Quartic NLW/NLKG: Difference between revisions
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* Scaling is <math>s_c = d/2 - 2/3</math>. | * Scaling is <math>s_c = d/2 - 2/3</math>. | ||
* For <math>d>2</math> LWP is known for <math>s \geq d/2 - 2/3</math> by Strichartz estimates. This is sharp by scaling argumentsin both the focusing and defocusing cases [CtCoTa-p2] | * For <math>d>2</math> LWP is known for <math>s \geq d/2 - 2/3</math> by Strichartz estimates. This is sharp by scaling argumentsin both the focusing and defocusing cases [[CtCoTa-p2]] | ||
* For <math>d=2</math> LWP is known for <math>s \geq 5/12</math> by Strichartz estimates. This is sharp by concentration arguments in the focusing case; the defocusing case is open. | * For <math>d=2</math> LWP is known for <math>s \geq 5/12</math> by Strichartz estimates. This is sharp by concentration arguments in the focusing case; the defocusing case is open. | ||
** In the defocusing case one has GWP for <math>s > 2/3</math> [Fo-p] | ** In the defocusing case one has GWP for <math>s > 2/3</math> [[Fo-p]] | ||
* For <math>d=1</math> one has LWP for <math>s\geq 1/4</math> by energy estimates and Sobolev (solution is in <math>L^4_x</math>). Below this regularity one cannot even make sense of the solution as a distribution. | * For <math>d=1</math> one has LWP for <math>s\geq 1/4</math> by energy estimates and Sobolev (solution is in <math>L^4_x</math>). Below this regularity one cannot even make sense of the solution as a distribution. | ||
[[Category:Wave]] | |||
[[Category:Equations]] | |||
Latest revision as of 07:08, 2 August 2006
- Scaling is .
- For LWP is known for by Strichartz estimates. This is sharp by scaling argumentsin both the focusing and defocusing cases CtCoTa-p2
- For LWP is known for by Strichartz estimates. This is sharp by concentration arguments in the focusing case; the defocusing case is open.
- In the defocusing case one has GWP for Fo-p
- For one has LWP for by energy estimates and Sobolev (solution is in ). Below this regularity one cannot even make sense of the solution as a distribution.
