Dispersive limit from the Kawahara to the KdV equation

We investigate the limit behavior of the solutions to the Kawahara equation $$u_t +u_{3x} + \varepsilon u_{5x} + u u_x =0,$$ as $0<\varepsilon \to 0$. In this equation, the terms $u_{3x}$ and $\varepsilon u_{5x}$ do compete together and do cancel each other at frequencies of order $1/\sqrt{\varepsilon}$. This prohibits the use of a standard dispersive approach for this problem. Nervertheless, by combining different dispersive approaches according to the range of spaces frequencies, we succeed in proving that the solutions to this equation converges in $C([0,T];H^1(\R))$ towards the solutions of the KdV equation for any fixed $T>0$.Comment: There was something incorrect in the section 3 of the first version. This version is correcte

On the Well-posedness of the Schr\"odinger-Korteweg-de Vries system

We prove that the Cauchy problem for the Schr\"odinger-Korteweg-de Vries system is locally well-posed for the initial data belonging to the Sovolev spaces $L^2(\R)\times H^{-{3/4}}(\R)$. The new ingredient is that we use the $\bar{F}^s$ type space, introduced by the first author in \cite{G}, to deal with the KdV part of the system and the coupling terms. In order to overcome the difficulty caused by the lack of scaling invariance, we prove uniform estimates for the multiplier. This result improves the previous one by Corcho and Linares.Comment: 16 page

Liouville-type theorems for the stationary MHD equations in 2D

This note is devoted to investigating Liouville type properties of the two dimensional stationary incompressible Magnetohydrodynamics equations. More precisely, under smallness conditions only on the magnetic field, we show that there are no non-trivial solutions to MHD equations either the Dirichlet integral or some $L^p$ norm of the velocity-magnetic fields are finite. In particular, these results generalize the corresponding Liouville type properties for the 2D Navier-Stokes equations, such as Gilbarg-Weinberger \cite{GW1978} and Koch-Nadirashvili-Seregin-Sverak \cite{KNSS}, to the MHD setting

Map Interface for Control of Smart Home Appliances

As homeowners increasingly adopt and install smart appliances and devices, differentiating such appliances and devices by names has become difficult. For example, a typical house may have tens of appliances such as lights, computers, televisions, audio units, game consoles, heaters, air-conditioners, etc. Attempting to control such appliances/devices by assigning names can become tedious and error-prone. This disclosure describes techniques that visually situate home appliances on an indoor map. To control an appliance, a user can quickly select an appliance by its location on the map

Energy Savings Using Virtual Assistants and Smart Home Appliances

This disclosure describes techniques to monitor and control smart home devices through virtual assistants. Use of the techniques can result in substantial energy savings