Giandomenico Orlandi

Vortex rings for the Gross-Pitaevskii equation

Abstract. We provide a mathematical proof of the existence of traveling vortex rings solutions to the Gross–Pitaevskii (GP) equation in dimension N ≥ 3. We also extend the asymptotic analysis of the free field Ginzburg–Landau equation to a larger class of equations, including the Ginzburg– Landau equation for superconductivity as well as the traveling wave equation for GP. In particular we rigorously derive a curvature equation for the concentration set (i.e. line vortices if N = 3).

Collisions and phase-vortex interactions in dissipative Ginzburg-Landau dynamics

In this paper we describe a natural framework for the vortex dynamics in the parabolic complex Ginzburg-Landau equation in R. This general setting does not rely on any assumption of well-preparedness and has the advantage to be valid even after collision times. We analyze carefully collisions leading to annihilation. A new phenomenon is identified, the phase-vortex interaction, related to persistence of low frequency oscillations, and leading to an unexpected drift in the motion of vortices.

APPROXIMATIONS WITH VORTICITY BOUNDS FOR THE GINZBURG–LANDAU FUNCTIONAL

We propose an approximation scheme for complex-valued functions defined on a smooth domain Ω: the approximating functions have a Ginzburg–Landau energy of the same magnitude as the initial function, but they possess moreover improved bounds on vorticity. As an application, we obtain a variant of a Jacobian estimate first established by Jerrard and Soner. This variant was conjectured by Bourgain, Brezis and Mironescu.

W1,p estimates for solutions to the Ginzburg–Landau equation with boundary data in H1/2

We consider complex-valued solutions ue of the Ginzburg–Landau on a smooth bounded simply connected domain Ω of RN, N⩾2 (here e is a parameter between 0 and 1). We assume that ue=ge on ∂Ω, where |ge|=1 and ge is uniformly bounded in H1/2(∂Ω). We also assume that the Ginzburg–Landau energy Ee(ue) is bounded by M0|loge|, where M0 is some given constant. We establish, for every 1⩽p<N/(N−1), uniform W1,p bounds for ue (independent of e). These types of estimates play a central role in the asymptotic analysis of ue as e→0.

Hamilton-Jacobi-Bellman Equation for a Time-Optimal Control Problem in the Space of Probability Measures

In this paper we formulate a time-optimal control problem in the space of probability measures endowed with the Wasserstein metric as a natural generalization of the correspondent classical problem in \({\mathbb {R}}^d\) where the controlled dynamics is given by a differential inclusion. The main motivation is to model situations in which we have only a probabilistic knowledge of the initial state. In particular we prove first a Dynamic Programming Principle and then we give an Hamilton-Jacobi-Bellman equation in the space of probability measures which is solved by a generalization of the minimum time function in a suitable viscosity sense.

Parabolic equations in time-dependent domains

We show existence and uniqueness results for nonlinear parabolic equations in noncylindrical domains with possible jumps in the time variable.

On the regularity of timelike extremal surfaces

We study a class of timelike weakly extremal surfaces in flat Minkowski space

Optimal mass transportation-based models for neuronal fibers

\mathbb R^{1+n}

Fiber bundles and regular approximation of codimension-one cycles.

, characterized by the fact that they admit a

Limiting models in condensed matter Physics and gradient flows of 1-homogeneous functional

C^1