard equation with singular potential and degenerate mobility in a bounded domain Ω ⊂ R d , d ≤ 3. We first prove the existence of maximal strong solutions in weighted (in time) L p spaces. Then we establish further regularity properties of the solution through maximal regularity theory. Finally, we revisit the separation property in an appendix. [ABSTRACT FROM AUTHOR]

We devise a first-order in time convex splitting scheme for a nonlocal Cahn–Hilliard–Oono type equation with a transport term and subject to homogeneous Neumann boundary conditions. However, we prove the stability of our scheme when the time step is sufficiently small, according to the velocity field and the interaction kernel. Furthermore, we prove the consistency of this scheme and the convergence to the exact solution. Finally, we give some numerical simulations which confirm our theoretical results and demonstrate the performance of our scheme not only for phase separation, but also for crystal nucleation, for several choices of the interaction kernel. [ABSTRACT FROM AUTHOR]

An introduction is presented in which the editor discusses various reports within the issue including scientist Mauro Fabrizio, mathematical models and thermodynamics.

We consider the nonlocal Cahn–Hilliard equation with singular potential and constant mobility. Well-posedness and regularity of weak solutions are studied. Then we establish the validity of the strict separation property in dimension two. Further regularity results as well as the existence of regular finite dimensional attractors and the convergence of a weak solution to a single equilibrium are also provided. Finally, regularity results and the strict separation property are also proven for the two-dimensional nonlocal Cahn–Hilliard–Navier–Stokes system with singular potential. [ABSTRACT FROM AUTHOR]

We consider a diffuse interface model which describes the motion of an incompressible isothermal mixture of two immiscible fluids. This model consists of the Navier-Stokes equations coupled with a convective nonlocal Cahn-Hilliard equation. Several results were already proven by two of the present authors. However, in the two-dimensional case, the uniqueness of weak solutions was still open. Here we establish such a result even in the case of degenerate mobility and singular potential. Moreover, we show the weak-strong uniqueness in the case of viscosity depending on the order parameter, provided that either the mobility is constant and the potential is regular or the mobility is degenerate and the potential is singular. In the case of constant viscosity, on account of the uniqueness results, we can deduce the connectedness of the global attractor whose existence was obtained in a previous paper. The uniqueness technique can be adapted to show the validity of a smoothing property for the difference of two trajectories which is crucial to establish the existence of an exponential attractor. The latter is established even in the case of variable viscosity, constant mobility and regular potential. [ABSTRACT FROM AUTHOR]

Phase-field crystal models are used to describe several pattern formation phenomena like crystallization of liquid, diffusion defects and glass formation. The prototypical equation is obtained as the conserved gradient flow associated with a free-energy functional of Swift–Hohenberg type. Here we consider a variant of the phase-field crystal equation proposed by P. Galenko et al. to account for fast dynamics. This version is characterized by the fact that the gradient flow depends on the past history of the particle density through a memory kernel k ε , ε > 0 being a relaxation time. Therefore the resulting nonlinear evolution equation is an integro-differential equation of sixth order which is equipped with the physically relevant periodic boundary conditions. We show that this problem generates a dissipative dynamical system on a suitable phase space. Moreover, we prove the existence of a robust family of exponential attractors which is Hölder continuous with respect to ε . [ABSTRACT FROM AUTHOR]

Abstract: A well-known diffuse interface model consists of the Navier–Stokes equations nonlinearly coupled with a convective Cahn–Hilliard type equation. This system describes the evolution of an incompressible isothermal mixture of binary fluids and it has been investigated by many authors. Here we consider a variant of this model where the standard Cahn–Hilliard equation is replaced by its nonlocal version. More precisely, the gradient term in the free energy functional is replaced by a spatial convolution operator acting on the order parameter φ, while the potential F may have any polynomial growth. Therefore the coupling with the Navier–Stokes equations is difficult to handle even in two spatial dimensions because of the lack of regularity of φ. We establish the global existence of a weak solution. In the two-dimensional case we also prove that such a solution satisfies the energy identity and a dissipative estimate, provided that F fulfills a suitable coercivity condition. [Copyright &y& Elsevier]

Abstract: We consider phase-field systems of Caginalp type on a three-dimensional bounded domain. The order parameter fulfills a dynamic boundary condition, while the (relative) temperature is subject to a homogeneous boundary condition of Dirichlet, Neumann or Robin type. Moreover, the two equations are nonlinearly coupled through a quadratic growth function. Here we extend several results which have been proven by some of the authors for the linear coupling. More precisely, we demonstrate the existence and uniqueness of global solutions. Then we analyze the associated dynamical system and we establish the existence of global as well as exponential attractors. We also discuss the convergence of given solutions to a single equilibrium. [Copyright &y& Elsevier]

Here we propose and analyze a mathematical model that aims to describe the marble sulphation process occurring in a given material. The model accounts for rugosity as well as for damaging effects. This model is characterized by some technical difficulties that seem hard to overcome from a theoretical viewpoint. Therefore, we introduce some physically reasonable modifications in order to establish the existence of a suitable notion of solution on a given time interval. Numerical simulations are presented and discussed, also in view of further research. [ABSTRACT FROM AUTHOR]