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