Cubic NLW/NLKG on R2: Difference between revisions

Latest revision as of 04:53, 2 August 2006

Scaling is $s_{c}=0$ .
LWP for $s\geq 1/4$ $s\geq 1/4$ by Strichartz estimates (see e.g. LbSo1995)
- This is sharp by concentration examples in the focusing case; the defocusing case is still open.
GWP for $s\geq 1/2$ $s\geq 1/2$ for defocussing NLKG Bo1999 and for defocussing NLW Fo-p
- For $s\geq 1$ this is clear from energy conservation (for both NLKG and NLW).
- In the focussing case there is blowup from large data by the ODE method.
Remark: This is a symplectic flow with the symplectic form of $H^{1/2}$ , as in the one-dimensional case.

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 * Scaling is <math>s_c = 0</math>.
-* LWP for <math>s \geq 1/4</math> by Strichartz estimates (see e.g. [[Bibliography#LbSo1995|LbSo1995]])
+* LWP for <math>s \geq 1/4</math> by [[Strichartz estimate]]s (see e.g. [[LbSo1995]])
 ** This is sharp by concentration examples in the focusing case; the defocusing case is still open.
-* GWP for <math>s \geq 1/2</math> for defocussing NLKG [[Bibliography#Bo1999|Bo1999]] and for defocussing NLW [Fo-p]
+* GWP for <math>s \geq 1/2</math> for defocussing NLKG [[Bo1999]] and for defocussing NLW [[Fo-p]]
-** For <math>s \geq 1</math> this is clear from energy conservation (for both NLKG and NLW)..
+** For <math>s \geq 1</math> this is clear from energy conservation (for both NLKG and NLW).
 ** In the focussing case there is blowup from large data by the ODE method.
-* ''Remark''<nowiki>: This is a symplectic flow with the symplectic form of <math>H^{1/2}</math>, as in </nowiki>[#nlw-3_on_R the one-dimensional case].
+* ''Remark'': This is a symplectic flow with the symplectic form of <math>H^{1/2}</math>, as in [[cubic NLW/NLKG on R|the one-dimensional case]].
+[[Category:Wave]]
-----   [[Category:Equations]]
+[[Category:Equations]]