scholarly journals CONSTRUCTING HYPERBOLIC SYSTEMS IN THE ASHTEKAR FORMULATION OF GENERAL RELATIVITY

2000 ◽  
Vol 09 (01) ◽  
pp. 13-34 ◽  
Author(s):  
GEN YONEDA ◽  
HISA-AKI SHINKAI
Keyword(s):  
General Relativity ◽  
Hyperbolic System ◽  
Hyperbolic Systems ◽  
Equations Of Motion ◽  
Well Posedness ◽  
Symmetric Hyperbolic System ◽  
The Cauchy Problem ◽  
Long Time ◽  
Vacuum Spacetime ◽  
Gauge Conditions

Hyperbolic formulations of the equations of motion are essential technique for proving the well-posedness of the Cauchy problem of a system, and are also helpful for implementing stable long time evolution in numerical applications. We, here, present three kinds of hyperbolic systems in the Ashtekar formulation of general relativity for Lorentzian vacuum spacetime. We exhibit several (I) weakly hyperbolic, (II) diagonalizable hyperbolic, and (III) symmetric hyperbolic systems, with each their eigenvalues. We demonstrate that Ashtekar's original equations form a weakly hyperbolic system. We discuss how gauge conditions and reality conditions are constrained during each step toward constructing a symmetric hyperbolic system.

The well-posedness theory for Euler–Poisson fluids with non-zero heat conduction

2014 ◽  
Vol 11 (04) ◽  
pp. 679-703
Author(s):  
Jiang Xu
Keyword(s):  
Heat Conduction ◽  
Besov Spaces ◽  
Hyperbolic Systems ◽  
Type Inequality ◽  
Classical Solutions ◽  
Well Posedness ◽  
Poisson Equations ◽  
The Cauchy Problem ◽  
Critical Besov Spaces ◽  
Temperature Equation

This paper is devoted to the Euler–Poisson equations for fluids with non-zero heat conduction, arising in semiconductor science. Due to the thermal effect of the temperature equation, the local well-posedness theory by Xu and Kawashima (2014) for generally symmetric hyperbolic systems in spatially critical Besov spaces does not directly apply. To deal with this difficulty, we develop a generalized version of the Moser-type inequality by using Bony's decomposition. With a standard iteration argument, we then establish the local well-posedness of classical solutions to the Cauchy problem for intial data in spatially Besov spaces.

scholarly journals NONLINEAR WELL-POSEDNESS AND RATES OF DECAY IN THERMOELASTICITY WITH SECOND SOUND

2008 ◽  
Vol 05 (01) ◽  
pp. 25-43 ◽  
Author(s):  
REINHARD RACKE ◽  
YA-GUANG WANG
Keyword(s):  
Hyperbolic Systems ◽  
Decay Rates ◽  
Energy Estimates ◽  
Well Posedness ◽  
Time Behavior ◽  
Second Sound ◽  
Small Solution ◽  
Nonlinear Thermoelasticity ◽  
The Cauchy Problem ◽  
Rates Of Decay

The Cauchy problem in nonlinear thermoelasticity with second sound in one space dimension is considered. Due to Cattaneo's law, replacing Fourier's law for heat conduction, the system is hyperbolic. The local well-posedness as a strictly hyperbolic system is investigated first, and then the relation between energy estimates for non-symmetric hyperbolic systems and well-posedness are discussed. For the global small solution, the long time behavior is described and the decay rates of the L2-norm are obtained.

scholarly journals On the well-posedness and stability for the fourth-order Schrödinger equation with nonlinear derivative term

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Kelin Li ◽  
Huafei Di
Keyword(s):  
Schrödinger Equation ◽  
Fourth Order ◽  
Schrodinger Equation ◽  
Blow Up ◽  
A Priori Estimates ◽  
A Priori ◽  
Well Posedness ◽  
The Cauchy Problem ◽  
Long Time ◽  
Derivative Term

