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In this paper, we establish the global existence of smooth solutions togeneral 4D quasilinear wave equations satisfying the first null condition withthe short pulse initial data. Although the global existence of small datasolutions to 4D quasilinear wave equations holds true without any requirementof null conditions, yet for short pulse data, in general, it is sufficient andnecessary to require the fulfillment of the first null condition to have globalsmooth solutions. It is noted that short pulse data are extensions of a classof spherically symmetric data, for which the smallness restrictions are imposedon angular directions and along the outgoing directional derivative$\partial_t+\partial_r$, but the largeness is kept for the incoming directionalderivative $\partial_t-\partial_r$. We expect that here methods can be appliedto study the global smooth solution or blowup problem with short pulse initialdata for the general 2D and 3D quasilinear wave equations when thecorresponding null conditions hold or not. On the other hand, as some directapplications of our main results, one can show that for the short pulse initialdata, the smooth solutions to the 4D irrotational compressible Euler equationsfor Chaplygin gases, 4D nonlinear membrane equations and 4D relativisticmembrane equations exist globally since their nonlinearities satisfy the firstnull condition; while the smooth solutions to the 4D irrotational compressibleEuler equations for polytropic gases generally blow up in finite time since thecorresponding first null condition does not hold.

Global smooth solutions to 4D quasilinear wave equations with short pulse initial data