Dynamics of screw dislocations: a generalised minimising-movements scheme approach

The gradient flow structure of the model introduced in [CG99] for the dynamics of screw dislocations is investigated by means of a generalised minimising-movements scheme approach. The assumption of a finite number of available glide directions, together with the "maximal dissipation criterion" that governs the equations of motion, results into solving a differential inclusion rather than an ODE. This paper addresses how the model in [CG99] is connected to a time-discrete evolution scheme which explicitly confines dislocations to move each time step along a single glide direction. It is proved that the time-continuous model in [CG99] is the limit of these time-discrete minimising-movement schemes when the time step converges to 0. The study presented here is a first step towards a generalisation of the setting in [AGS08, Chap. 2 and 3] that allows for dissipations which cannot be described by a metric.Comment: 17 pages, 2 figures http://cvgmt.sns.it/paper/2781

Equivalence of gradient flows and entropy solutions for singular nonlocal interaction equations in 1D

We prove the equivalence between the notion of Wasserstein gradient flow for a one-dimensional nonlocal transport PDE with attractive/repulsive Newtonian potential on one side, and the notion of entropy solution of a Burgers-type scalar conservation law on the other. The solution of the former is obtained by spatially differentiating the solution of the latter. The proof uses an intermediate step, namely the $L^2$ gradient flow of the pseudo-inverse distribution function of the gradient flow solution. We use this equivalence to provide a rigorous particle-system approximation to the Wasserstein gradient flow, avoiding the regularization effect due to the singularity in the repulsive kernel. The abstract particle method relies on the so-called wave-front-tracking algorithm for scalar conservation laws. Finally, we provide a characterization of the sub-differential of the functional involved in the Wasserstein gradient flow

Identification of a response amplitude operator for ships Citation for published version (APA): Identification of a Response Amplitude Operator for Ships

Abstract At the European Study Group Mathematics with Industry 2012 in Eindhoven, the Maritime Research Institute Netherlands (MARIN) presented the problem of identifying the response amplitude operator (RAO) for a ship, given input information on the amplitudes of the sea waves and output information on the movement of the ship. We approach the problem from a threefold perspective: a direct least-squares approach, an approach based on truncated Fourier series, and an approach using low-dimensional measures of the RAO. We give a few recommendations for possible further investigations

Equivalence of gradient flows and entropy solutions for singular nonlocal interaction equations in 1D

We prove the equivalence between the notion of Wasserstein gradient flow for a one-dimensional nonlocal transport PDE with attractive/repulsive Newtonian potential on one side, and the notion of entropy solution of a Burgers-type scalar conservation law on the other. The solution of the former is obtained by spatially differentiating the solution of the latter. The proof uses an intermediate step, namely the L2 gradient flow of the pseudo-inverse distribution function of the gradient flow solution. We use this equivalence to provide a rigorous particle-system approximation to the Wasserstein gradient flow, avoiding the regularization effect due to the singularity in the repulsive kernel. The abstract particle method relies on the so-called wave-front-tracking algorithm for scalar conservation laws. Finally, we provide a characterization of the subdifferential of the functional involved in the Wasserstein gradient flow