Uniform bound of Sobolev norms of solutions to 3D nonlinear wave equations with null condition

This article concerns the time growth of Sobolev norms of classical solutions to the 3D quasi-linear wave equations with the null condition. Given initial data in Hs×Hs−1 with compact supports, the global well-posedness theory has been established independently by Klainerman [13] and Christodoulou [3], respectively, for a relatively large integer s. However, the highest order Sobolev energy, namely, the Hs energy of solutions may have a logarithmic growth in time. In this paper, we show that the Hs energy of solutions is also uniformly bounded for s⩾5. The proof employs the generalized energy method of Klainerman, enhanced by weighted L2 estimates and the ghost weight introduced by Alinhac.