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$$u\_t-D\_x^\alpha u\_{x} + u\_{xyy} = uu\_x,\quad (t,x,y)\in\R^3,\quad 1\le \alpha\le 2,$$is locally well-posed in the spaces $E^s$, s\textgreater{}\frac 2\alpha-\frac 34, endowed with the norm$\|f\|\_{E^s} = \|\langle |\xi|^\alpha+\mu^2\rangle^s\hat{f}\|\_{L^2(\R^2)}.$As a consequence, we get the global well-posedness in the energy space $E^{1/2}$ as soon as \alpha\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.Comment: arXiv admin note: text overlap with arXiv:1205.0169 by other author

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

We prove the local well-posedness of the three-dimensional Zakharov-Kuznetsov equation $\partial_tu+\Delta\partial_xu+ u\partial_xu=0$ 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

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

In this note we study the generalized 2D Zakharov-Kuznetsov equations $\partial_tu+\Delta\partial_xu+u^k\partial_xu=0$ 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 $k\ge 4$

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

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 syste

WELL-POSEDNESS IN H(1) FOR GENERALIZED BENJAMIN-ONO EQUATIONS ON THE CIRCLE

International audienceWe prove the local-well posedness of the generalized Benjamin-Ono equations in H(1)(T)

On some wave equations in dispersive or dispersive-dissipative media

Dans cette thèse nous nous intéressons aux propriétés qualitatives et quantitatives des solutions de quelques équations d'ondes en milieux dispersifs ou dispersifs-dissipatifs. Dans une première partie, nous étudions le problème de Cauchy associé aux équations de Benjamin-Ono généralisées. A l'aide de transformées de jauge, combinées avec des outils d'analyse harmonique, nous prouvons des résultats concernant le caractère localement bien posé pour des données initiales de régularité minimale dans l'échelle des espaces de Sobolev. Dans une seconde partie, nous étudions le problème de Cauchy pour des versions dissipatives des équations de Benjamin-Ono et de Korteweg-de Vries. Nous mettons en évidence l'influence des effets dissipatifs sur ces équations en donnant des résultats optimaux sur leur caractère bien ou mal posé. Ceux-ci sont obtenus en travaillant dans des espaces de type Bourgain adaptés à la partie dispersive-dissipative. Pour finir nous étudions le comportement asymptotique des solutions des équations de KdV dissipatives, lorsque celles-ci existent pour tout temps, en calculant explicitement les premiers termes du développement asymptotique dans de nombreux espaces de SobolevThis thesis deals with the qualitative and quantitative properties of solutions to some wave equations in dispersive or dispersive-dissipative media. In the first part, we study the Cauchy problem for the generalized Benjamin-Ono equations. By means of gauge transforms combined with some harmonic analysis tools, we prove some local well-posedness results for initial data with minimal regularity in Sobolev spaces. In the second part, we study the Cauchy problem for some dissipative versions of the Benjamin-Ono and Korteweg-de Vries equations. We show the influence of the dissipative effects and prove sharp well and ill-posedness results. This is obtained by working in suitable Bourgain's spaces, adapted to the dispersive-dissipative part of the equation. Finally, we study the asymptotic behavior of solutions to the dissipative KdV equations. We explicitly compute the first terms of the asymptotic expansion in Sobolev spacesPARIS-EST-Université (770839901) / SudocSudocFranceF