Hörmander’s method for the characteristic Cauchy problem and conformal scattering for a nonlinear wave equation

Abstract

The purpose of this note is to prove the existence of a conformal scattering operator for the cubic defocusing wave equation on a non-stationary background. The proof essentially relies on solving the characteristic initial value problem by the method developed by Hörmander. This method consists in slowing down the propagation speed of the waves to transform a characteristic initial value problem into a standard Cauchy problem. 2010 Mathematics Subject Classification. 35L05, 35P25, 53A30, 35Q75. 1 Conformal techniques, scattering, asymptotic behaviour. The well-posedness of the Cauchy problem for this equation, and in this geometric setting, has been addressed in [CC84]. The existence of a scattering operator on Minkowski space-time has been considered, on flat space-times in [KT06], for general semi-linear wave equations, and by conformal methods on flat space-time in [BSZ90]. The existence of a scattering operator, under more constraining assumptions, has been considered by the author in [Jou12], and this work extends this previous work to the generic cubic wave equation. In particular, we remove an artificial assumption in the decay of the coefficient of the non-linearity. The purpose of this assumption was to compensate for the blow-up of the Sobolev constant associated with the Sobolev embeddings of H 1 into L 6 in dimension 3.Since the metric is not stationary, this geometrical setting is a priori not amenable to standard analytic techniques to prove the existence of a scattering operator. To construct this

Conformal scattering theories for tensorial wave equations on Schwarzschild spacetime

“…The convergence (42) is valid by the energy decay (41) obtained in theorem 2. Combining this convergence and proposition 2, we obtain the energy identity up to i + such as equality (43).…”

“…In particular, the conformal scattering theory (i.e. the geometric scattering theory) has been studied extensively from the early works by Friedlander [25][26][27][28][29], Baez et al [8], Hörmander [37] to recent ones by Mason and Nicolas [49], Joudioux [41,42], Nicolas [59], Mokdad [53,54], Taujanskas [72] and Pham [63,65].…”

Cauchy and Goursat problems for the generalized spin zero rest-mass equations on Minkowski spacetime

“…The result of Hörmander was extended for the scalar wave equation by Nicolas [19] with the following minor modifications: the C 1 -metric, the continuous coefficients of the derivatives of the first order and the terms of order zero have locally L ∞ -coefficients. We refer [6,15,21,24] for…”

Peeling of Dirac field on Kerr spacetime

“…The results of Hörmander were extended for the scalar wave equation by Nicolas [32] with the following minor modifications: the C 1 -metric, the continuous coefficients of the derivatives of the first order and the terms of order zero have locally L ∞ -coefficients. We refer [17,27,28,34,50,51,52] for the appllications of the generalized results of Hörmander to solve the Goursat problem for the massless spin, tensorial equations, linear and semiliear wave equations. Here we will show that how the Goursat problem is valid for the spin wave equations in the future I + (S) of S in BI (recall that S is the spacelike hypersurface in BI such that it pass I + strictly in the past of the support data).…”