A Definition of Total Energy-Momenta and the Positive Mass Theorem on Asymptotically Hyperbolic 3-Manifolds. I

Two sets of asymptotically hyperbolic initial data are defined, which correspond to the spatial infinity in asymptotically AdS spacetimes and to the null infinity in asymptotically Minkowski spacetimes respectively. The positive mass theorem involving the total energy, the total linear momentum and the total angular momentum is established for these initial data sets.Research partially supported by National Natural Science Foundation of China under grant 10231050 and the innovation project of the Chinese Academy of Sciences.     

AnS matrix theory for classical nonlinear physics

The basic concepts appropriate for anS matrix theory for classical nonlinear physics are formulated here. These concepts are illustrated by a discussion of shock wave diffraction patterns. Other information concerning solutions of non-linear conservation laws is surveyed, so that a coherent picture of this theory can be seen. Within thisS matrix framework, a number of open problems as well as a few solved ones will be discussed.Dedicated to John A. Wheeler on the occasion of his 75th birthday.Supported in part by the National Science Foundation, grant DMS-831229.Supported in part by the Aplied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under contract DE-AC02-76ER03077.Supported in part by the Army Research Office, grant DAAG29-83-K-0007.Work supported by the U.S. Department of Energy.     

Large-time behavior of solutions of initial and initial-boundary value problems of a general system of hyperbolic conservation laws

We study the asymptotic behavior of the solution of the initial and initial-boundary value problem of hyperbolic conservation laws when the initial and boundary data have bounded total variation. It is shown that the solution converges to the linear superposition of traveling waves, shock waves and rarefaction waves. The strength and speed of these waves depend only on the values of the data at infinity.Results obtained at the Courant Institute of Mathematical Sciences, New York University while the author was a Visiting Member at the Institute; this work was supported by the National Science Foundation, Grant NSF-MCS 76-07039On leave from the University of Maryland, College Park, USA     

The stability of rotating vortex patches

In this paper we examine the nonlinear and linear stability of variousrotating vortex patches. These patches include the Kirchhoff ellipse, the Kelvin waves, and the co-rotating uniformm vortices. These are achieved by using relative variational methods and spectral analysis. Thus, we extend Arnol'd's idea for stability problems in [1965, 1969] to a non-smooth symmetric setting and also relate that to the usual linear stability analysis.Supported by National Science Foundation under grant DMS-8501746Dedicated to the memory of my grandmother, Lang-Chang Lee Wan     

Hydrodynamical limit for a Hamiltonian system with weak noise

Starting from a general hamiltonian system with superstable pairwise potential, we construct a stochastic dynamics by adding a noise term which exchanges the momenta of nearby particles. We prolve that, in the scaling limit, the time conserved quantities, energy, momenta and density, satisfy the Euler equation of conservation laws up to a fixed timet provided that the Euler equation has a smooth solution with a given initial data up to timet. The strength of the noise term is chosen to be very small (but nonvanishing) so that it disappears in the scaling limit.Research partially supported by U.S. National Science Foundation grants DMS 89001682, DMS 920-1222 and a grant from ARO, DAAL03-92-G-0317Research partially supported by U.S. National Science Foundation grants DMS-9101196, DMS-9100383, and PHY-9019433-A01, Sloan Foundation Fellowship and David and Lucile Packard Foundation Fellowship     

Ergosphere instability

We consider stationary asymptotically flat spacetimes having an ergosphere but with no horizon. In the framework of linear perturbation theory such configurations are unstable or marginally unstable to scalar and electromagnetic perturbations.Research supported in part by the National Science Foundation under grant MPS 74-17456 with the University of Chicago and grant MPS 74-7456 at the University of Wisconsin-Milwaukee     

Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping

We consider a model of hyperbolic conservation laws with damping and show that the solutions tend to those of a nonlinear parabolic equation time-asymptotically. The hyperbolic model may be viewed as isentropic Euler equations with friction term added to the momentum equation to model gas flow through a porous media. In this case our result justifies Darcy's law time-asymptotically. Our model may also be viewed as an elastic model with damping.Research supported in part by Energy Dept. grant DEFG 02-88-ER25053Research supported in part by NSF grant DMS 90-0226 and Army grant DAAL 03-91-G0017     

A Rigorous Treatment of Energy Extraction from a Rotating Black Hole

The Cauchy problem is considered for the scalar wave equation in the Kerr geometry. We prove that by choosing a suitable wave packet as initial data, one can extract energy from the black hole, thereby putting supperradiance, the wave analogue of the Penrose process, into a rigorous mathematical framework. We quantify the maximal energy gain. We also compute the infinitesimal change of mass and angular momentum of the black hole, in agreement with Christodoulou’s result for the Penrose process. The main mathematical tool is our previously derived integral representation of the wave propagator. Research supported in part by the Deutsche Forschungsgemeinschaft. Research supported by NSERC grant # RGPIN 105490-2004. Research supported in part by the Humboldt Foundation and the National Science Foundation, Grant No. DMS-0603754. Research supported in part by the NSF, Grant No. 33-585-7510-2-30.     

