Stefania Gatti

HYPERBOLIC RELAXATION OF THE VISCOUS CAHN–HILLIARD EQUATION IN 3-D

We consider a modified version of the viscous Cahn–Hilliard equation governing the relative concentration u of one component of a binary system. This equation is characterized by the presence of the additional inertial term ωutt 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 eω,δ, which is a robust perturbation of an exponential attractor of the Cahn–Hilliard equation, namely the symmetric Hausdorff distance between eω,δ and e0, 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.

A construction of a robust family of exponential attractors

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

Memory relaxation of first order evolution equations

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, the formal limit of which, as the scaling parameter (or relaxation time) e 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 e 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 e → 0. The abstract result is then applied to Allen–Cahn and Cahn–Hilliard type equations.

A ONE-DIMENSIONAL WAVE EQUATION WITH NONLINEAR DAMPING

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.

Continuous families of exponential attractors for singularly perturbed equations with memory

For a family of semigroups S-epsilon(t): H-epsilon -> H-epsilon depending on a perturbation parameter epsilon is an element of [0,1], where the perturbation is allowed to become singular at epsilon = 0, we establish a general theorem on the existence of exponential attractors epsilon(epsilon) satisfying a suitable Holder continuity property with respect to the symmetric Hausdorff distance at every epsilon is an element of [0,1]. The result is applied to the abstract evolution equations with memorypartial derivative(t)x(t) + integral(infinity)(0) k(epsilon)(s)B-0(x(t - s))ds + B1(x(t)) = 0, epsilon is an element of (0, 1],where k(epsilon)(s) = (1/epsilon)k(s/epsilon) is the resulting of a convex summable kernel k with unit mass. Such a family can be viewed as a memory perturbation of the equationpartial derivative(t)x(t) + B-0(x(t)) + B-1(x(t)) = 0,formally obtained in the singular limit epsilon -> 0.

Uniform decay properties of linear Volterra integro-differential equations

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 describing the motion of linearly viscoelastic solids. We provide sufficient conditions for the decay to hold, without invoking differential inequalities involving the convolution kernel μ.

An existence result for an inverse problem for a quasilinear parabolic equation

In this paper we are concerned with a quasilinear parabolic equation with nonhomogenous 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.

Trajectory and global attractors for evolution equations with memory

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

ATTRACTORS FOR A CAGINALP MODEL WITH A LOGARITHMIC POTENTIAL AND COUPLED DYNAMIC BOUNDARY CONDITIONS

We study the longtime behavior of the Caginalp phase-field model with a logarithmic potential and dynamic boundary conditions for both 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.

Decay rates of Volterra equations on ℝN

AbstractThis note is concerned with the linear Volterra equation of hyperbolic type