Global Regularity for General Non-Linear Wave Equations I. (6 + 1) and Higher Dimensions

Abstract

Following work of Tataru [Tataru, D. (1998). Local and global results for wave maps I. Comm. Partial Differential Equations 23(9–10):1781–1793; Tataru, D. (1999). On the equation in 5 + 1 dimensions. Math. Res. Lett. 6 (5–6):469–485], we solve the division problem for wave equations with generic quadratic non-linearities in high dimensions. Specifically, we show that non-linear wave equations which can be written as systems involving equations of the form and are well-posed with scattering in (6 + 1) and higher dimensions if the Cauchy data are small in the scale invariant ℓ1 Besov space . This paper is the first in a series of works where we discuss the global regularity properties of general non-linear wave equations for all dimensions 4 ≤ n.