Dafermos and Rodnianski introduced an -weighted vector field method to obtain boundedness and decay of solutions to wave equations on a Lorentzian background, avoiding to use global vector field multipliers and commutators with weights in and thus proving to be more robust in dealing with black holes such as in Schwarzschild and Kerr background metrics. The approach has found numerous applications; see, for instance, Dafermos-Rodnianski, Moschidis, or Keir for many insightful discussions. In this blog post, I will give some basic details of this approach, based on lecture notes of S. Klainerman, focusing mostly on the flat Minkowski spacetime.

Posted in General Relativity, Uncategorized | Tagged r^p approach, Wave equations | Leave a Comment »

In his famous 1981 paper, Mourre gave a sufficient condition for a self-adjoint operator to assure the absence of its singular continuous spectrum. More precisely, consider a self-adjoint operator on a Hilbert space (e.g., with the usual norm), and assume that there is a self-adjoint operator , called a conjugate operator of on an interval , so that

for some positive constant and some compact operator on , where denotes the spectral projection of onto , the commutator , and the inequality is understood in the sense of self-adjoint operators. Then, Mourre’s main results are

- the point spectrum of in the interior of is finite.
- for any closed interval and any , the operator is bounded on uniformly as , where .

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.

Posted in fluid dynamics | Tagged conjugate operator method, enhanced viscous dissipation, Euler, inviscid damping, Navier-Stokes | Leave a Comment »

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 . 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 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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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.

Posted in New papers | Tagged Green function, nonlinear stability, source defects | Leave a Comment »

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.

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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 , which reads

on , with the relativistic velocity .

Posted in Kinetic theory, New papers, Plasma Physics | Tagged Non-relativistic limit, Stable Penrose, Vlasov-Maxwell, Vlasov-Poisson | Leave a Comment »