Nonzero solutions of perturbed Hammerstein integral equations with deviated arguments and applications

We provide a theory to establish the existence of nonzero solutions of perturbed Hammerstein integral equations with deviated arguments, being our main ingredient the theory of fixed point index. Our approach is fairly general and covers a variety of cases. We apply our results to a periodic boundary value problem with reflections and to a thermostat problem. In the case of reflections we also discuss the optimality of some constants that occur in our theory. Some examples are presented to illustrate the theory.Comment: 3 figures, 23 page

Bifurcation From Infinity And Multiplicity Of Solutions For Nonlinear Periodic Boundary Value Problems

We are concerned with multiplicity and bifurcation results for solutions of nonlinear second order differential equations with general linear part and periodic boundary conditions. We impose asymptotic conditions on the nonlinearity and let the parameter vary. We then proceed to establish a priori estimates and prove multiplicity results (for large-norm solutions) when the parameter belongs to a (nontrivial) continuum of real numbers. Our results extend and complement those in the literature. The proofs are based on degree theory, continuation methods, and bifurcation from infinity techniques

Some mathematical problems in numerical relativity

The main goal of numerical relativity is the long time simulation of highly nonlinear spacetimes that cannot be treated by perturbation theory. This involves analytic, computational and physical issues. At present, the major impasses to achieving global simulations of physical usefulness are of an analytic/computational nature. We present here some examples of how analytic insight can lend useful guidance for the improvement of numerical approaches.Comment: 17 pages, 12 graphs (eps format

Coercivity and stability results for an extended Navier-Stokes system

In this article we study a system of equations that is known to {\em extend} Navier-Stokes dynamics in a well-posed manner to velocity fields that are not necessarily divergence-free. Our aim is to contribute to an understanding of the role of divergence and pressure in developing energy estimates capable of controlling the nonlinear terms. We address questions of global existence and stability in bounded domains with no-slip boundary conditions. Even in two space dimensions, global existence is open in general, and remains so, primarily due to the lack of a self-contained $L^2$ energy estimate. However, through use of new $H^1$ coercivity estimates for the linear equations, we establish a number of global existence and stability results, including results for small divergence and a time-discrete scheme. We also prove global existence in 2D for any initial data, provided sufficient divergence damping is included.Comment: 29 pages, no figure