• DOI:10.1007/s10231-018-0805-1
• Corpus ID: 53615345

Dynamics of the nonlinear Klein–Gordon equation in the nonrelativistic limit

@article{Pasquali2017DynamicsOT,
  title={Dynamics of the nonlinear Klein–Gordon equation in the nonrelativistic limit},
  author={Stefano Pasquali},
  journal={Annali di Matematica Pura ed Applicata (1923 -)},
  year={2017},
  volume={198},
  pages={903-972}
}

• S. Pasquali
• Published 5 March 2017
• Mathematics
• Annali di Matematica Pura ed Applicata (1923 -)

We study the nonlinear Klein–Gordon (NLKG) equation on a manifold M in the nonrelativistic limit, namely as the speed of light c tends to infinity. In particular, we consider a higher-order normalized approximation of NLKG (which corresponds to the NLS at order $$r=1$$r=1) and prove that when M is a smooth compact manifold or $$\mathbb {R}^d$$Rd, the solution of the approximating equation approximates the solution of the NLKG locally uniformly in time. When $$M=\mathbb {R}^d$$M=Rd, $$d \ge 2$$d≥2, we also prove that for $$r \ge 2$$r≥2 small radiation solutions of the order-r normalized equation approximate solutions of the nonlinear NLKG up to times of order $$\mathscr {O}(c^{2(r-1)})$$O(c2(r-1)). We also prove a global existence result uniform with respect to c for the NLKG equation on $$\mathbb {R}^3$$R3 with cubic nonlinearity for small initial data and Strichartz estimates for the Klein–Gordon equation with potential on $$\mathbb {R}^3$$R3.