The null-condition well-posedness conjecture for quasilinear wave equations

Consider a quasilinear wave equation (GNLW) with coefficients $g^{ij}(u,p)$, where $p$ denotes the first-derivative variables. The equation satisfies the null condition when

$$\frac{\partial g^{ij}(u,p)}{\partial p_k}\xi_i\xi_j\xi_k=0$$

whenever

$$g^{ij}(u,p)\xi_i\xi_j=0.$$

Null-condition well-posedness conjecture. If the null condition holds, then (GNLW) is well-posed in $H^s\times H^{s-1}$ for some $s<\frac n2+\frac34$ when $n=2$, respectively for some $s<\frac n2+\frac12$ when $n=3$. The conjecture proposes that the null structure permits well-posedness below the regularity threshold suggested by the available general quasilinear wave estimates; the source presents these improvements as an open direction.