Energetic methods for capturing sharp convergence- and metastable-relaxation-rates of gradient flows.
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Scattering via Morawetz estimates for the non-radial inhomogeneous nonlinear Schr ̈odinger equation.
Abstract: The concentration-compactness-rigidity method, pioneered by Kenig and Merle, has become standard in the study of global well-posedness and scattering in the context of dispersive and wave equations. Albeit powerful, it requires building some heavy machinery in order to obtain the desired space-time bounds. In this talk, we present a simpler method, based on Tao’s scattering criterion and on Dodson-Murphy’s Virial/Morawetz inequalities, first proved for the 3d cubic nonlinear Schr ̈odinger (NLS) equation. Tao’s criterion is, in some sense, universal, and it is expected to work in...Read More
Abstract: Wave energy converters (WECs) are devices that convert the energy associated with a moving ocean wave into electrical energy. In this talk we present a mathematical model of a particular wave energy converter, the so-called oscillating water columns in shallow water regime. This model can be reformulated as two transmission problems: one is related to the wave motion over the stepped topography and the other one is related to the wave-structure interaction where a fixed partially immersed structure is installed. We analyze the evolution of the contact line between the surface of...Read More
Abstract: We analyse a nonconvex variational model from micromagnetics with a nonlocal energy functional, depending on a small parameter epsilon > 0. The model gives rise to transition layers, called Néel walls, and we study their behaviour in the limit epsilon -> 0. The analysis has some similarity to the theory of Ginzburg-Landau vortices. In particular, it gives rise to a renormalised energy that determines the interaction (attraction or repulsion) between Néel walls to leading order. But while Ginzburg-Landau vortices show attraction for winding numbers of the same sign and...Read More
Abstract: Consider a real scalar wave equation in dimension 1+1 with a positive external potential having non-degenerate isolated zeros. I will speak about the problem of construction of weakly interacting pure multi-solitons, that is solutions converging exponentially in time to a superposition of Lorentz-transformed solitons (“kinks”), in the case of distinct velocities. In a joint work with Gong Chen from the University of Toronto, we prove that these solutions form a 2K-dimensional smooth manifold in the space of solutions, where K is the number of the kinks. This manifold...Read More
Abstract: In this work we consider the numerical approximation of nonlocal integro differential parabolic equations via neural networks. These equations appear in many recent applications, including finance, biology and others, and have been recently studied in great generality starting from the work of Caffarelli and Silvestre. Based in the work by Hure, Pham and Warin, we generalize their Euler scheme and consistency result for Backward Forward Stochastic Differential Equations to the nonlocal case. We rely on Lévy processes and a new neural network approximation of thenonlocal part to...Read More