Gradient NLW on Curved Background in 4+1 Dimensions

Abstract

We obtain a sharp local well-posedness result for the Gradient Nonlinear Wave Equation on a nonsmooth curved background. In the process we introduce variable coefficient versions of Bourgain's X s,b spaces, and use a trilinear multiscale wave packet decomposition in order to prove a key trilinear estimate.

Wave maps on (1+2)-dimensional curved spacetimes

“…For wave maps with variable coefficients, Geba [9] established local wellposedness in the subcritical regime s > n 2 when 3 ≤ n ≤ 5, building on previous work of Geba and Tataru [11]. More recently, Lawrie constructed global-in-time solutions on perturbations of R 1+4 Minkowski space for small data in the critical space H 2 × H 1 (R 4 ), and Lawrie-Oh-Shahshashani obtained analogous small-data results on R × H n , n ≥ 4 [24].…”

“…A variable coefficient version of the X s,b spaces, defined in physical space, was first introduced 1 in [37], and then further developed in [11] in order to study semilinear wave equations on curved backgrounds with a generic quadratic derivative nonlinearity. These spaces were later utilized by Geba [9] in his treatment of energy-subcritical wave maps in dimensions 3 ≤ d ≤ 5.…”

On global regularity for systems of nonlinear wave equations with the null-condition

“…In order to invoke the theory developed by Geba-Tataru [3], we need to work in the context of R 3 . For that purpose, we localize the functionsũ I to one of the stereographic coordinate charts U, V. Thus denoting by χ 1,2 a smooth partition of unity subordinate to U, V, we have…”

“…Here we quickly recall the norms used in [3], [2]. For s ą 3 2 and 1 2 ă θ ă s´1, we introduce on L 2 pr´π, πsˆR 3 q the norms…”

“…Our goal here is to explore a somewhat different approach to this problem, namely the combination of the Penrose compactification approach with a new local 1 well-posedness result for nonlinear waves on curved backgrounds by Geba [2] and Geba-Tataru [3]. The latter has its origins in the harmonic analysis approach to optimal local well-posedness, such as [8].…”