877 research outputs found
Well-posedness for the viscous shallow water equations in critical spaces
In this paper, we prove the existence and uniqueness of the solutions for the
2D viscous shallow water equations with low regularity assumptions on the
initial data as well as the initial height bounded away from zero.Comment: 32 page
Well-posedness of the Viscous Boussinesq System in Besov Spaces of Negative Order Near Index
This paper is concerned with well-posedness of the Boussinesq system. We
prove that the () dimensional Boussinesq system is well-psoed for
small initial data () either in
or in
if
, and , where
(, , )
is the logarithmically modified Besov space to the standard Besov space
. We also prove that this system is well-posed for small initial
data in
.Comment: 18 page
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
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
Laplacian Growth, Elliptic Growth, and Singularities of the Schwarz Potential
The Schwarz function has played an elegant role in understanding and in
generating new examples of exact solutions to the Laplacian growth (or "Hele-
Shaw") problem in the plane. The guiding principle in this connection is the
fact that "non-physical" singularities in the "oil domain" of the Schwarz
function are stationary, and the "physical" singularities obey simple dynamics.
We give an elementary proof that the same holds in any number of dimensions for
the Schwarz potential, introduced by D. Khavinson and H. S. Shapiro [17]
(1989). A generalization is also given for the so-called "elliptic growth"
problem by defining a generalized Schwarz potential. New exact solutions are
constructed, and we solve inverse problems of describing the driving
singularities of a given flow. We demonstrate, by example, how \mathbb{C}^n -
techniques can be used to locate the singularity set of the Schwarz potential.
One of our methods is to prolong available local extension theorems by
constructing "globalizing families". We make three conjectures in potential
theory relating to our investigation
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
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