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We prove a stable shock formation result for a large class of systems ofquasilinear wave equations in two spatial dimensions. We give a precisedescription of the dynamics all the way up to the singularity. Our main theoremapplies to systems of two wave equations featuring two distinct wave speeds andvarious quasilinear and semilinear nonlinearities, while the solutions understudy are (non-symmetric) perturbations of simple outgoing plane symmetricwaves. The two waves are allowed to interact all the way up to the singularity.Our approach is robust and could be used to prove shock formation results forother related systems with many unknowns and multiple speeds, in varioussolution regimes, and in higher spatial dimensions. However, a fundamentalaspect of our framework is that it applies only to solutions in which the"fastest wave" forms a shock while the remaining solution variables do not.  Our approach is based on an extended version of the geometric vectorfieldmethod developed by D. Christodoulou in his study of shock formation for scalarwave equations as well as the framework developed in our recent joint work withJ. Luk, in which we proved a shock formation result for a quasilinearwave-transport system featuring a single wave operator. A key new difficultythat we encounter is that the geometric vectorfields that we use to commute theequations are, by necessity, adapted to the wave operator of the(shock-forming) fast wave and therefore exhibit very poor commutationproperties with the slow wave operator, much worse than their commutationproperties with a transport operator. To overcome this difficulty, we rely on afirst-order reformulation of the slow wave equation, which, though somewhatlimiting in the precision it affords, allows us to avoid uncontrollablecommutator terms.

annals of pde

Shock formation for $2D$ quasilinear wave systems featuring multiple  speeds: Blowup for the fastest wave, with non-trivial interactions up to the  singularity