Improved multilinear estimates and global regularity for general nonlinear wave equations in $(1+3)$ dimensions

TL;DR

This paper develops improved multilinear estimates to analyze the long-time behavior of solutions to nonlinear wave and Dirac equations with singular nonlinearities in (1+3) dimensions, establishing scattering results for critical Sobolev data.

Contribution

It introduces new multilinear estimates that handle singular nonlinearities, enabling the proof of global regularity and scattering for a broad class of nonlinear wave and Dirac equations.

Findings

Established scattering for critical Sobolev data.

Developed techniques to relax singularities using angular momentum operators.

Extended results to various nonlinearities in wave and Dirac equations.

Abstract

This paper is devoted to the investigation of long-time behaviour of solutions to wave equations with quadratic nonlinearity and cubic Dirac equations with Hartree-type nonlinearity. We consider the nonlinearity here with enough simplicity so that we can treat it as a toy model and simultaneously with enough generality so that we can apply our result to wave and Dirac equations with various nonlinearities. The challenging point is that nonlinearity possesses singularity near the origin. Our strategy is to relax such a singularity by exploiting fully an angular momentum operator. In this manner we establish scattering for the critical Sobolev data.