Quadratic NLW/NLKG: Difference between revisions
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* Scaling is <math>s_c = d/2 - 2</math>. | * 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 [[ | * 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 1/4</math> by Strichartz estimates [[ | * 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=3</math> LWP is known for <math>s > 0</math> by Strichartz estimates [[ | * 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> [[ | ** One has ill-posedness for <math>s=0</math> ([[Lb1996]]). This is related to the failure of endpoint Strichartz when <math>d=3</math>. | ||
* For <math>d=1,2</math> LWP is known for <math>s\geq 0</math> by Strichartz estimates (or energy estimates and Sobolev in the <math>d=1</math> case). | * For <math>d=1,2</math> LWP is known for <math>s\geq 0</math> by Strichartz estimates (or energy estimates and Sobolev in the <math>d=1</math> case). | ||
** For s<0 one has rather severe ill-posedness generically, indeed cannot even interpret the non-linearity <math>f^2</math> as a distribution [CtCoTa-p2]. | ** For s<0 one has rather severe ill-posedness generically, indeed cannot even interpret the non-linearity <math>f^2</math> as a distribution ([[CtCoTa-p2]]). | ||
** In the two-speed | ** In the [[two-speed wave equations|two-speed case]] one can improve this to <math>s>-1/4</math> for non-linearities of the form <math>F = uv</math> and <math>G = uv</math> ([[Tg-p]]). | ||
[[Category:Wave]] | |||
[[Category:Equations]] | |||
Revision as of 21:37, 30 July 2006
- Scaling is .
- For LWP is known for by Strichartz estimates (LbSo1995). This is sharp by scaling arguments.
- For LWP is known for by Strichartz estimates (LbSo1995).This is sharp from Lorentz invariance (concentration) considerations.
- For LWP is known for by Strichartz estimates (LbSo1995).
- One has ill-posedness for (Lb1996). This is related to the failure of endpoint Strichartz when .
- For LWP is known for by Strichartz estimates (or energy estimates and Sobolev in the case).
- For s<0 one has rather severe ill-posedness generically, indeed cannot even interpret the non-linearity as a distribution (CtCoTa-p2).
- In the two-speed case one can improve this to for non-linearities of the form and (Tg-p).
