We consider the Cauchy problem for the scalar wave equation in the Kerr
geometry for smooth initial data supported outside the event horizon. We prove
that the solutions decay in time in L^\infty_loc. The proof is based on a
representation of the solution as an infinite sum over the angular momentum
modes, each of which is an integral of the energy variable on the real line.
This integral representation involves solutions of the radial and angular ODEs
which arise in the separation of variables.Comment: 44 pages, 5 figures, minor correction
We discuss the asymptotic behavior of solutions of weakly coupled parabolic equations describing systems undergoing diffusion, convection and nonlinear interaction in a bounded spatial domain, fl. If the system admits a bounded invariant region E of phase space then we isolate a parameter r which depends upon the size of 11, the lower bound for the diffusion matrix, the magnitude of convection and a measure of the strength or sensitivity of the reaction. For r > 0 we show that every solution with initial values in E and subject to homogeneous Neumann boundary conditions decays exponentially to a spatially homogeneous function of time. This limiting function is a solution of an ordinary differential equation whose o)-limit sets are determined by the reaction mechanism alone. This result may be interpreted as giving a sufficient condition for the validity of the "lumped parameter" approximation of distributed systems by solutions of ordinary differential equations. In particular we show that all attractors of the ODE are stable as solutions of the PDE as well.
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.
We consider the Cauchy problem for the massive Dirac equation in the non-extreme Kerr-Newman geometry outside the event horizon. We derive an integral representation for the Dirac propagator involving the solutions of the ODEs which arise in Chandrasekhar's separation of variables. It is proved that for initial data in L ∞ loc near the event horizon with L 2 decay at infinity, the probability of the Dirac particle to be in any compact region of space tends to zero as t goes to infinity. This means that the Dirac particle must either disappear in the black hole or escape to infinity. e-print archive:
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