632 research outputs found
Low regularity solutions of two fifth-order KdV type equations
The Kawahara and modified Kawahara equations are fifth-order KdV type
equations and have been derived to model many physical phenomena such as
gravity-capillary waves and magneto-sound propagation in plasmas. This paper
establishes the local well-posedness of the initial-value problem for Kawahara
equation in with and the local well-posedness
for the modified Kawahara equation in with .
To prove these results, we derive a fundamental estimate on dyadic blocks for
the Kawahara equation through the multiplier norm method of Tao
\cite{Tao2001} and use this to obtain new bilinear and trilinear estimates in
suitable Bourgain spaces.Comment: 17page
Analycity and smoothing effect for the coupled system of equations of Korteweg - de Vries type with a single point singularity
We study that a solution of the initial value problem associated for the
coupled system of equations of Korteweg - de Vries type which appears as a
model to describe the strong interaction of weakly nonlinear long waves, has
analyticity in time and smoothing effect up to real analyticity if the initial
data only has a single point singularity at $x=0.
Convergence Rates in L^2 for Elliptic Homogenization Problems
We study rates of convergence of solutions in L^2 and H^{1/2} for a family of
elliptic systems {L_\epsilon} with rapidly oscillating oscillating coefficients
in Lipschitz domains with Dirichlet or Neumann boundary conditions. As a
consequence, we obtain convergence rates for Dirichlet, Neumann, and Steklov
eigenvalues of {L_\epsilon}. Most of our results, which rely on the recently
established uniform estimates for the L^2 Dirichlet and Neumann problems in
\cite{12,13}, are new even for smooth domains.Comment: 25 page
Boundary regularity for the Poisson equation in reifenberg-flat domains
This paper is devoted to the investigation of the boundary regularity for the
Poisson equation {{cc} -\Delta u = f & \text{in} \Omega u= 0 & \text{on}
\partial \Omega where belongs to some and is a
Reifenberg-flat domain of More precisely, we prove that given an
exponent , there exists an such that the
solution to the previous system is locally H\"older continuous provided
that is -Reifenberg-flat. The proof is based on
Alt-Caffarelli-Friedman's monotonicity formula and Morrey-Campanato theorem
Global well-posedness of the KP-I initial-value problem in the energy space
We prove that the KP-I initial value problem is globally well-posed in the
natural energy space of the equation
An alternative approach to regularity for the Navier-Stokes equations in critical spaces
In this paper we present an alternative viewpoint on recent studies of
regularity of solutions to the Navier-Stokes equations in critical spaces. In
particular, we prove that mild solutions which remain bounded in the space
do not become singular in finite time, a result which was proved
in a more general setting by L. Escauriaza, G. Seregin and V. Sverak using a
different approach. We use the method of "concentration-compactness" +
"rigidity theorem" which was recently developed by C. Kenig and F. Merle to
treat critical dispersive equations. To the authors' knowledge, this is the
first instance in which this method has been applied to a parabolic equation.
We remark that we have restricted our attention to a special case due only to
a technical restriction, and plan to return to the general case (the
setting) in a future publication.Comment: 41 page
Homoclinic orbits and chaos in a pair of parametrically-driven coupled nonlinear resonators
We study the dynamics of a pair of parametrically-driven coupled nonlinear
mechanical resonators of the kind that is typically encountered in applications
involving microelectromechanical and nanoelectromechanical systems (MEMS &
NEMS). We take advantage of the weak damping that characterizes these systems
to perform a multiple-scales analysis and obtain amplitude equations,
describing the slow dynamics of the system. This picture allows us to expose
the existence of homoclinic orbits in the dynamics of the integrable part of
the slow equations of motion. Using a version of the high-dimensional Melnikov
approach, developed by Kovacic and Wiggins [Physica D, 57, 185 (1992)], we are
able to obtain explicit parameter values for which these orbits persist in the
full system, consisting of both Hamiltonian and non-Hamiltonian perturbations,
to form so-called Shilnikov orbits, indicating a loss of integrability and the
existence of chaos. Our analytical calculations of Shilnikov orbits are
confirmed numerically
Null sets of harmonic measure on NTA domains: Lipschitz approximation revisited
We show the David-Jerison construction of big pieces of Lipschitz graphs
inside a corkscrew domain does not require its surface measure be upper Ahlfors
regular. Thus we can study absolute continuity of harmonic measure and surface
measure on NTA domains of locally finite perimeter using Lipschitz
approximations. A partial analogue of the F. and M. Riesz Theorem for simply
connected planar domains is obtained for NTA domains in space. As a consequence
every Wolff snowflake has infinite surface measure.Comment: 22 pages, 6 figure
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