Vittorino Pata

Lévy type characterization of stable laws for free random variables

We give a description of stable probability measures relative to free additive convolution. The definition of domain of attraction is given, and a proof is provided of the noncommutative analogue of Levy Theorem.

Reaction-diffusion with memory in the minimal state framework

We consider the integrodifferential equation ∂tu−Δu− ∫ ∞ 0 κ(s)Δu(t− s) ds+ φ(u) = f arising in the Coleman-Gurtin theory of heat conduction with hereditary memory. Within a novel abstract framework, based on the notion of minimal state, we prove the existence of global and exponential attractors of optimal regularity and finite fractal dimension for the related semigroup of solutions.

On the time-dependent Cattaneo law in space dimension one

We consider the one-dimensional wave equation e u tt - u xx + 1 + e f ( u ) ] u t + f ( u ) = h where e = e ( t ) is a decreasing function vanishing at infinity, providing a model for heat conduction of Cattaneo type with thermal resistance decreasing in time. Within the theory of processes on time-dependent spaces, we prove the existence of an invariant time-dependent attractor, which converges in a suitable sense to the attractor of the classical Fourier equation u t - u xx + f ( u ) = h formally arising in the limit t ? ∞ .

Navier–Stokes–Voigt Equations with Memory in 3D Lacking Instantaneous Kinematic Viscosity

We consider a Navier–Stokes–Voigt fluid model where the instantaneous kinematic viscosity has been completely replaced by a memory term incorporating hereditary effects, in presence of Ekman damping. Unlike the classical Navier–Stokes–Voigt system, the energy balance involves the spatial gradient of the past history of the velocity rather than providing an instantaneous control on the high modes. In spite of this difficulty, we show that our system is dissipative in the dynamical systems sense and even possesses regular global and exponential attractors of finite fractal dimension. Such features of asymptotic well-posedness in absence of instantaneous high modes dissipation appear to be unique within the realm of dynamical systems arising from fluid models.

A regularity criterion for the Navier-Stokes equations in terms of the pressure gradient

The incompressible three-dimensional Navier-Stokes equations are considered. A new regularity criterion for weak solutions is established in terms of the pressure gradient.