arXiv subjects

Francis Ribaud

Publications and source records attributed to Francis Ribaud.

6 recordsLinked to original sources

arXiv2016

Local and global well-posedness results for the Benjamin-Ono-Zakharov-Kuznetsov equation

We show that the initial value problem associated to the dispersive generalized Benjamin-Ono-Zakharov-Kuznetsov equation$$ u\_t-D\_x^αu\_{x} + u\_{xyy} = uu\_x,\quad (t,x,y)\in\R^3,\quad 1\le α\le 2,$$is locally well-posed in the spaces $E^s$, $s\textgreater{}\frac 2α-\frac 34$, endowed with the norm$\|f\|\_{E^s} = \|\langle |ξ|^α+μ^2\rangle^s\hat{f}\|\_{L^2(\R^2)}.$As a consequence, we get the global well-posedness in the energy space $E^{1/2}$ as soon as $α\textgreater{}\frac 85$. The proof is based on the approach of the short time Bourgain spaces developed by Ionescu, Kenig and Tataru \cite{IKT} combined with new Strichartz estimates and a modified energy.

math.AP↗

arXiv2011

A note on the Cauchy problem for the 2D generalized Zakharov-Kuznetsov equations

In this note we study the generalized 2D Zakharov-Kuznetsov equations $\partial_tu+Δ\partial_xu+u^k\partial_xu=0$ for $k\ge 2$. By an iterative method we prove the local well-posedness of these equations in the Sobolev spaces $H^s(\mathbb{R}^2)$ for $s>1/4$ if $k=2$, $s>5/12$ if $k=3$ and $s>1-2/k$ if $k\ge 4$.

math.AP↗

arXiv2011

Well-posedness results for the 3D Zakharov-Kuznetsov equation

We prove the local well-posedness of the three-dimensional Zakharov-Kuznetsov equation $\partial_tu+Δ\partial_xu+ u\partial_xu=0$ in the Sobolev spaces $H^s(\R^3)$, $s>1$, as well as in the Besov space $B^{1,1}_2(\R^3)$. The proof is based on a sharp maximal function estimate in time-weighted spaces.

math.AP↗

arXiv2010

The Logarithmic Sobolev Constant of The Lamplighter

We give estimates on the logarithmic Sobolev constant of some finite lamplighter graphs in terms of the spectral gap of the underlying base. Also, we give examples of application.

math.PR↗

arXiv2009

Global existence for a system of non-linear and non-local transport equations describing the dynamics of dislocation densities

In this paper, we study the global in time existence problem for the Groma-Balogh model describing the dynamics of dislocation densities. This model is a two-dimensional model where the dislocation densities satisfy a system of transport equations such that the velocity vector field is the shear stress in the material, solving the equations of elasticity. This shear stress can be expressed as some Riesz transform of the dislocation densities. The main tool in the proof of this result is the existence of an entropy for this system

math-ph↗

arXiv2006

Well-posedness in H^1 for the (generalized) Benjamin-Ono equation on the circle

We prove the local well posedness of the Benjamin-Ono equation and the generalized Benjamin-Ono equation in $ H^1(\T) $. This leads to a global well-posedness result in $ H^1(\T)$ for the Benjamin-Ono equation.

math.AP↗