I moved to this NEW blog address, hosted by Penn State University, more secure and no ads. Sorry for the inconvenience.

I moved to this NEW blog address, hosted by Penn State University, more secure and no ads. Sorry for the inconvenience.

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In a recent joint work with Daniel Han-Kwan (CMLS, Ecole polytechnique) and Frédéric Rousset (Paris-Sud University), we give an alternative proof of the Landau damping for screened Vlasov-Poisson system near stable homogenous equilibria on the whole space, a result that was first established by Bedrossian, Masmoudi and Mouhot, for data with finite Sobolev regularity (they remarked that 36 derivatives were sufficient). In this work, via a Lagrangian approach, we prove the Landau damping for data with essentially Lipschitz regularity, yielding the time decay estimates that are essentially sharp, being the same as those for free transport, up to a logarithmic correction. Below, I shall briefly present the result and sketch the proof.

Continue Reading »Posted in Kinetic theory | Tagged Landau damping, Vlasov-Poisson | Leave a Comment »

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 »