Page
%P
-
Article
Global well-posedness for the derivative nonlinear Schrödinger equation
This paper is dedicated to the study of the derivative nonlinear Schrödinger equation on the real line. The local well-posedness of this equation in the Sobolev spaces
-
Article
Strichartz Estimates and Fourier Restriction Theorems on the Heisenberg Group
This paper is dedicated to the proof of Strichartz estimates on the Heisenberg group \({\mathop {\mathbb H}\nolimits }^d\) ...
-
Chapter and Conference Paper
Logarithmic Littlewood-Paley Decomposition and Applications to Orlicz Spaces
This paper is devoted to the construction of a logarithmic Littlewood-Paley decomposition. The approach we adopted to carry out this construction is based on the notion introduced in (Bahouri, Trends Math pp 1...
-
Article
On the Stability in Weak Topology of the Set of Global Solutions to the Navier–Stokes Equations
Let X be a suitable function space and let \({\mathcal{G} \subset X}\) be the set of divergence free vector fields ge...
-
Chapter and Conference Paper
On the Elements Involved in the Lack of Compactness in Critical Sobolev Embedding
The goal of this paper is to emphasize ideas coming from microlocal analysis to refine the study of lack of compactness in critical Sobolev embedding. In particular, we shall highlight that the elements involv...
-
Article
Refined Sobolev Inequalities in Lorentz Spaces
We establish refined Sobolev inequalities between the Lorentz spaces and homogeneous Besov spaces. The sharpness of these inequalities is illustrated on several examples, in particular based on non-uniformly o...
-
Book
-
Chapter
Littlewood–Paley Theory
In Chapter 2 we give a detailed presentation on Littlewood-Paley decomposition and define homogeneous and nonhomogeneous Besov spaces. We should emphasize that we have replaced the usual definition of homogene...
-
Chapter
Quasilinear Symmetric Systems
Chapter 4 is devoted to solving linear and quasilinear symmetric systems with data in Sobolev spaces. Blow-up criteria and results concerning the continuity of the flow map are also given. The case of data wit...
-
Chapter
Anisotropic Viscosity
In order to emphasize the robustness of the tools that have been introduced hitherto in this book, we present in Chapter 6 a nonlinear system of partial differential equations with degenerate parabolicity. In ...
-
Chapter
Strichartz Estimates and Applications to Semilinear Dispersive Equations
Chapter 8 is devoted to Strichartz estimates for dispersive equations with a focus on Schrödinger and wave equations. After proving a dispersive inequality (i.e., decay in time of the L ...
-
Chapter
Basic Analysis
Chapter 1 is devoted to a self-contained elementary presentation of classical Fourier analysis results. Even though none of the results are new, some of the proofs that we present are not the standard ones and...
-
Chapter
Transport and Transport-Diffusion Equations
In Chapter 3, we give a very complete theory of strong solutions for transport and transport-diffusion equations. In particular, we provide a priori estimates which are the key to solving nonlinear systems com...
-
Chapter
The Incompressible Navier–Stokes System
In Chapter 5 we take advantage of the tools introduced in the previous chapters to establish most of the classical results concerning the well-posedness of the incompressible Navier–Stokes system for data with...
-
Chapter
Euler System for Perfect Incompressible Fluids
Chapter 7 is the natural continuation of the previous chapter: The diffusion term is removed, leading to the study of the Euler system for inviscid incompressible fluids. Here, we state local (in dimension d≥3) a...
-
Chapter
The Compressible Navier–Stokes System
In Chapter 10 we present a more complicated system of partial differential equations coming from fluid mechanics, the so-called barotropic compressible Navier–Stokes equations. Those equations are of mixed hyperb...
-
Chapter
Smoothing Effect in Quasilinear Wave Equations
Chapter 9 is devoted to the study of a class of quasilinear wave equations which can be seen as a toy model for the Einstein equations. First, by taking advantage of energy methods in the spirit of those of Ch...
-
Chapter
The Heat Kernel and Frequency Localized Functions on the Heisenberg Group
The goal of this paper is to study the action of the heat operator on the Heisenberg group H d , and in particular to characterize Besov spaces of negative index on H ...
-
Chapter
Trace theorem on the Heisenberg group on homogeneous hypersurfaces
We prove in this work the trace and trace lifting theorem for Sobolev spaces on the Heisenberg groups for homogeneous hypersurfaces.
-
Article
Espaces de Besov et estimations de Strichartz généralisées sur le groupe de Heisenberg
In this paper, we prove dispersive and Strichartz inequalities on the Heisenberg group. The proof involves the analysis of Besov-type spaces on the Heisenberg group.
