Global solutions of the relativistic Euler equations

Abstract

Abstract. We demonstrate the existence of solutions with shocks for the equations describing a perfect fluid in special relativity, namely, divT= 0, where T ~ = (p + pcZ)ulU j + prl ij is the stress energy tensor for the fluid. Here, p denotes the pressure, u the 4-velocity, p the mass-energy density of the fluid, t/~ the flat Minkowski metric, and c the speed of light. We assume that the equation of state is given by p --a2p, where o -2, 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(p) is non-increasing on approximate weak solutions generated by Glimm's 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 (p(x ~ x 1), v(x~ 1)) themselves satisfy the Lorentz invariant estimates Var{ln(p(x~ < Vo and Var fln ~ + v(x~ _ v(xO,.)j < V1 for all t x ~ > 0, where Vo and V~ are Lorentz invariant constants that depend only on the total variation of the initial data, and v is the classical velocity. The equation of state p = (c2/3)p describes a gas of highly relativistic particles in several important general relativistic models which describe the evolution of stars.

The “Big Wave” Theory for Dark Energy

“…Since the one fact most certain about the standard model is that our universe arose from an earlier hot dense epoch in which all sources of energy were in the form of radiation, and since it is approximately uniform on the largest scale but highly oscillatory on smaller scales 3 , one might reasonably conjecture that decay to a non-interacting expanding wave occurred during the radiation phase of the standard model, via the highly nonlinear evolution driven by the large sound speed, and correspondingly large modulus of genuine nonlinearity 4 , present when p = ρc 2 /3, cf. [11]. Our analysis has shown that FRW is just one point in a family of non-interacting, self-similar expansion waves, and as a result we conclude that some further explanation is required as to why, on some length scale, decay during the radiation phase of the standard model would not proceed to a member of the family satisfying a = 1.…”

Shock Waves in Plane Symmetric Spacetimes

“…In this case all the assumptions are satisfied. They are also satisfied for an equation of state leading to a linear relation p = Kµ with K > 0, as considered in [17] and [3]. This is true in spite of the fact that if a full thermodynamic treatment of this case is attempted the internal energy is negative at low densities and so its physical interpretation is questionable.…”

“…First, are there global existence theorems for weak solutions of the special relativistic Euler equations? Some positive answers have been given in [17], [4] and [7]. Second, do these results extend to the case of a selfgravitating fluid?…”

Instabilities of the Riemann problem in relativistic hydrodynamics

“…Solutions of the Riemann problem for system I were obtained by Smoller and Temple for the case without velocities tangent to the initial discontinuity in [9]. Solutions valid for arbitrary velocities (also tangent to the discontinuity) were presented by Piȩtka and the author in [5].…”