In an influential 1964 article, P. Lax studied 2 × 2 genuinely nonlinear strictly hyperbolic PDE systems (in one spatial dimension). Using the method of Riemann invariants, he showed that a large set of smooth initial data lead to bounded solutions whose first spatial derivatives blow up in finite time, a phenomenon known as wave breaking. In the present article, we study the Cauchy problem for two classes of quasilinear wave equations in two spatial dimensions that are closely related to the systems studied by Lax. When the data have one-dimensional symmetry, Lax's methods can be applied to the wave equations to show that a large set of smooth initial data lead to wave breaking. Here we study solutions with initial data that are close, as measured by an appropriate Sobolev norm, to data belonging to a distinguished subset of Lax's data: the data corresponding to simple plane waves. Our main result is that under suitable relative smallness assumptions, the Lax-type wave breaking for simple plane waves is stable. The key point is that we allow the data perturbations to break the symmetry. Moreover, we give a detailed, constructive description of the asymptotic behavior of the solution all the way up to the first singularity, which is a shock driven by the intersection of null (characteristic) hyperplanes. We also outline how to extend our results to the compressible irrotational Euler equations. To derive our results, we use Christodoulou's framework for studying shock formation to treat a new solution regime in which wave dispersion is not present.

In his 2007 monograph, D. Christodoulou proved a remarkable result giving a detailed description of shock formation, for small H s -initial conditions (s sufficiently large), in solutions to the relativistic Euler equations in three space dimensions. His work provided a significant advancement over a large body of prior work concerning the long-time behavior of solutions to higher-dimensional quasilinear wave equations, initiated by F. John in the mid 1970's and continued by S. Klainerman, T. Sideris, L. Hörmander, H. Lindblad, S. Alinhac, and others. Our goal in this paper is to give an overview of his result, outline its main new ideas, and place it in the context of the above mentioned earlier work. We also introduce the recent work of J. Speck, which extends Christodoulou's result to show that for two important classes of quasilinear wave equations in three space dimensions, small-data shock formation occurs precisely when the quadratic nonlinear terms fail the classic null condition.

In the present paper, the characterization of the Kerr metric found by Marc Mars is extended to the Kerr-Newman family. A simultaneous alignment of the Maxwell field, the Ernst two-form of the pseudo-stationary Killing vector field, and the Weyl curvature of the metric is shown to imply that the space-time is locally isometric to domains in the Kerr-Newman metric. The paper also presents an extension of Ionescu and Klainerman's null tetrad formalism to explicitly include Ricci curvature terms.

École Polytechnique Fédérale de Lausanne, Switzerland a)Under mild assumptions, we remove all traces of the axiom of choice from the construction of the maximal globally hyperbolic Cauchy development in general relativity. The construction relies on the notion of direct union manifolds, which we review. The construction given is very general: any physical theory with a suitable geometric representation (in particular all classical fields), and such that a strong notion of "local existence and uniqueness" of solutions for the corresponding initial value problem is available, is amenable to the same treatment.A celebrated theorem on the local Cauchy problem for Einstein's equations is that of Choquet-Bruhat and Geroch (1969) which asserts that every initial data set leads to a unique maximal globally hyperbolic Cauchy development. In their original proof (as well as many subsequent treatments, see e.g. Ringström (2009)) the authors appealed to Zorn's Lemma in their construction of the space-time manifold, which led to the common misconception that the proof is non-constructive as the argument seemingly depends on the axiom of choice.In this paper, we will show that, insofar as the actual construction of the space-time manifold is concerned, the use of axiom of choice is not necessary. As it turns out, however, the manifold constructed will not, in general, be second countable, making geometry and analysis somewhat awkward on the space-time. One can circumvent this difficulty in two ways: firstly in many situations assuming the axiom of countable choice (or even weaker statements (Howard and Rubin, 1998) such as "every countable union of countable sets is countable") can allow us to recover statements about countability of a basis for the topology; secondly, an option that the author hopes to emphasize here, is that sometimes adding some additional structures (in a manner that is natural and physical) to the definition of a space-time will allow us to sidestep the issue of choice entirely.Recently the same question, in the context of general relativity, has been treated exhaustively in a pre-print by Sbierski (2013). Our approach here offers two minor improvements: