TY - JOUR

T1 - Scattering from Infinity of the Maxwell Klein Gordon Equations in Lorenz Gauge

AU - He, Lili

N1 - Funding Information:
The author would like to thank her advisor, Hans Lindblad, for suggesting this problem, many helpful discussions and his constructive criticism of the manuscript.
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2021/9

Y1 - 2021/9

N2 - We prove global existence backwards from the scattering data posed at infinity for the Maxwell Klein Gordon equations in Lorenz gauge satisfying the weak null condition. The asymptotics of the solutions to the Maxwell Klein Gordon equations in Lorenz gauge were shown to be wave like at null infinity and homogeneous towards timelike infinity in Candy et al. (Commun Math Phys 367(2):683–716, 2019) and expressed in terms of radiation fields, and thus our scattering data will be given in the form of radiation fields in the backward problem. We give a refinement of the asymptotics results in Candy et al. (2019), and then making use of this refinement, we find a global solution which attains the prescribed scattering data at infinity. Our work starts from the approach in [21] and is more delicate since it involves with nonlinearities with fewer derivatives. Our result corresponds to “existence of scattering states” in the scattering theory. The method of proof relies on a suitable construction of the approximate solution from the scattering data, a weighted conformal Morawetz energy estimate and a spacetime version of Hardy inequality.

AB - We prove global existence backwards from the scattering data posed at infinity for the Maxwell Klein Gordon equations in Lorenz gauge satisfying the weak null condition. The asymptotics of the solutions to the Maxwell Klein Gordon equations in Lorenz gauge were shown to be wave like at null infinity and homogeneous towards timelike infinity in Candy et al. (Commun Math Phys 367(2):683–716, 2019) and expressed in terms of radiation fields, and thus our scattering data will be given in the form of radiation fields in the backward problem. We give a refinement of the asymptotics results in Candy et al. (2019), and then making use of this refinement, we find a global solution which attains the prescribed scattering data at infinity. Our work starts from the approach in [21] and is more delicate since it involves with nonlinearities with fewer derivatives. Our result corresponds to “existence of scattering states” in the scattering theory. The method of proof relies on a suitable construction of the approximate solution from the scattering data, a weighted conformal Morawetz energy estimate and a spacetime version of Hardy inequality.

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U2 - 10.1007/s00220-021-04105-y

DO - 10.1007/s00220-021-04105-y

M3 - Article

AN - SCOPUS:85106432215

SN - 0010-3616

VL - 386

SP - 1747

EP - 1801

JO - Communications In Mathematical Physics

JF - Communications In Mathematical Physics

IS - 3

ER -