Quadratic NLW/NLKG: Difference between revisions

Revision as of 13:47, 16 January 2007

Scaling is $s_{c}=d/2-2$ .
For $d>4$ LWP is known for $s\geq {\frac {d}{2}}d/2-2$ 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 $d=3$ $d=3$ LWP is known for $s>0$ $s>0$ 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 $d=1,2$ $d=1,2$ LWP is known for $s\geq 0$ $s\geq 0$ by Strichartz estimates (or energy estimates and Sobolev in the $d=1$ $d=1$ case).
- For s<0 one has rather severe ill-posedness generically, indeed cannot even interpret the non-linearity $f^{2}$ 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).

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 * Scaling is <math>s_c = d/2 - 2</math>.
-* For <math>d>4</math> LWP is known for <math>s \geq d/2 - 2</math> by Strichartz estimates ([[LbSo1995]]). This is sharp by scaling arguments.
+* For <math>d>4</math> LWP is known for <math>s \geq \frac{d}{2} d/2 - 2</math> by Strichartz estimates ([[LbSo1995]]). This is sharp by scaling arguments.
-* For <math>d=4</math> LWP is known for <math>s \geq 1/4</math> by Strichartz estimates ([[LbSo1995]]).This is sharp from Lorentz invariance (concentration) considerations.
+* For <math>d=4</math> LWP is known for <math>s \geq \frac{1}{4}</math> by Strichartz estimates ([[LbSo1995]]).This is sharp from Lorentz invariance (concentration) considerations.
 * For <math>d=3</math> LWP is known for <math>s > 0</math> by Strichartz estimates ([[LbSo1995]]).
 ** One has ill-posedness for <math>s=0</math> ([[Lb1996]]). This is related to the failure of endpoint Strichartz when <math>d=3</math>.