Long time solutions for wave maps with large data

A Note on Semi-linear Wave Equations

“…This part is quite similar to [21], the only difference is that we are in the 3-space dimensional Minkowski spacetime R × R 3 . We shall use the same notations as in [21]. We have the optical functions:…”

“…The parameter u is confined in [0, δ] where δ is a small parameter which will be determined later. As in [21], the corresponding cones are pictured as follows (See cone 2.1). When we derive estimates in section 3 and 4, u ∈ [0, δ] where δ will be sufficiently small.…”

“…We will follow the main steps of [21]. The Cauchy data will be finally given on t = u 0 + δ and the solution will exist at least for t ∈ [u 0 + δ, −1].…”

“…Inspired by the recent work [21], we study long time solutions of semi-linear wave equations by a different approach. Following the "short-pulse" method, which was first introduced by Christodoulou [3], and extended by Klainerman and Rodnianski [11], we establish the following long-time existence result for equation (1):…”

“…In [21], Wang and Yu constructed a solution for 2+1 wave maps with S 2 as its target, by using a bootstrap argument. Since the characteristic initial data (so called shortpulse) is chosen to be highlyoscillating, they can close the bootstrap.…”