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Register a new userBoundary conditions for coupled quasilinear wave equations with application to isolated systems
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We consider the initial-boundary value problem for systems of quasilinear wave equations on domains of the form $[0,T] times Sigma$, where $Sigma$ is a compact manifold with smooth boundaries $partialSigma$. By using an appropriate reduction to a first order symmetric hyperbolic system with maximal dissipative boundary conditions, well posedness of such problems is established for a large class of boundary conditions on $partialSigma$. We show that our class of boundary conditions is sufficiently general to allow for a well posed formulation for different wave problems in the presence of constraints and artificial, nonreflecting boundaries, including Maxwells equations in the Lorentz gauge and Einsteins gravitational equations in harmonic coordinates. Our results should also be useful for obtaining stable finite-difference discretizations for such problems.
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We classify positive solutions to a class of quasilinear equations with Neumann or Robin boundary conditions in convex domains. Our main tool is an integral formula involving the trace of some relevant quantities for the problem. Under a suitable condition on the nonlinearity, a relevant consequence of our results is that we can extend to weak solutions a celebrated result obtained for stable solutions by Casten and Holland and by Matano.
A maximally rotating Kerr black hole is said to be extremal. In this paper we introduce the corresponding restrictions for isolated and dynamical horizons. These reduce to the standard notions for Kerr but in general do not require the horizon to be either stationary or rotationally symmetric. We consider physical implications and applications of these results. In particular we introduce a parameter e which characterizes how close a horizon is to extremality and should be calculable in numerical simulations.
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The Binary Black Hole Grand Challenge Alliance: A. M. Abrahams
, L.n Rezzolla
, M. E. Rupright
1997
We present a method for extracting gravitational radiation from a three-dimensional numerical relativity simulation and, using the extracted data, to provide outer boundary conditions. The method treats dynamical gravitational variables as nonspherical perturbations of Schwarzschild geometry. We discuss a code which implements this method and present results of tests which have been performed with a three dimensional numerical relativity code.
Hormander proved global existence of solutions for sufficiently small initial data for scalar wave equations in $(1+4)-$dimensions of the form $Box u = Q(u, u, u)$ where $Q$ vanishes to second order and $(partial_u^2 Q)(0,0,0)=0$. Without the latter condition, only almost global existence may be guaranteed. The first author and Sogge considered the analog exterior to a star-shaped obstacle. Both results relied on writing the lowest order terms $upartial_alpha u = frac{1}{2}partial_alpha u^2$ and as such do not immediately generalize to systems. The current study remedies such and extends both results to the case of multiple speed systems.
We prove global stability results of {sl DiPerna-Lions} renormalized solutions for the initial boundary value problem associated to some kinetic equations, from which existence results classically follow. The (possibly nonlinear) boundary conditions are completely or partially diffuse, which includes the so-called Maxwell boundary conditions, and we prove that it is realized (it is not only a boundary inequality condition as it has been established in previous works). We are able to deal with Boltzmann, Vlasov-Poisson and Fokker-Planck type models. The proofs use some trace theorems of the kind previously introduced by the author for the Vlasov equations, new results concerning weak-weak convergence (the renormalized convergence and the biting $L^1$-weak convergence), as well as the Darroz`es-Guiraud information in a crucial way.
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