Dirac and Klein-Gordon equations: Convergence of solutions in the nonrelativistic limit

The convergence of solutions of the Dirac and Klein-Gordon equations to solutions of the Pauli and Schrödinger equations in the non-relativistic limit is discussed. An abstract theory of these equations is developed which is general enough to allow physical space to be an arbitrary complete Riemannian manifold.Research partially supported by National Science Foundation grant MCS-77-13070     

Symmetry and related properties via the maximum principle

We prove symmetry, and some related properties, of positive solutions of second order elliptic equations. Our methods employ various forms of the maximum principle, and a device of moving parallel planes to a critical position, and then showing that the solution is symmetric about the limiting plane. We treat solutions in bounded domains and in the entire space.Supported in part by the National Science Foundation, grant no. PHY-78-08066Partially supported by the U.S. Army Research Office, grant no. DAA 29-78-G-0127     

A Half-space Problem for the Boltzmann Equation with Specular Reflection Boundary Condition

There are many open problems on the stability of nonlinear wave patterns to the Boltzmann equation even though the corresponding stability theory has been comparatively well-established for the gas dynamical systems. In this paper, we study the nonlinear stability of a rarefaction wave profile to the Boltzmann equation with the boundary effect imposed by specular reflection for both the hard sphere model and the hard potential model with angular cut-off. The analysis is based on the property of the solution and its derivatives which are either odd or even functions at the boundary coming from specular reflection, and the decomposition on both the solution and the Boltzmann equation introduced in [24, 26] for energy method.Research supported by the RGC Competitive Earmarked Research Grant, CityU 1142/01P.Research supported by the JSPS Research Fellowship for Foreign Researchers, the National Natural Science Foundation of China (10329101, 10431060), the National Key Program for Basic Research of China under grant 2002CCA03700, and the grant from the Chinese Academy of Sciences entitled Yin Jin Guo Wai Jie Chu Ren Cai Ji Jin.     

General method of calculation of any hadronic decay in the3 P 0 model

The³ P ₀ pair creation model of hadron decays is generalized to be applicable to the decay of any hadron. The wave function of the decaying hadron is expanded in terms of two clusters. The transition amplitude is derived for any combination of angular momenta, and for general wave functions in momentum space, expanded in terms of Gaussians times polynomials.Work supported by the Natural Science and Engineering Research Council of Canada, the National Science Foundation under grant # PHY-8714654, and by the Department of Energy under grant # DE-AC05-84ER40150Supported in part by the Natural Sciences and Engineering Research Council of Canada     

The Euler-Poincaré equations and double bracket dissipation

This paper studies the perturbation of a Lie-Poisson (or, equivalently an Euler-Poincaré) system by a special dissipation term that has Brockett's double bracket form. We show that a formally unstable equilibrium of the unperturbed system becomes a spectrally and hence nonlinearly unstable equilibrium after the perturbation is added. We also investigate the geometry of this dissipation mechanism and its relation to Rayleigh dissipation functions. This work complements our earlier work (Bloch, Krishnaprasad, Marsden and Ratiu [1991, 1994]) in which we studied the corresponding problem for systems with symmetry with the dissipation added to the internal variables; here it is added directly to the group or Lie algebra variables. The mechanisms discussed here include a number of interesting examples of physical interest such as the Landau-Lifschitz equations for ferromagnetism, certain models for dissipative rigid body dynamics and geophysical fluids, and certain relative equilibria in plasma physics and stellar dynamics.Research partially supported by the National Science Foundation PYI grant DMS-91-57556, and AFOSR grant F49620-93-1-0037.Research partially supported by the AFOSR University Research Initiative Program under grants AFOSR-87-0073 and AFOSR-90-0105 and by the National Science Foundation's Engineering Research Centers Program NSFD CDR 8803012.Research partially supported by, DOE contract DE-FG03-92ER-25129, a Fairchild Fellowship at Caltech, and the Fields Institute for Research in the Mathematical Sciences.Research partially supported by NSF Grant DMS 91-42613, DOE contract DE-FG03-92ER-25129, the Fields Institute, the Erwin Schrödinger Institute, and the Miller Institute of the University of California.     

