Joel Smoller has done research in shock-wave theory, Navier-Stokes equations, systems of reaction-diffusion equations, dynamical systems (Conley Index Theory), and bifurcation theory (symmetry-breaking bifurcations). For the last several years he has been working in General Relativity (GR); specifically, shock-waves in (GR), with J.B. Temple, Einstein-Yang/Mills equations, (with A.G. Wasserman, and S.-T. Yau), Einstein-Dirac-Maxwell equations, and Einstein-Dirac-Yang/Mills equations with F. Finster, and S.-T. Yau, and decay of solutions to the Dirac equation and the scalar wave equation in a Kerr-Newman (rotating black-hole) geometry, also with Niky Kamran. With Ronghua Pan he has shown that singularities develop for the relativistic Euler equations. Jointly with Tao Luo he has proved the existence of steady solutions of the compressible Euler-Poisson equations with prescribed angular velocity in bounded domains. Such equations model a rotating Newtonian star. Smoller continues to be active in these areas.

Recently, Smoller has been concerned with problems involving gravity, as described by Einstein's Theory of General Relativity (GR), on two different scales: (A) elementary particles, (which involves the coupling of gravity to other "fundamental" forces), and (B) astrophysical shock-waves. The first, a continuation of work with S.-T. Yau, (Harvard) and Felix Finster, (University of Regensburg) studies the equations which describe the nonlinear coupling of gravity to both the Dirac equation, (describing quantum-mechanical particles), and to a Yang/Mills field, (describing nuclear forces), the EDYM equations. The second, an on-going project with Blake Temple, (University of California, Davis), is concerned with the construction of shock-wave solutions of Einstein's equations, and the application of such solutions to problems in cosmology and astrophysics.

In part (A), Smoller and his collaborators studied the interaction of Dirac particles (fermions) with non-abelian gauge fields and gravity. First, we showed that for arbitrarily weak gravitational interaction, we can find (linearly) stable static, spherically symmetric bound-state (particle-like) solutions. Second, we considered static, spherically symmetric black-hole solutions of the EDYM equations, and we proved that the spinors (wave functions), cannot be normalized. (One interpretation of this latter result is that the Dirac particles either enter the black-hole, or go off to infinity.) Both of these results are quite surprising. Finally, we showed that time-periodic solutions of the Dirac equation, in a (non-extreme) Reissner-Nordstrom black-hole background, cannot exist. This result was extended, together with Niky Kamran (McGill University), to the case of "rotating", (axi-symmetric), black-holes in the presence of quantum mechanical particles, and we again showed that time-periodic solutions of the Dirac equation in an axi-symmetric black-hole geometry cannot exist. We have also proved that solutions of the Cauchy problem for the Dirac equation, with initial data outside of the event horizon, decay generically at a rate t^-5/6. In addition we have obtained probability estimates for when the Dirac particle enters the black hole, or escapes to infinity. We have found an integral representation for the propagator for the wave equation in a rotating black-hole geometry. This representation was used to prove that solutions of the wave equation, with smooth, compactly supported initial data, decay to zero uniformly on compact sets.

In part (B), Smoller, together with Blake Temple, have constructed shock-wave solutions of Einstein's equations for a perfect fluid, which can model explosions. We applied these results to propose an alternative scenario to the usual Big-Bang theory. We showed that our dynamical shock-wave model is consistent with both the observed values of the Hubble constant and the cosmic background microwave radiation temperature.

We have very recently constructed exact shock wave solutions of the Einstein equations for perfect fluid which we interpret as the extension of the Oppenheimer-Snyder (OS) model to the case of non-zero pressure, inside the Black Hole. To accomplish this, we derive equations that describe the general matching of a critically expanding, (k = 0), Friedmann-Robertson-Walker (FRW) metric to a new metric, the Tolman-Oppenheimer-Volkoff (TOV) metric inside the Black Hole, across a shock wave interface. In the cosmological interpretation of the FRW metric, the shock wave must lie at least one Hubble length from the center of the FRW spacetime. In this case we (rigorously) demonstrate the existence of a new class of global solutions in which the expanding FRW universe emerges behind a subluminal blast wave that explodes outward from the FRW center at the instant of the Big Bang. The shock wave then continues to expand, satisfying the entropy condition for shocks, all the way out until it weakens to the point where it continues naturally to an OS interface that emerges from the White Hole event horizon of an ambiant Schwarzschild metric. These shock wave solutions differ qualitatively from the OS solution, and indicate a new cosmological model in which the Big Bang arises from a localized explosion occurring inside the Black Hole of an asymptotically flat Schwarzschild spacetime. In this model the equation of state p=[(c^2)/3](ρ), correct for early Big Bang physics, is distinguished by the differential equations: for this equation of state alone does the shock wave emerge from the Big Bang at finite non-zero speed, the speed of light. We have recently also incorporated "inflation" into our shock-wave cosmological model.

With David Hoff (Indiana University), we proved that weak solutions of the Navier-Stokes equations for compressible fluid flow in one space dimension do not exhibit vacuum states, provided that no vacuum states are present initially.

With Ronguha Pan (GA Tech) we have proved that solutions of the compressible relativistic Euler equations, in (3+1) dimensions, develop singularities in finite time, for both finite and infinite initial energy. In the finite energy case, we show that any smooth solution with compactly supported initial data develops a singularity in finite time. For the infinite initial energy case, the smooth solution becomes singular provided that the initial data is a smooth compact perturbation of a constant state, and the radial component of the initial "momentum" is sufficiently large.

Tao Luo (Georgetown University) and Smoller have studied the compressible Euler-Poisson equations in bounded domains with prescribed angular velocity. This models a rotating Newtonian star consisting of a compressible perfect fluid with a general equation of state. When the domain is a ball, and the angular velocity is constant, we obtain both existence and non-existence results, depending on the adiabatic gas constant. We also obtain several interesting properties of the solutions, some of them quite surprising.

 
Some of Smoller's most recent publications can be downloaded by going to this website: http://xxx.lanl.gov/archive/gr-qc. Then select the time period. Type in Joel Smoller and click on Do search.

 

This website was created by Alex Smoller, and was last updated on March 10, 2006.