Blake Temple

Global solutions of the relativistic Euler equations

AbstractWe demonstrate the existence of solutions with shocks for the equations describing a perfect fluid in special relativity, namely, divT=0, whereTij=(p+ρc2)uiuj+pηij is the stress energy tensor for the fluid. Here,p denotes the pressure,u the 4-velocity, φ the mass-energy density of the fluid,ηij the flat Minkowski metric, andc the speed of light. We assume that the equation of state is given byp=σ2ρ, whereσ2, the sound speed, is constant. For these equations, we construct bounded weak solutions of the initial value problem in two dimensional Minkowski spacetime, for any initial data of finite total variation. The analysis is based on showing that the total variation of the variable ln(ρ) is non-increasing on approximate weak solutions generated by Glimms method, and so this quantity, unique to equations of this type, plays a role similar to an energy function. We also show that the weak solutions (ρ(x0,x1),v(x0,x1)) themselves satisfy the Lorentz invariant estimates Var{ln(ρ(x0,·)}<V0 and

Nonlinear resonance in systems of conservation laws

\left\{ {In\frac{{c + v(x^0 , \cdot )}}{{c - v(x^0 , \cdot )}}} \right\}< V_1

Shock-wave cosmology inside a black hole

for allx0≧0, whereV0 andV1 are Lorentz invariant constants that depend only on the total variation of the initial data, andv is the classical velocity. The equation of statep=(c2/3)ρ describes a gas of highly relativistic particles in several important general relativistic models which describe the evolution of stars.

On weak continuity and the Hodge decomposition

We introduce a generalized solution of the Riemann problem for a general resonant nonlinear balance law, and we prove the convergence of the 2 x 2 Godunov numerical method based on these solutions. In particular, we obtain generic conditions that guarantee a canonical structure for the elementary waves in the solution of the Riemann problem, and an interesting multiplicity of time-asymptotic wave patterns is observed and characterized.

The large time stability of sound waves

The Riemann problem for a general inhomogeneous system of conservation laws is solved in a neighborhood of a state at which one of the nonlinear waves in the problem takes on a zero speed. The inhomogeneity is modeled by a linearly degenerate field. The solution of the Riemann problem determines the nature of wave interactions, and thus the Riemann problem serves as a canonical form for nonlinear systems of conservation laws. Generic conditions on the fluxes are stated and it is proved that under these conditions, the solution of the Riemann problem exists, is unique, and has a fixed structure; this demonstrates that, in the above sense, resonant inhomogeneous systems generically have the same canonical form. The wave curves for these systems are only Lipschitz continuous in a neighborhood of the states where the wave speeds coincide, and so, in contrast to strictly hyperbolic systems, the implicit function theorem cannot be applied directly to obtain existence and uniqueness. Here we show that existence ...

Stability of Godunov’s method for a class of 2×2 systems of conservation laws

The purpose of this paper is to classify the solutions of Riemann problems near a hyperbolic singularity in a nonlinear system of conservation laws. Hyperbolic singularities play the role in the theory of Riemann problems that rest points play in the theory of ordinary differential equations: Indeed, generically, only a finite number of structures can appear in a neighborhood of such a singularity. In this, the first of three papers, the program of classification is discussed in general and the simplest structure that occurs is characterized.

A Bound on the Total Variation of the Conserved Quantities for Solutions of a General Resonant Nonlinear Balance Law

We construct a class of global exact solutions of the Einstein equations that extend the Oppeheimer–Snyder model to the case of nonzero pressure, inside the black hole, by incorporating a shock wave at the leading edge of the expansion of the galaxies, arbitrarily far beyond the Hubble length in the Friedmann–Robertson–Walker (FRW) spacetime. Here the expanding FRW universe emerges be-hind a subluminous blast wave that explodes outward from the FRW center at the instant of the big bang. The total mass behind the shock decreases as the shock wave expands, and the entropy condition implies that the shock wave must weaken to the point where it settles down to an Oppenheimer–Snyder interface, (bounding a finite total mass), that eventually emerges from the white hole event horizon of an ambient Schwarzschild spacetime. The entropy condition breaks the time symmetry of the Einstein equations, selecting the explosion over the implosion. These shock-wave solutions indicate a cosmological model in which the big bang arises from a localized explosion occurring inside the black hole of an asymptotically flat Schwarzschild spacetime.

The structure of asymptotic states in a singular system of conservation laws

We address the problem of determining the weakly continuous polynomials for sequences of functions that satisfy general linear first-order differential constraints. We prove that wedge products are weakly continuous when the differential constraints are given by exterior derivatives. This is sufficient for reproducing the Div-Curl Lemma of Murat and Tartar, the null Lagrangians in the calculus of variations and the weakly continuous polynomials for Maxwells equations. This result was derived independently by Tartar who stated it in a recent survey article [7]. Our proof is explicit and uses the Hodge decomposition.