Cubic NLW/NLKG on R2
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- Scaling is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s_c = 0} .
- LWP for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s \geq 1/2}
for defocussing NLKG Bo1999 and for defocussing NLW Fo-p
- For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^{1/2}} , as in the one-dimensional case.
