641 research outputs found
Tunnel effect for semiclassical random walk
We study a semiclassical random walk with respect to a probability measure
with a finite number n_0 of wells. We show that the associated operator has
exactly n_0 exponentially close to 1 eigenvalues (in the semiclassical sense),
and that the other are O(h) away from 1. We also give an asymptotic of these
small eigenvalues. The key ingredient in our approach is a general
factorization result of pseudodifferential operators, which allows us to use
recent results on the Witten Laplacian
Resolvent estimates for normally hyperbolic trapped sets
We give pole free strips and estimates for resolvents of semiclassical
operators which, on the level of the classical flow, have normally hyperbolic
smooth trapped sets of codimension two in phase space. Such trapped sets are
structurally stable and our motivation comes partly from considering the wave
equation for Kerr black holes and their perturbations, whose trapped sets have
precisely this structure. We give applications including local smoothing
effects with epsilon derivative loss for the Schr\"odinger propagator as well
as local energy decay results for the wave equation.Comment: Further changes to erratum correcting small problems with Section 3.5
and Lemma 4.1; this now also corrects hypotheses, explicitly requiring
trapped set to be symplectic. Erratum follows references in this versio
Conormal distributions in the Shubin calculus of pseudodifferential operators
We characterize the Schwartz kernels of pseudodifferential operators of
Shubin type by means of an FBI transform. Based on this we introduce as a
generalization a new class of tempered distributions called Shubin conormal
distributions. We study their transformation behavior, normal forms and
microlocal properties.Comment: 23 page
The influence of fractional diffusion in Fisher-KPP equations
We study the Fisher-KPP equation where the Laplacian is replaced by the
generator of a Feller semigroup with power decaying kernel, an important
example being the fractional Laplacian. In contrast with the case of the stan-
dard Laplacian where the stable state invades the unstable one at constant
speed, we prove that with fractional diffusion, generated for instance by a
stable L\'evy process, the front position is exponential in time. Our results
provide a mathe- matically rigorous justification of numerous heuristics about
this model
Derivation of the Zakharov equations
This paper continues the study of the validity of the Zakharov model
describing Langmuir turbulence. We give an existence theorem for a class of
singular quasilinear equations. This theorem is valid for well-prepared initial
data. We apply this result to the Euler-Maxwell equations describing
laser-plasma interactions, to obtain, in a high-frequency limit, an asymptotic
estimate that describes solutions of the Euler-Maxwell equations in terms of
WKB approximate solutions which leading terms are solutions of the Zakharov
equations. Because of transparency properties of the Euler-Maxwell equations,
this study is led in a supercritical (highly nonlinear) regime. In such a
regime, resonances between plasma waves, electromagnetric waves and acoustic
waves could create instabilities in small time. The key of this work is the
control of these resonances. The proof involves the techniques of geometric
optics of Joly, M\'etivier and Rauch, recent results of Lannes on norms of
pseudodifferential operators, and a semiclassical, paradifferential calculus
On the massive wave equation on slowly rotating Kerr-AdS spacetimes
The massive wave equation is
studied on a fixed Kerr-anti de Sitter background
. We first prove that in the Schwarzschild case
(a=0), remains uniformly bounded on the black hole exterior provided
that , i.e. the Breitenlohner-Freedman bound holds. Our proof
is based on vectorfield multipliers and commutators: The usual energy current
arising from the timelike Killing vector field (which fails to be
non-negative pointwise) is shown to be non-negative with the help of a Hardy
inequality after integration over a spacelike slice. In addition to , we
construct a vectorfield whose energy identity captures the redshift producing
good estimates close to the horizon. The argument is finally generalized to
slowly rotating Kerr-AdS backgrounds. This is achieved by replacing the Killing
vectorfield with for an
appropriate , which is also Killing and--in contrast to the
asymptotically flat case--everywhere causal on the black hole exterior. The
separability properties of the wave equation on Kerr-AdS are not used. As a
consequence, the theorem also applies to spacetimes sufficiently close to the
Kerr-AdS spacetime, as long as they admit a causal Killing field which is
null on the horizon.Comment: 1 figure; typos corrected, references added, introduction revised; to
appear in CM
Global well-posedness for a Smoluchowski equation coupled with Navier-Stokes equations in 2D
We prove global existence for a nonlinear Smoluchowski equation (a nonlinear
Fokker-Planck equation) coupled with Navier-Stokes equations in two dimensions.
The proof uses a deteriorating regularity estimate and the tensorial structure
of the main nonlinear terms
On the well-posedness for the Ideal MHD equations in the Triebel-Lizorkin spaces
In this paper, we prove the local well-posedness for the Ideal MHD equations
in the Triebel-Lizorkin spaces and obtain blow-up criterion of smooth
solutions. Specially, we fill a gap in a step of the proof of the local
well-posedness part for the incompressible Euler equation in \cite{Chae1}.Comment: 16page
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