54 research outputs found
Well-Posed Initial-Boundary Evolution in General Relativity
Maximally dissipative boundary conditions are applied to the initial-boundary
value problem for Einstein's equations in harmonic coordinates to show that it
is well-posed for homogeneous boundary data and for boundary data that is small
in a linearized sense. The method is implemented as a nonlinear evolution code
which satisfies convergence tests in the nonlinear regime and is robustly
stable in the weak field regime. A linearized version has been stably matched
to a characteristic code to compute the gravitational waveform radiated to
infinity.Comment: 5 pages, 6 figures; added another convergence plot to Fig. 2 + minor
change
The Newtonian Limit for Asymptotically Flat Solutions of the Vlasov-Einstein System
It is shown that there exist families of asymptotically flat solutions of the
Einstein equations coupled to the Vlasov equation describing a collisionless
gas which have a Newtonian limit. These are sufficiently general to confirm
that for this matter model as many families of this type exist as would be
expected on the basis of physical intuition. A central role in the proof is
played by energy estimates in unweighted Sobolev spaces for a wave equation
satisfied by the second fundamental form of a maximal foliation.Comment: 24 pages, plain TE
Global classical solutions for partially dissipative hyperbolic system of balance laws
This work is concerned with (-component) hyperbolic system of balance laws
in arbitrary space dimensions. Under entropy dissipative assumption and the
Shizuta-Kawashima algebraic condition, a general theory on the well-posedness
of classical solutions in the framework of Chemin-Lerner's spaces with critical
regularity is established. To do this, we first explore the functional space
theory and develop an elementary fact that indicates the relation between
homogeneous and inhomogeneous Chemin-Lerner's spaces. Then this fact allows to
prove the local well-posedness for general data and global well-posedness for
small data by using the Fourier frequency-localization argument. Finally, we
apply the new existence theory to a specific fluid model-the compressible Euler
equations with damping, and obtain the corresponding results in critical
spaces.Comment: 39 page
An existence theorem in the calculus of variations based on Sobolev's imbedding theorems
Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46175/1/205_2004_Article_BF00266572.pd
The Nonlinear Future-Stability of the FLRW Family of Solutions to the Euler-Einstein System with a Positive Cosmological Constant
In this article, we study small perturbations of the family of
Friedmann-Lema\^itre-Robertson-Walker cosmological background solutions to the
1 + 3 dimensional Euler-Einstein system with a positive cosmological constant.
These background solutions describe an initially uniform quiet fluid of
positive energy density evolving in a spacetime undergoing accelerated
expansion. Our nonlinear analysis shows that under the equation of state
pressure = c_s^2 * energy density, with 0 < c_s^2 < 1/3, the background
solutions are globally future-stable. In particular, we prove that the
perturbed spacetime solutions, which have the topological structure [0,infty) x
T^3, are future causally geodesically complete. These results are extensions of
previous results derived by the author in a collaboration with I. Rodnianski,
in which the fluid was assumed to be irrotational. Our novel analysis of a
fluid with non-zero vorticity is based on the use of suitably-defined energy
currents.Comment: Accepted for publication in Selecta Mathematica, 78 pages. arXiv
admin note: significant text overlap with arXiv:0911.550
Multidimensional Conservation Laws: Overview, Problems, and Perspective
Some of recent important developments are overviewed, several longstanding
open problems are discussed, and a perspective is presented for the
mathematical theory of multidimensional conservation laws. Some basic features
and phenomena of multidimensional hyperbolic conservation laws are revealed,
and some samples of multidimensional systems/models and related important
problems are presented and analyzed with emphasis on the prototypes that have
been solved or may be expected to be solved rigorously at least for some cases.
In particular, multidimensional steady supersonic problems and transonic
problems, shock reflection-diffraction problems, and related effective
nonlinear approaches are analyzed. A theory of divergence-measure vector fields
and related analytical frameworks for the analysis of entropy solutions are
discussed.Comment: 43 pages, 3 figure
Entropy Stable Finite Volume Approximations for Ideal Magnetohydrodynamics
This article serves as a summary outlining the mathematical entropy analysis of the ideal magnetohydrodynamic (MHD) equations. We select the ideal MHD equations as they are particularly useful for mathematically modeling a wide variety of magnetized fluids. In order to be self-contained we first motivate the physical properties of a magnetic fluid and how it should behave under the laws of thermodynamics. Next, we introduce a mathematical model built from hyperbolic partial differential equations (PDEs) that translate physical laws into mathematical equations. After an overview of the continuous analysis, we thoroughly describe the derivation of a numerical approximation of the ideal MHD system that remains consistent to the continuous thermodynamic principles. The derivation of the method and the theorems contained within serve as the bulk of the review article. We demonstrate that the derived numerical approximation retains the correct entropic properties of the continuous model and show its applicability to a variety of standard numerical test cases for MHD schemes. We close with our conclusions and a brief discussion on future work in the area of entropy consistent numerical methods and the modeling of plasmas
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