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In his famous 1981 paper, Mourre gave a sufficient condition for a self-adjoint operator {H} to assure the absence of its singular continuous spectrum. More precisely, consider a self-adjoint operator {H} on a Hilbert space {\mathcal{H}} (e.g., {L^2} with the usual norm), and assume that there is a self-adjoint operator {A}, called a conjugate operator of {H} on an interval {I\subset \mathbb{R}}, so that

\displaystyle P_I i[H,A] P_I \ge \theta_I P_I + P_I K P_I \ \ \ \ \ (1)

for some positive constant {\theta_I} and some compact operator {K} on {\mathcal{H}}, where {P_I} denotes the spectral projection of {H} onto {I}, the commutator {[H,A] = HA - AH}, and the inequality is understood in the sense of self-adjoint operators. Then, Mourre’s main results are

  • the point spectrum of {H} in the interior of {I} is finite.
  • for any closed interval {J \Subset I \cap \sigma_c(H)} and any {z\in J}, the operator {\langle A \rangle^{-1} (H - z - i\epsilon)^{-1} \langle A\rangle^{-1}} is bounded on {\mathcal{H}} uniformly as {\epsilon \rightarrow 0}, where {\langle A \rangle = \sqrt{1+|A|^2}}.

The Mourre’s theory is proven to be very useful in the study of spectral and scattering theory for Schrödinger operators and other dispersive PDEs. For instance, it yields the limiting absorbing principle, which in turn gives the Kato’s local smoothing estimate and the scattering RAGE’s theorem; for instance, see this blog post of T. Tao.

Below, I shall give a sketch of the proof of the Mourre’s beautiful theorem, then derive some local decay estimates on solutions to Schrödinger equations, and discuss some quick applications to linear damping in fluids.

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In 1904, Prandtl introduced his famous boundary layer theory in order to describe the behavior of solutions of incompressible Navier Stokes equations near a boundary as the viscosity goes to 0. His Ansatz was that the solution of Navier Stokes equations can be described as a solution of Euler equations, plus a boundary layer corrector, plus a vanishing error term in L^\infty in the inviscid limit. In this post, I briefly announce my recent work with E. Grenier (ENS Lyon) on the Prandtl’s boundary layer theory, where we prove

  • the Prandtl’s Ansatz is false for shear profiles that are unstable to Rayleigh equations;
  • the Prandtl’s asymptotic expansion is invalid for shear profiles that are monotone and stable to Rayleigh equations.

 

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This is part of my lecture notes on Kinetic Theory of Gases, taught at Penn State last semester, Fall 2017. In this part, I’d like to introduce this nice Bardos-Degond 1985’s global solutions to the Vlasov-Poisson system. Of course, the global smooth solutions are already constructed, without any restriction on size of initial data (e.g., Pfaffelmoser, Schaeffer ’91; see also the previous lecture), however they give no information on their asymptotic behavior at large time. Now, for initial data that are sufficiently small near zero, Bardos and Degond were able to construct global smooth solutions that decay in large time. To my knowledge, this was the first result where dispersion is rigorously shown for kinetic equations (they appear to be motivated by similar results for nonlinear wave equations where dispersion was (still is) the key to deduce the global behavior at the large time; e.g., Klainerman, Ponce, Shatah, among others, in the early 80s).

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Patterns are ubiquitous in nature, and understanding their formation and their dynamical behavior is always challenging and of great interest. Examples include patterns in fluids (e.g., Rayleigh-Benard convection between two flat plates, Taylor-Couette flow between rotating cylinders, surface waves in hydrothermal fluid flows,…), as well as in nonlinear optics, oscillatory chemical reactions and excitable biological media. Many of them arise from linear instabilities of an homogenous equilibrium, having space, time, or space-time periodic coherent structures such as wave trains (spatially periodic travelling waves). In presence of boundaries or defects, complex patterns form and thus break the symmetry or the periodic structures. Below, I shall briefly discuss some defect structures and my recent work on the subject.

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Today, I give a Graduate Student Seminar lecture whose goal is to introduce to the first and second year graduate students at Penn State a few topics of research in Fluid Dynamics. There are many recent exciting developments in the field, which I only have time to present a few (many students haven’t taken any PDE course!). You may find the slides of my lecture here (up to many details delivered on the board!). You may also enjoy my similar lecture on Kinetic Theory of Gases, also aiming at first and second year students.

In this note, I briefly explain my recent joint work with D. Han-Kwan (CNRS, Ecole polytechnique) and F. Rousset (Paris-Sud) on the non-relativistic limit of Vlassov-Maxwell. Precisely, we consider the relativistic Vlasov-Maxwell system, modeling the dynamics of electrons with electron density distribution {f(t,x,v)}, which reads

\displaystyle \partial_t f + \hat v \cdot \nabla_x f + (E + \epsilon \hat v \times B)\cdot \nabla_v f = 0

on {\mathbb{T}^3\times \mathbb{R}^3}, with the relativistic velocity {\hat v = v/\sqrt{1+ \epsilon^2 |v|^2}}.

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As an analogue of the previous post dealing with the classical particles, in this post, I shall formally discuss how similar models for quantum particles arrive. These particles behave like a wave and their dynamics is governed by the Schrödinger equation. We start the chapter with some basic quantum mechanics.

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