<p style='text-indent:20px;'>Considered herein is the well-posedness and stability for the Cauchy problem of the fourth-order Schrödinger equation with nonlinear derivative term <inline-formula><tex-math id="M1">\begin{document}$ iu_{t}+\Delta^2 u-u\Delta|u|^2+\lambda|u|^pu = 0 $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M2">\begin{document}$ t\in\mathbb{R} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ x\in \mathbb{R}^n $\end{document}</tex-math></inline-formula>. First of all, for initial data <inline-formula><tex-math id="M4">\begin{document}$ \varphi(x)\in H^2(\mathbb{R}^{n}) $\end{document}</tex-math></inline-formula>, we establish the local well-poseness and finite time blow-up criterion of the solutions, and give a rough estimate of blow-up time and blow-up rate. Secondly, under a smallness assumption on the initial value <inline-formula><tex-math id="M5">\begin{document}$ \varphi(x) $\end{document}</tex-math></inline-formula>, we demonstrate the global well-posedness of the solutions by applying two different methods, and at the same time give the scattering behavior of the solutions. Finally, based on founded a priori estimates, we investigate the stability of solutions by the short-time and long-time perturbation theories, respectively.</p>

scholarly journals Global Well-Posedness and Long Time Decay of Fractional Navier-Stokes Equations in Fourier-Besov Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Weiliang Xiao ◽  
Jiecheng Chen ◽  
Dashan Fan ◽  
Xuhuan Zhou
Keyword(s):  
Besov Spaces ◽  
Critical Case ◽  
Stokes Equations ◽  
Global Solutions ◽  
Navier Stokes ◽  
Well Posedness ◽  
Time Decay ◽  
Navier Stokes Equations ◽  
The Cauchy Problem ◽  
Long Time

We study the Cauchy problem of the fractional Navier-Stokes equations in critical Fourier-Besov spacesFB˙p,q1-2β+3/p′. Some properties of Fourier-Besov spaces have been discussed, and we prove a general global well-posedness result which covers some recent works in classical Navier-Stokes equations. Particularly, our result is suitable for the critical caseβ=1/2. Moreover, we prove the long time decay of the global solutions in Fourier-Besov spaces.

scholarly journals STRONGLY HYPERBOLIC SYSTEMS IN GENERAL RELATIVITY

2004 ◽  
Vol 01 (02) ◽  
pp. 251-269 ◽  
Author(s):  
OSCAR A. REULA
Keyword(s):  
General Relativity ◽  
Numerical Modeling ◽  
Evolution Equations ◽  
Hyperbolic Systems ◽  
Propagation Speed ◽  
Energy Estimates ◽  
Well Posedness ◽  
Evolution System ◽  
Hyperbolic Case ◽  
First Order

We discuss several topics related to the notion of strong hyperbolicity which are of interest in general relativity. After introducing the concept and showing its relevance we provide some covariant definitions of strong hyperbolicity. We then prove that if a system is strongly hyperbolic with respect to a given hypersurface, then it is also strongly hyperbolic with respect to any nearby surface. We then study for how much these hypersurfaces can be deformed and discuss then causality, namely what the maximal propagation speed in any given direction is. In contrast with the symmetric hyperbolic case, for which the proof of causality is geometrical and direct, relaying in energy estimates, the proof for general strongly hyperbolic systems is indirect for it is based in Holmgren's theorem. To show that the concept is needed in the area of general relativity we discuss two results for which the theory of symmetric hyperbolic systems shows to be insufficient. The first deals with the hyperbolicity analysis of systems which are second order in space derivatives; they include certain versions of the ADM and the BSSN families of equations. This analysis is considerably simplified by introducing pseudo-differential first-order evolution equations. Well-posedness for some members of the latter family systems is established by showing they satisfy the strong hyperbolicity property. Furthermore it is shown that many other systems of such families are only weakly hyperbolic, implying they should not be used for numerical modeling. The second result deals with systems having constraints. The question posed is which hyperbolicity properties, if any, are inherited from the original evolution system by the subsidiary system satisfied by the constraint quantities. The answer is that, subject to some condition on the constraints, if the evolution system is strongly hyperbolic then the subsidiary system is also strongly hyperbolic and the causality properties of both are identical.

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