On the Yudovich solutions for the ideal MHD equations

In this paper, we address the problem of weak solutions of Yudovich type for the inviscid MHD equations in two dimensions. The local-in-time existence and uniqueness of these solutions sound to be hard to achieve due to some terms involving Riesz transforms in the vorticity-current formulation. We shall prove that the vortex patches with smooth boundary offer a suitable class of initial data for which the problem can be solved. However this is only done under a geometric constraint by assuming the boundary of the initial vorticity to be frozen in a magnetic field line. We shall also discuss the stationary patches for the incompressible Euler system $(E)$ and the MHD system. For example, we prove that a stationary simply connected patch with rectifiable boundary for the system $(E)$ is necessarily the characteristic function of a disc.Comment: 40 page

On the trivial solutions for the rotating patch model

In this paper we study the clockwise simply connected rotating patches for Euler equations. By using the moving plane method we prove that Rankine vortices are the only solutions to this problem in the class of {\it slightly} convex domains. We discuss in the second part of the paper the case where the angular velocity $\Omega=\frac12$ and we show without any geometric condition that the set of the V-states is trivial and reduced to the Rankine vortices.Comment: 14 page

The impact of stock spams on volatility

This paper is dedicated to study the impact of stock spams through the analysis of the variations of volatility. We use the methodology of event studies on a sample of hundred ten firms. The results show positive and significant changes in volatility during 12 days of the event window; a widening of the variation [lowest price - highest price] was noticed following the consignment of messages by the spammers. The sending of stock spams affected the behaviour of investors, indicating thus that the spamming activity is a lucrative business.Stock spam, event studies, volatility, penny stock

On the inviscid Boussinesq system with rough initial data

We deal with the local well-posedness theory for the two-dimensional inviscid Boussinesq system with rough initial data of Yudovich type. The problem is in some sense critical due to some terms involving Riesz transforms in the vorticity-density formulation. We give a positive answer for a special sub-class of Yudovich data including smooth and singular vortex patches. For the latter case we assume in addition that the initial density is constant around the singular part of the patch boundary.Comment: 26 page