700 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
Well-posedness and stability results for the Gardner equation
In this article we present local well-posedness results in the classical
Sobolev space H^s(R) with s > 1/4 for the Cauchy problem of the Gardner
equation, overcoming the problem of the loss of the scaling property of this
equation. We also cover the energy space H^1(R) where global well-posedness
follows from the conservation laws of the system. Moreover, we construct
solitons of the Gardner equation explicitly and prove that, under certain
conditions, this family is orbitally stable in the energy space.Comment: 1 figure. Accepted for publication in Nonlin.Diff Eq.and App
Frequency Precision of Oscillators Based on High-Q Resonators
We present a method for analyzing the phase noise of oscillators based on
feedback driven high quality factor resonators. Our approach is to derive the
phase drift of the oscillator by projecting the stochastic oscillator dynamics
onto a slow time scale corresponding physically to the long relaxation time of
the resonator. We derive general expressions for the phase drift generated by
noise sources in the electronic feedback loop of the oscillator. These are
mixed with the signal through the nonlinear amplifier, which makes them
{cyclostationary}. We also consider noise sources acting directly on the
resonator. The expressions allow us to investigate reducing the oscillator
phase noise thereby improving the frequency precision using resonator
nonlinearity by tuning to special operating points. We illustrate the approach
giving explicit results for a phenomenological amplifier model. We also propose
a scheme for measuring the slow feedback noise generated by the feedback
components in an open-loop driven configuration in experiment or using circuit
simulators, which enables the calculation of the closed-loop oscillator phase
noise in practical systems
Weighted maximal regularity estimates and solvability of non-smooth elliptic systems II
We continue the development, by reduction to a first order system for the
conormal gradient, of \textit{a priori} estimates and solvability for
boundary value problems of Dirichlet, regularity, Neumann type for divergence
form second order, complex, elliptic systems. We work here on the unit ball and
more generally its bi-Lipschitz images, assuming a Carleson condition as
introduced by Dahlberg which measures the discrepancy of the coefficients to
their boundary trace near the boundary. We sharpen our estimates by proving a
general result concerning \textit{a priori} almost everywhere non-tangential
convergence at the boundary. Also, compactness of the boundary yields more
solvability results using Fredholm theory. Comparison between classes of
solutions and uniqueness issues are discussed. As a consequence, we are able to
solve a long standing regularity problem for real equations, which may not be
true on the upper half-space, justifying \textit{a posteriori} a separate work
on bounded domains.Comment: 76 pages, new abstract and few typos corrected. The second author has
changed nam
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