Global symmetry restoration in high temperature Higgs theories

A simple high temperature expansion is developed for lattice gauge theories with scalar matter fields. The expansion is used to prove the absence of global symmetry breaking for sufficiently high temperature.Research supported in part by U.S. National Science Foundation grant PHY 8117463Research supported in part by the Department of Energy under grant No. DE-AC02-76ER03072Alfred P. Sloan Research Fellow     

The Hartree-Fock theory for Coulomb systems

For neutral atoms and molecules and positive ions and radicals, we prove the existence of solutions of the Hartree-Fock equations which minimize the Hartree-Fock energy. We establish some properties of the solutions including exponential falloff.Research partially supported by U.S. National Science Foundation Grant MCS-75-21684Research partially supported by U.S. National Science Foundation under Grants MPS-75-11864 and MPS-75-20638. On leave from Departments of Mathematics and Physics, Princeton University, Princeton, NJ08540, USA     

Inertial range scaling of laminar shear flow as a model of turbulent transport

Asymptotic scaling behavior, characteristic of the inertial range, is obtained for a fractal stochastic system proposed as a model for turbulent transport.Research supported in part by the U.S. Department of Energy, contract DE-FG02-90ER25084Research supported in part by the U.S. Department of Energy, contract DE-FG02-90ER25084, the National Science Foundation, grant DMS-8901884, the Army Research Office, grant DAAL03-K-0017     

Well-Posedness for the Dumbbell Model of Polymeric Fluids

The dumbbell model is a coupled hydrodynamic-kinetic model for polymeric fluids in which the configurations of the dumbbells are described by stochastic differential equations. We prove well-posedness of this model by deriving directly a priori estimates on the stochastic model. Our results can be used to analyze stochastic simulation methods such as the ones that are based on Brownian configuration fields.Supported by ONR grant N00014-01-1-0674 and National Science Foundation of China through a Class B Award for Distinguished Young Scholars 10128102.Partially supported by the special funds for Major State Research Projects G1999032804 and National Science Foundation of China for Distinguished Young Scholars 10225103.     

Stability of a Composite Wave of Two Viscous Shock Waves for the Full Compressible Navier-Stokes Equation

In this paper we investigate the asymptotic stability of a composite wave consisting of two viscous shock waves for the full compressible Navier-Stokes equation. By introducing a new linear diffusion wave special to this case, we successfully prove that if the strengths of the viscous shock waves are suitably small with same order and also the initial perturbations which are not necessarily of zero integral are suitably small, the unique global solution in time to the full compressible Navier-Stokes equation exists and asymptotically tends toward the corresponding composite wave whose shifts (in space) of two viscous shock waves are uniquely determined by the initial perturbations. We then apply the idea to study a half space problem for the full compressible Navier-Stokes equation and obtain a similar result. Research is supported in part by NSFC Grant No. 10471138, NSFC-NSAF Grant No. 10676037 and 973 project of China, Grant No. 2006CB805902, in part by Japan Society for the Promotion of Science, the Invitation Fellowship for Research in Japan (Short-Term). Research is supported in part by Grant-in-Aid for Scientific Research (B) 19340037, Japan.     

On determinants of Laplacians on Riemann surfaces

Determinants of Laplacians on tensors and spinors of arbitrary weights on compact hyperbolic Riemann surfaces are computed in terms of values of Selberg zeta functions at half integer points.Research supported in part by the U.S. Department of EnergyResearch supported in part by the National Science Foundation under Grant DMS-84-02710     

Linear Stability in an Ideal Incompressible Fluid

 We give an explicit construction of approximate eigenfunctions for a linearized Euler operator in dimensions two and three with periodic boundary conditions, and an estimate from below for its spectral bound in terms of an appropriate Lyapunov exponent. As a consequence, we prove that in dimension 2 the spectral and growth bounds for the corresponding group are equal. Therefore, the linear hydrodynamic stability of a steady state for the Euler equations in dimension 2 is equivalent to the fact that the spectrum of the linearized operator is pure imaginary. In dimension 3 we prove the estimate from below for the spectral bound that implies the same equality for every example where the relevant Lyapunov exponents could be effectively computed. For the kinematic dynamo operator describing the evolution of a magnetic field in an ideally conducting incompressible fluid we prove that the growth bound equals the spectral bound in dimensions 2 and 3. Received: 20 May 2002 / Accepted: 5 September 2002 Published online: 10 January 2003 RID="*" ID="*" The first author was partially supported by the Twinning Program of the National Academy of Sciences and National Science Foundation, and by the Research Council and Research Board of the University of Missouri. RID="**" ID="**" The second author was partially supported by the National Science Foundation grant DMS 9876947 and CRDF grant RM1-2084. Acknowledgements. The authors thank Susan Friedlander for useful discussions. Communicated by P. Constantin