Ricerc@Sapienza

In this paper we study a perturbative approach to the problem of quantization of probability distributions in the plane.

We introduce a gradient flow formulation of linear Boltzmann equations. Under a diffusive scaling we derive a diffusion equation by using the machinery of gradient flows.

I present some results obtained together with D. Benedetto and L. Bertini on a gradient flow formulation of linear kinetic equations, in terms of an entropy dissipation inequality. The setting includes the current as a dynamical variable. As an application I discuss the diffusive limit of linear Boltzmann equations and show that the rescaled entropy inequality asymptotically provides the corresponding inequality for heat equation.

Free energies with many small wiggles, arising from small scale micro-structural changes, appear often in phase transformations, protein folding and friction problems. In this paper we investigate gradient flows with energies E_epsilon given by the superposition of a convex functional and fast small
oscillations. We apply the time-discrete minimising movement scheme to capture the effect of the local minimizers ofE_epsilon in the limit equation as epsilon tends to zero.

Minimizing movements are investigated for an energy which is the superposition of a convex functional and fast small oscillations. Thus a minimizing movement scheme involves a temporal parameter τ and a spatial parameter ε, with τ describing the time step and the frequency of the oscillations being proportional to 1/ε. The extreme cases of fast time scales τ ≪ ε and slow time scales ε ≪ τ have been investigated in [4]. In this paper, the intermediate (critical) case of finite ratio ε/τ > 0 is studied.