119,782 research outputs found
Global Solutions for the One-Dimensional Vlasov-Maxwell System for Laser-Plasma Interaction
We analyse a reduced 1D Vlasov--Maxwell system introduced recently in the
physical literature for studying laser-plasma interaction. This system can be
seen as a standard Vlasov equation in which the field is split in two terms: an
electrostatic field obtained from Poisson's equation and a vector potential
term satisfying a nonlinear wave equation. Both nonlinearities in the Poisson
and wave equations are due to the coupling with the Vlasov equation through the
charge density. We show global existence of weak solutions in the
non-relativistic case, and global existence of characteristic solutions in the
quasi-relativistic case. Moreover, these solutions are uniquely characterised
as fixed points of a certain operator. We also find a global energy functional
for the system allowing us to obtain -nonlinear stability of some
particular equilibria in the periodic setting
Global existence and blow-up of solutions for a general class of doubly dispersive nonlocal nonlinear wave equations
This study deals with the analysis of the Cauchy problem of a general class
of nonlocal nonlinear equations modeling the bi-directional propagation of
dispersive waves in various contexts. The nonlocal nature of the problem is
reflected by two different elliptic pseudodifferential operators acting on
linear and nonlinear functions of the dependent variable, respectively. The
well-known doubly dispersive nonlinear wave equation that incorporates two
types of dispersive effects originated from two different dispersion operators
falls into the category studied here. The class of nonlocal nonlinear wave
equations also covers a variety of well-known wave equations such as various
forms of the Boussinesq equation. Local existence of solutions of the Cauchy
problem with initial data in suitable Sobolev spaces is proven and the
conditions for global existence and finite-time blow-up of solutions are
established.Comment: 17 page
Thresholds for global existence and blow-up in a general class of doubly dispersive nonlocal wave equations
In this article we study global existence and blow-up of solutions for a
general class of nonlocal nonlinear wave equations with power-type
nonlinearities, , where the
nonlocality enters through two pseudo-differential operators and . We
establish thresholds for global existence versus blow-up using the potential
well method which relies essentially on the ideas suggested by Payne and
Sattinger. Our results improve the global existence and blow-up results given
in the literature for the present class of nonlocal nonlinear wave equations
and cover those given for many well-known nonlinear dispersive wave equations
such as the so-called double-dispersion equation and the traditional
Boussinesq-type equations, as special cases.Comment: 17 pages. Accepted for publication in Nonlinear Analysis:Theory,
Methods & Application
A note on a strongly damped wave equation with fast growing nonlinearities
A strongly damped wave equation including the displacement depending
nonlinear damping term and nonlinear interaction function is considered. The
main aim of the note is to show that under the standard dissipativity
restrictions on the nonlinearities involved the initial boundary value problem
for the considered equation is globally well-posed in the class of sufficiently
regular solutions and the semigroup generated by the problem possesses a global
attractor in the corresponding phase space. These results are obtained for the
nonlinearities of an arbitrary polynomial growth and without the assumption
that the considered problem has a global Lyapunov function
A host of traveling waves in a model of three-dimensional water-wave dynamics
We describe traveling waves in a basic model for three-dimensional water-wave
dynamics in the weakly nonlinear long-wave regime. Small solutions that are
periodic in the direction of translation (or orthogonal to it) form an
infinite-dimensional family. We characterize these solutions through spatial
dynamics, by reducing a linearly ill-posed mixed-type initial-value problem to
a center manifold of infinite dimension and codimension. A unique global
solution exists for arbitrary small initial data for the two-component bottom
velocity, specified along a single line in the direction of translation (or
orthogonal to it). A dispersive, nonlocal, nonlinear wave equation governs the
spatial evolution of bottom velocity.Comment: 22 pages with 1 figure, LaTeX2e with amsfonts, epsfig package
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