31,093 research outputs found
Variable Support Control for the Wave Equation: A Multiplier Approach
We study the controllability of the multidimensional wave equation in a
bounded domain with Dirichlet boundary condition, in which the support of the
control is allowed to change over time. The exact controllability is reduced to
the proof of the observability inequality, which is proven by a multiplier
method. Besides our main results, we present some applications
Estimates for the kinetic transport equation in hyperbolic Sobolev spaces
We establish smoothing estimates in the framework of hyperbolic Sobolev
spaces for the velocity averaging operator of the solution of the
kinetic transport equation. If the velocity domain is either the unit sphere or
the unit ball, then, for any exponents and , we find a characterisation
of the exponents and , except possibly for an endpoint case,
for which is bounded from space-velocity
to space-time . Here, and are the classical
and hyperbolic derivative operators, respectively. In fact, we shall provide an
argument which unifies these velocity domains and the velocity averaging
estimates in either case are shown to be equivalent to mixed-norm bounds on the
cone multiplier operator acting on . We develop our ideas further in
several ways, including estimates for initial data lying in certain Besov
spaces, for which a key tool in the proof is the sharp decoupling
theorem recently established by Bourgain and Demeter. We also show that the
level of permissible smoothness increases significantly if we restrict
attention to initial data which are radially symmetric in the spatial variable.Comment: 23 pages; some additional arguments added to the proof of Theorem 1.3
in the case d=3; to appear in Journal de Math\'ematiques Pures et
Appliqu\'ee
Holder Continuous Solutions of Active Scalar Equations
We consider active scalar equations , where is a divergence-free velocity field, and
is a Fourier multiplier operator with symbol . We prove that when is
not an odd function of frequency, there are nontrivial, compactly supported
solutions weak solutions, with H\"older regularity . In fact,
every integral conserving scalar field can be approximated in by
such solutions, and these weak solutions may be obtained from arbitrary initial
data. We also show that when the multiplier is odd, weak limits of
solutions are solutions, so that the -principle for odd active scalars may
not be expected.Comment: 61 page
Decay estimates for variable coefficient wave equations in exterior domains
In this article we consider variable coefficient, time dependent wave
equations in exterior domains. We prove localized energy estimates if the
domain is star-shaped and global in time Strichartz estimates if the domain is
strictly convex.Comment: 15 pages. In the new version, some typos are fixed and a minor
correction was made to the proof of Lemma 1
Regular and Singular Pulse and Front Solutions and Possible Isochronous Behavior in the Short-Pulse Equation: Phase-Plane, Multi-Infinite Series and Variational Approaches
In this paper we employ three recent analytical approaches to investigate the
possible classes of traveling wave solutions of some members of a family of
so-called short-pulse equations (SPE). A recent, novel application of
phase-plane analysis is first employed to show the existence of breaking kink
wave solutions in certain parameter regimes. Secondly, smooth traveling waves
are derived using a recent technique to derive convergent multi-infinite series
solutions for the homoclinic (heteroclinic) orbits of the traveling-wave
equations for the SPE equation, as well as for its generalized version with
arbitrary coefficients. These correspond to pulse (kink or shock) solutions
respectively of the original PDEs.
Unlike the majority of unaccelerated convergent series, high accuracy is
attained with relatively few terms. And finally, variational methods are
employed to generate families of both regular and embedded solitary wave
solutions for the SPE PDE. The technique for obtaining the embedded solitons
incorporates several recent generalizations of the usual variational technique
and it is thus topical in itself. One unusual feature of the solitary waves
derived here is that we are able to obtain them in analytical form (within the
assumed ansatz for the trial functions). Thus, a direct error analysis is
performed, showing the accuracy of the resulting solitary waves. Given the
importance of solitary wave solutions in wave dynamics and information
propagation in nonlinear PDEs, as well as the fact that not much is known about
solutions of the family of generalized SPE equations considered here, the
results obtained are both new and timely.Comment: accepted for publication in Communications in Nonlinear Science and
Numerical Simulatio
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