Revisiting the characteristic initial value problem for the vacuum Einstein field equations

Using the Newman–Penrose formalism we study the characteristic initial value problem in vacuum General Relativity. We work in a gauge suggested by Stewart, and following the strategy taken in the work of Luk, demonstrate local existence of solutions in a neighbourhood of the set on which data are given. These data are given on intersecting null hypersurfaces. Existence near their intersection is achieved by combining the observation that the field equations are symmetric hyperbolic in this gauge with the results of Rendall. To obtain existence all the way along the null-hypersurfaces themselves, a bootstrap argument involving the Newman–Penrose variables is performed.

Improved existence for the characteristic initial value problem with the conformal Einstein field equations

We adapt Luk’s analysis of the characteristic initial value problem in general relativity to the asymptotic characteristic problem for the conformal Einstein field equations to demonstrate the local existence of solutions in a neighbourhood of the set on which the data are given. In particular, we obtain existence of solutions along a narrow rectangle along null infinity which, in turn, corresponds to an infinite domain in the asymptotic region of the physical spacetime. This result generalises work by Kánnár on the local existence of solutions to the characteristic initial value problem by means of Rendall’s reduction strategy. In analysing the conformal Einstein equations we make use of the Newman–Penrose formalism and a gauge due to J. Stewart.

Characteristic initial value problem for spherically symmetric barotropic flow

We study the equations of motion for a barotropic fluid in spherical symmetric flow. Making use of the Riemann invariants, we consider the characteristic form of these equations. In a first part, we show that the resulting constraint equations along characteristics can be solved globally away from the center of symmetry. In a second part, given data on two intersecting characteristics, we show existence and uniqueness of a smooth solution in a neighborhood in the future of these characteristics.

Linear Hyperbolic Functional-Differential Equations with Essentially Bounded Right-Hand Side

Theorems on the unique solvability and nonnegativity of solutions to the characteristic initial value problemu1,1(t,x)=l0(u)(t,x)+l1(u1,0)(t,x)+l2(u0,1)(t,x)+q(t,x), u(t,c)=α(t)fort∈[a,b], u(a,x)=β(x) for x∈[c,d]given on the rectangle[a,b]×[c,d]are established, where the linear operatorsl0,l1,l2map suitable function spaces into the space of essentially bounded functions. General results are applied to the hyperbolic equations with essentially bounded coefficients and argument deviations.