Quadratic NLW/NLKG: Difference between revisions

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* Scaling is <math>s_c = \frac{d}{2} - 2</math>.
[Thanks to Chengbo Wang for some corrections \u2013 Ed.]
* For <math>d>4</math> LWP is known for <math>s \geq \frac{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{1}{4}</math> by Strichartz estimates ([[LbSo1995]]).This is sharp from Lorentz invariance (concentration) considerations.
* Scaling is <math>s_c = d/2 - 2</math>.
* For <math>d=3</math> LWP is known for <math>s > 0</math> by Strichartz estimates ([[LbSo1995]]).
* For <math>d>4</math> LWP is known for <math>s \geq d/2 - 2</math> by Strichartz estimates [[Bibliography#LbSo1995|LbSo1995]]. This is sharp by scaling arguments.
** 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=4</math> LWP is known for <math>s \geq 1/4</math> by Strichartz estimates [[Bibliography#LbSo1995|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 [[Bibliography#LbSo1995|LbSo1995]].
** One has ill-posedness for <math>s=0</math> [[Bibliography#Lb1996|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 case ([#two-speed see Overview]) 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].
** 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:Equations]]
[[Category:Wave]]
[[Category:Equations]]

Latest revision as of 13:48, 16 January 2007

  • 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).
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