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We prove global existence for solutions arising from small initial data for alarge class of quasilinear wave equations satisfying the `weak null condition'of Lindblad and Rodnianski, significantly enlarging upon the class of equationsfor which global existence is known. In addition to the usual weak nullcondition, we require a certain hierarchical structure in the semilinear terms.Included in this class are the Einstein equations in harmonic coordinates, so aspecial case of our results is a new proof of the stability of Minkowski space.Our proof also applies to the coupled Einstein-Maxwell system in harmoniccoordinates and Lorenz gauge, as well as to various model scalar wave equationswhich do not satisfy the null condition. Our proof also applies to theEinstein(-Maxwell) equations if, after writing the equations as a set ofnonlinear wave equations, we then `forget' about the gauge conditions. Themethods we use allow us to treat initial data which only has a small`degenerate energy', involving a weight that degenerates at null infinity, sothe usual (unweighted) energy might be unbounded. We also demonstrate aconnection between the weak null condition and geometric shock formation,showing that equations satisfying the weak null condition can exhibit `shockformation at infinity', of which we provide an explicit example. The methodsthat we use are very robust, including a generalisation of the p-weightedenergy method of Dafermos and Rodnianski, adapted to the dynamic geometry. Thismeans that our proof applies in a wide range of situations, including those inwhich the metric remains close to, but never approaches the flat metric in somespatially bounded domain, and those in which the `geometric' null infinity andthe `background' null infinity differ dramatically, for example, when thesolution exhibits shock formation at null infinity.

The weak null condition and global existence using the p-weighted energy method