8,706 research outputs found
Oscillations of vector differential equations of hyperbolic type with functional arguments
Vector hyperbolic differential equations with functional arguments are studied, and oscillations of solutions of certain boundary value problems are investigated. The approach used is to reduce the multi-dimensional oscillation problems to the nonexistence of positive solutions of scalar functional differential inequalities by employing the concept of H-oscillation introduced by Domšlak, where H denotes some unit vector
A new class of Fermionic Projectors: M{\o}ller operators and mass oscillation properties
Recently, a new functional analytic construction of quasi-free states for a
self-dual CAR algebra has been presented in \cite{Felix2}. This method relies
on the so-called strong mass oscillation property. We provide an example where
this requirement is not satisfied, due to the nonvanishing trace of the
solutions of the Dirac equation on the horizon of Rindler space, and we propose
a modification of the construction in order to weaken this condition. Finally,
a connection between the two approaches is built.Comment: 21 pages, accepted for publication in Letters in Mathematical Physics
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An adaptive pseudospectral method for discontinuous problems
The accuracy of adaptively chosen, mapped polynomial approximations is studied for functions with steep gradients or discontinuities. It is shown that, for steep gradient functions, one can obtain spectral accuracy in the original coordinate system by using polynomial approximations in a transformed coordinate system with substantially fewer collocation points than are necessary using polynomial expansion directly in the original, physical, coordinate system. It is also shown that one can avoid the usual Gibbs oscillation associated with steep gradient solutions of hyperbolic pde's by approximation in suitably chosen coordinate systems. Continuous, high gradient solutions are computed with spectral accuracy (as measured in the physical coordinate system). Discontinuous solutions associated with nonlinear hyperbolic equations can be accurately computed by using an artificial viscosity chosen to smooth out the solution in the mapped, computational domain. Thus, shocks can be effectively resolved on a scale that is subgrid to the resolution available with collocation only in the physical domain. Examples with Fourier and Chebyshev collocation are given
Weak Continuity and Compactness for Nonlinear Partial Differential Equations
We present several examples of fundamental problems involving weak continuity
and compactness for nonlinear partial differential equations, in which
compensated compactness and related ideas have played a significant role. We
first focus on the compactness and convergence of vanishing viscosity solutions
for nonlinear hyperbolic conservation laws, including the inviscid limit from
the Navier-Stokes equations to the Euler equations for homentropy flow, the
vanishing viscosity method to construct the global spherically symmetric
solutions to the multidimensional compressible Euler equations, and the
sonic-subsonic limit of solutions of the full Euler equations for
multidimensional steady compressible fluids. We then analyze the weak
continuity and rigidity of the Gauss-Codazzi-Ricci system and corresponding
isometric embeddings in differential geometry. Further references are also
provided for some recent developments on the weak continuity and compactness
for nonlinear partial differential equations.Comment: 29 page
A Non-Perturbative Construction of the Fermionic Projector on Globally Hyperbolic Manifolds II - Space-Times of Infinite Lifetime
The previous functional analytic construction of the fermionic projector on
globally hyperbolic Lorentzian manifolds is extended to space-times of infinite
lifetime. The construction is based on an analysis of families of solutions of
the Dirac equation with a varying mass parameter. It makes use of the so-called
mass oscillation property which implies that integrating over the mass
parameter generates decay of the Dirac wave functions at infinity. We obtain a
canonical decomposition of the solution space of the massive Dirac equation
into two subspaces, independent of observers or the choice of coordinates. The
constructions are illustrated in the examples of ultrastatic space-times and de
Sitter space-time.Comment: 29 pages, LaTeX, minor improvements (published version
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