Improved well-posedness for the quadratic derivative nonlinear wave equation in 2D

Abstract: In this paper we consider the Cauchy problem for the nonlinear wave equation (NLW) with quadratic derivative nonlinearities in two space dimensions. Following Gr\"{u}nrock's result in 3D, we take the data in the Fourier-Lebesgue spaces $^{H}s^r$, which coincide with the Sobolev spaces of the same regularity for $r=2$, but scale like lower regularity Sobolev spaces for $1<r\<2$. We show local well-posedness (LWP) for the range of exponents $s\>1+\frac{3}{2r}$, $1<r\leq 2$. On one end this recovers the sharp result on the Sobolev scale, $H^{\frac{7}{4}+}$, while on the other end establishes the $^{H}{\frac{5}{2}}^{1+}$ result, which scales like the Sobolev $H^{\frac{3}{2}+}$, thus, corresponding to a $\frac{1}{4}$ derivative improvement on the Sobolev scale.