We consider the ARZ traffic flow model, and two particular solutions of it: a constant flow of cars, or a congestion modelled by a travelling wave. In both cases, we give a criteria to show the stability of these flows, both at the linear and non-linear level.
In this talk we will introduce the recent results about the non-linear stability of a shear layer profile for Navier-Stokes equations near a boundary. This question plays a major role in the study of the inviscid limit of Navier-Stokes equations in a bounded domain as the viscosity goes to 0. We mainly study the effect of cubic interactions on the growth of the linear instability here. In the case of the exponential profile and Blasius profile we obtain that the non-linearity tames the linear instability. We thus conjecture that small perturbations grow until they reach a magnitude O(ν1/4) only, forming small rolls in the critical layer near the boundary.
We consider the superquadratic diffusive Hamilton-Jacobi equation ut-Δu=|∇u|p with p > 2, in a smooth bounded domains of ℝn (n ≥ 2) under homogeneous Dirichlet conditions, which is known to undergo boundary gradient blow-up (BGBU) phenomena. First, for the elliptic problem in a half-space, we prove a Liouville-type classification, or symmetry result, which asserts that any solution has to be 1-dimensional. This turns out to be an efficient tool to study the BGBU behaviour for the parabolic problem. Namely, we show that in a neighbourhood of the boundary, at leading order, solutions display a global ODE type behaviour of the form uνν ~ -|uν|p, with domination of the normal derivatives upon the tangential derivatives. This leads to the existence of a universal, sharp blow-up profile in the normal direction at any BGBU point, and moreover implies that the behaviour in the tangential direction is more singular. In single-point BGBU cases, the tangential profile can be precisely determined under additional assumptions. Related results on the time rates will be also presented. The ODE type behaviour and its connection with the Liouville-type theorem can be considered as an analogue of the well-known results of Merle and Zaag (1998) for the subcritical semilinear heat equation, with the significant difference that for the latter, u itself blows up and the ODE behaviour is in the time direction (instead of the normal spatial direction).
Phase separation in a binary liquid (e.g. oil and vinegar) is a phenomenon which can be described as a competition between a entropy mixing effects and demixing effects due to the internal energy (i.e. the attraction of molecules of the same liquid), provided that, for instance, the temperature is low enough. Liquid-liquid phase separation has recently become a sort of new paradigm in Cell Biology. Quoting from E. Dolgin: "Not only is phase separation intuitive, but it seems to be everywhere. Droplets of proteins and RNAs are turning up in bacteria, fungi, plants and animals. Phase separation at the wrong place or time could create clogs or aggregate of molecules linked to neuro degenerative diseases, and poorly formed droplets could contribute to cancers and might help explain the ageing process." Well-known mathematical models for phase separation (e.g. in binary alloys) are given by the so-called Cahn-Hilliard equation or by the (conserved) Allen-Cahn equation. In the case of liquids, such equations must be suitably coupled with the Navier-Stokes equations for the averaged velocity of the binary mixture. This talk will be focused on Allen-Cahn-Navier-Stokes systems with some remarks on inviscid and pure transport cases.
This talk is devoted to studying homogenization error for non-stationary Stokes equations on perforated domains, which originally developed by J.-L. Lions. We now present a sharp error estimate in the sense of energy norms, where the main challenge is to control the boundary layers caused by the incompressibility condition. We start from a brief introduction to homogenization theory, and then move to the ideas of non-standard two-scale expansions. To obtain the optimal error, we introduce some refined regularity estimates for corrector without compatibility conditions between initial and boundary data, as well as, the well posedness of the effective equations in Bochner space. As a result, we further explain how we handle the boundary-layer correctors associated with Bogovskii's operator.
A classical problem that traces back to Helmholtz and Kirchhoff is the understanding of the dynamics of solutions to the Euler equations of an inviscid incompressible fluid when the vorticity of the solution is initially concentrated near isolated points in 2d or vortex lines in 3d. We discuss some recent results on the existence and asymptotic behavior of these solutions. We describe, with precise asymptotics, interacting vortices, and traveling helices. We rigorously establish the law of motion of 'leapfrogging vortex rings', originally conjectured by Helmholtz in 1858.
An increasing number of works combining neural networks and differential equations for spatio-temporal forecasting have been proposed for the last few years. Some of them show substantial improvements for the prediction of dynamical systems or videos compared to standard RNNs by defining the dynamics using learned ODEs. In this talk, we first present our recent approach for adapting such works for stochastic data. We introduce a novel dynamic model for stochastic video prediction which, unlike prior image-autoregressive models, decouples frame synthesis and dynamics. The dynamics of the model are governed in a latent space by a residual update rule, which is motivated by discretization schemes of differential equations. This endows our method with several desirable properties, such as temporal efficiency and latent space interpretability. Then, we will present a second method, more specifically focused on learning disentangled spatial and temporal representations of spatio-temporal phenomena, with the aim of more accurately predicting future tendencies from initial observations. We propose to model the evolution of partially observed spatiotemporal phenomena with unknown dynamics by taking inspiration from a formal method for the analytical resolution of PDEs: the functional separation of variables. We experimentally demonstrate the performance and broad applicability of our method against prior state-of-the-art models on physical and synthetic video datasets.
Surface waves determination from pressure measurements at the seabed leads to an ill-posed inversion problem. For two-dimensional steady waves in irrotational motion, using elementary complex analysis, the problem can be solved exactly in an analytic implicit form. For practical purposes, the implicit relations must be solved iteratively. We show that simple fixed-point iterations converge, even for extreme waves with angular crests.
You must be logged in to post a comment.