Spacetime bounds for the energy-critical nonlinear wave equation in three spatial dimensions

Results of Struwe, Grillakis, Struwe-Shatah, Kapitanski, Bahouri-Shatah, Bahouri-G'erard and Nakanishi have established global wellposedness, regularity, and scattering in the energy class for the energy-critical nonlinear wave equation $\mathrm{\square }u=u^5$ in ${\mathbb{R}}^{1+3}$, together with a spacetime bound $$\mathrm{\parallel }u{\mathrm{\parallel }}_{L_t^4{L}_x^{12}({\mathbb{R}}^{1+3})}\le M(E(u))$$ for some finite quantity M(E(u)) depending only on the energy E(u) of u. We reprove this result, and show that this quantity obeys a bound of at most exponential type in the energy, and specifically $M(E)\le C(1+E{)}^{C{E}^{105\mathrm{/}2}}$ for some absolute constant C > 0. The argument combines the quantitative local potential energy decay estimates of these previous papers with arguments used by Bourgain and the author for the analogous nonlinear Schr"odinger equation.