Almost critical local well-posedness for the space-time monopole equation in Lorenz gauge

Abstract

Recently, Candy and Bournaveas proved local well-posedness of the space-time monopole equation in Lorenz gauge for initial data in Hs with [Formula: see text]. The equation is L2-critical, and hence a [Formula: see text] derivative gap is left between their result and the scaling prediction. In this paper, we consider initial data in the Fourier–Lebesgue space [Formula: see text] for 1 < p ≤ 2 which coincides with Hs when p = 2 but scales like lower regularity Sobolev spaces for 1 < p < 2. In particular, we will see that as p → 1+, the critical exponent [Formula: see text], in which case [Formula: see text] is the critical space. We shall prove almost optimal local well-posedness to the space-time monopole equation in Lorenz gauge with initial data in the aforementioned spaces that correspond to p close to 1.