• Scaling is $s_c = \frac{d}{2} - 2$.
• For d > 4 LWP is known for $s \geq \frac{d}{2} - 2$ by Strichartz estimates (LbSo1995). This is sharp by scaling arguments.
• For d = 4 LWP is known for $s \geq \frac{1}{4}$ by Strichartz estimates (LbSo1995).This is sharp from Lorentz invariance (concentration) considerations.
• For d = 3 LWP is known for s > 0 by Strichartz estimates (LbSo1995).
• One has ill-posedness for s = 0 (Lb1996). This is related to the failure of endpoint Strichartz when d = 3.
• For d = 1,2 LWP is known for $s\geq 0$ by Strichartz estimates (or energy estimates and Sobolev in the d = 1 case).
• For s<0 one has rather severe ill-posedness generically, indeed cannot even interpret the non-linearity f2 as a distribution (CtCoTa-p2).
• In the two-speed case one can improve this to s > − 1 / 4 for non-linearities of the form F = uv and G = uv (Tg-p).