Global results for a Cauchy problem related to biharmonic wave maps

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.