140 research outputs found
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 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
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 . The new ingredient is that we use the
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
Dispersive limit from the Kawahara to the KdV equation
We investigate the limit behavior of the solutions to the Kawahara equation
as .
In this equation, the terms and do compete
together and do cancel each other at frequencies of order . 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 towards
the solutions of the KdV equation for any fixed .Comment: There was something incorrect in the section 3 of the first version.
This version is correcte
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