Fractional derivatives of composite functions and the Cauchy problem for the nonlinear half wave equation

TL;DR

This paper establishes new well-posedness results for the nonlinear half wave equation using innovative fractional chain rule techniques and space-time estimates, advancing understanding of fractional derivatives in PDEs.

Contribution

It introduces two novel approaches based on fractional chain rules and space-time estimates to prove well-posedness for the nonlinear half wave equation.

Findings

Proves local well-posedness in $H^{s}_{{ m rad}}({ m f R}^n)$ for $s>1/2$.

Develops a new fractional chain rule applicable to the problem.

Provides methods that can be applied to similar fractional PDEs.

Abstract

We show new results of wellposedness for the Cauchy problem for the half wave equation with power-type nonlinear terms. For the purpose, we propose two approaches on the basis of the contraction-mapping argument. One of them relies upon the $L_t^q L_x^\infty$ Strichartz-type estimate together with the chain rule of fairly general fractional orders. This chain rule has a significance of its own. Furthermore, in addition to the weighted fractional chain rule established in Hidano, Jiang, Lee, and Wang (arXiv:1605.06748v1 [math.AP]), the other approach uses weighted space-time $L^2$ estimates for the inhomogeneous equation which are recovered from those for the second-order wave equation. In particular, by the latter approach we settle the problem left open in Bellazzini, Georgiev, and Visciglia (arXiv:1611.04823v1 [math.AP]) concerning the local wellposedness in $H^{s}_{{\rm rad}}({\mathbb R}^n)$ with $s>1/2$.