34 research outputs found
Erratum to: "The Schrödinger–Maxwell system with Dirac mass" [Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (5) (2007) 773–793]
Abstract We correct the proof of [G.M. Coclite, H. Holden, The Schrodinger–Maxwell system with Dirac mass, Ann. Inst. H. Poincare Anal. Non Lineaire 24 (5) (2007) 773–793, Lemma 4.1]
Analytic solutions and Singularity formation for the Peakon b--Family equations
Using the Abstract Cauchy-Kowalewski Theorem we prove that the -family
equation admits, locally in time, a unique analytic solution. Moreover, if the
initial data is real analytic and it belongs to with , and the
momentum density does not change sign, we prove that the
solution stays analytic globally in time, for . Using pseudospectral
numerical methods, we study, also, the singularity formation for the -family
equations with the singularity tracking method. This method allows us to follow
the process of the singularity formation in the complex plane as the
singularity approaches the real axis, estimating the rate of decay of the
Fourier spectrum
Finite difference schemes for the symmetric Keyfitz-Kranzer system
We are concerned with the convergence of numerical schemes for the initial
value problem associated to the Keyfitz-Kranzer system of equations. This
system is a toy model for several important models such as in elasticity
theory, magnetohydrodynamics, and enhanced oil recovery. In this paper we prove
the convergence of three difference schemes. Two of these schemes is shown to
converge to the unique entropy solution. Finally, the convergence is
illustrated by several examples.Comment: 31 page
Stable standing waves for a class of nonlinear Schroedinger-Poisson equations
We prove the existence of orbitally stable standing waves with prescribed
-norm for the following Schr\"odinger-Poisson type equation \label{intro}
%{%{ll} i\psi_{t}+ \Delta \psi - (|x|^{-1}*|\psi|^{2}) \psi+|\psi|^{p-2}\psi=0
\text{in} \R^{3}, %-\Delta\phi= |\psi|^{2}& \text{in} \R^{3},%. when . In the case we prove the existence and
stability only for sufficiently large -norm. In case our approach
recovers the result of Sanchez and Soler \cite{SS} %concerning the existence
and stability for sufficiently small charges. The main point is the analysis of
the compactness of minimizing sequences for the related constrained
minimization problem. In a final section a further application to the
Schr\"odinger equation involving the biharmonic operator is given
Quasivariational solutions for first order quasilinear equations with gradient constraint
We prove the existence of solutions for an evolution quasi-variational
inequality with a first order quasilinear operator and a variable convex set,
which is characterized by a constraint on the absolute value of the gradient
that depends on the solution itself. The only required assumption on the
nonlinearity of this constraint is its continuity and positivity. The method
relies on an appropriate parabolic regularization and suitable {\em a priori}
estimates. We obtain also the existence of stationary solutions, by studying
the asymptotic behaviour in time. In the variational case, corresponding to a
constraint independent of the solution, we also give uniqueness results
A characteristic particle method for traffic flow simulations on highway networks
A characteristic particle method for the simulation of first order
macroscopic traffic models on road networks is presented. The approach is based
on the method "particleclaw", which solves scalar one dimensional hyperbolic
conservations laws exactly, except for a small error right around shocks. The
method is generalized to nonlinear network flows, where particle approximations
on the edges are suitably coupled together at the network nodes. It is
demonstrated in numerical examples that the resulting particle method can
approximate traffic jams accurately, while only devoting a few degrees of
freedom to each edge of the network.Comment: 15 pages, 5 figures. Accepted to the proceedings of the Sixth
International Workshop Meshfree Methods for PDE 201
Elliptic Perturbations for Hammerstein Equations with Singular Nonlinear Term
We consider a singular elliptic perturbation of a Hammerstein integral equation with singular nonlinear term at the origin. The compactness
of the solutions to the perturbed problem and, hence, the existence of a positive solution for the integral equation are proved. Moreover, these results are applied to nonlinear singular homogeneous Dirichlet problems
Positive solutions for an integro-differential equation with singular nonlinear term
The existence of a positive solution in a weighted Sobolev
space for a homogeneous semilinear elliptic integro-differential Dirichlet
problem is proved. The integral operator of the equation depends on a
nonlinear function with a singularity at the origin
Positive solutions for an integro-differential equation in all space with singular nonlinear term
We prove the existence of a positive solution in for a semilinear elliptic integro-differential problem in
The integral operator of the equation depends on a nonlinear function that is singular in the origin. Moreover, we prove
that the averages of the solution and its gradient on the balls
vanish as $R\to \infty.