Global results for a Cauchy problem related to biharmonic wave maps

TL;DR

This paper establishes the global existence and regularity of solutions to a derivative biharmonic wave equation with small initial data in a critical function space, extending results to certain target manifolds like the sphere.

Contribution

It proves the global well-posedness of a derivative biharmonic wave equation with non-generic quadratic nonlinearity in critical Besov spaces, including for biharmonic wave maps.

Findings

Global existence for small initial data in critical space

Persistence of higher regularity of solutions

Applicability to biharmonic wave maps on spheres

Abstract

We prove global existence of a derivative bi-harmonic wave equation with a non-generic quadratic nonlinearity and small initial data in the scaling critical space $\dot{B}^{2,1}_{\frac{d}{2}}(\mathbb{R}^d) \times \dot{B}^{2,1}_{\frac{d}{2}-2}(\mathbb{R}^d)$ for $ d \geq 3 $. Since the solution persists higher regularity of the initial data, we obtain a small data global regularity result for the biharmonic wave maps equation for a certain class of target manifolds including the sphere.