Quartic NLS: Difference between revisions
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* Scaling is <math>s_c = -1/6\,</math>. | * Scaling is <math>s_c = -1/6\,</math>. | ||
* For any quartic non-linearity one can obtain LWP for <math>s \ge 0\,</math> [[CaWe1990]] | * For any quartic non-linearity one can obtain LWP for <math>s \ge 0\,</math> [[CaWe1990]] | ||
** Below <math>L^2\,</math> we have ill-posedness by Gallilean invariance considerations in both the focusing [KnPoVe-p] and defocusing [CtCoTa-p2] cases. | ** Below <math>L^2\,</math> we have ill-posedness by Gallilean invariance considerations in both the focusing [[KnPoVe-p]] and defocusing [[CtCoTa-p2]] cases. | ||
* If the quartic non-linearity is of <math>\underline{uuuu}\,</math> type then one can obtain LWP for <math>s > -1/6\,.</math> For <math>|u|^4\,</math> one has LWP for <math>s > -1/8\,</math>, while for the other three types <math>u^4\,</math>, <math>u u u \underline{u}\,</math>, or <math>u \underline{uuu}\,</math> one has LWP for <math>s > -1/6\,</math> [[Gr-p2]]. | * If the quartic non-linearity is of <math>\underline{uuuu}\,</math> type then one can obtain LWP for <math>s > -1/6\,.</math> For <math>|u|^4\,</math> one has LWP for <math>s > -1/8\,</math>, while for the other three types <math>u^4\,</math>, <math>u u u \underline{u}\,</math>, or <math>u \underline{uuu}\,</math> one has LWP for <math>s > -1/6\,</math> [[Gr-p2]]. | ||
* In the Hamiltonian case (a non-linearity of type <math>|u|^3 u\,</math>) we have GWP for <math>s \ge 0\,</math> by <math>L^2\,</math> conservation. In the other cases it is not clear whether there is any reasonable GWP result, except possibly for very small data. | * In the Hamiltonian case (a non-linearity of type <math>|u|^3 u\,</math>) we have GWP for <math>s \ge 0\,</math> by <math>L^2\,</math> conservation. In the other cases it is not clear whether there is any reasonable GWP result, except possibly for very small data. | ||
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* Scaling is <math>s_c = 1/3\,.</math> | * Scaling is <math>s_c = 1/3\,.</math> | ||
* For any quartic non-linearity one can obtain LWP for <math>s \ge s_c\,</math> [[CaWe1990]]. | * For any quartic non-linearity one can obtain LWP for <math>s \ge s_c\,</math> [[CaWe1990]]. | ||
** For <math>s<s_c\,</math> we have ill-posedness, indeed the <math>H^s\,</math> norm can get arbitrarily large arbitrarily quickly [CtCoTa-p2]. In the focusing case we have instantaneous blowup from the virial identity and scaling. | ** For <math>s<s_c\,</math> we have ill-posedness, indeed the <math>H^s\,</math> norm can get arbitrarily large arbitrarily quickly [[CtCoTa-p2]]. In the focusing case we have instantaneous blowup from the virial identity and scaling. | ||
* In the Hamiltonian case (a non-linearity of type <math>|u|^3 u\,</math>) we have GWP for <math>s \ge 1\,</math> [[Ka1986]]. | * In the Hamiltonian case (a non-linearity of type <math>|u|^3 u\,</math>) we have GWP for <math>s \ge 1\,</math> [[Ka1986]]. | ||
** This has been improved to <math>s > 1-e\,</math> in [[CoKeStTkTa2003c]] in the defocusing Hamiltonian case. This result can of course be improved further. | ** This has been improved to <math>s > 1-e\,</math> in [[CoKeStTkTa2003c]] in the defocusing Hamiltonian case. This result can of course be improved further. | ||
Revision as of 04:36, 16 March 2007
Quartic NLS on
- 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 = -1/6\,} .
- For any quartic non-linearity one can obtain 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 \ge 0\,}
CaWe1990
- Below 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 L^2\,} we have ill-posedness by Gallilean invariance considerations in both the focusing KnPoVe-p and defocusing CtCoTa-p2 cases.
- If the quartic non-linearity is of type then one can obtain 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 > -1/6\,.} 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 |u|^4\,} one has 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 > -1/8\,} , while for the other three types 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 u^4\,} , 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 u u u \underline{u}\,} , or 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 u \underline{uuu}\,} one has 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 > -1/6\,} Gr-p2.
- In the Hamiltonian case (a non-linearity of type 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 |u|^3 u\,} ) we have 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 \ge 0\,} by 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 L^2\,} conservation. In the other cases it is not clear whether there is any reasonable GWP result, except possibly for very small data.
Quartic NLS on 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 T}
- For any quartic non-linearity one has 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>0\,} Bo1993.
- If the quartic non-linearity is 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 \underline{uuuu}\,} type then one can obtain 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 > -1/6\,} , Gr-p2.
- If the nonlinearity is 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 |u|^3 u\,} type one has GWP for random data whose Fourier coefficients decay like 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 1/|k|\,} (times a Gaussian random variable) Bo1995c. Indeed one has an invariant measure.
Quartic NLS on 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 R^2}
- 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 = 1/3\,.}
- For any quartic non-linearity one can obtain 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 \ge s_c\,}
CaWe1990.
- 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<s_c\,} we have ill-posedness, indeed the 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^s\,} norm can get arbitrarily large arbitrarily quickly CtCoTa-p2. In the focusing case we have instantaneous blowup from the virial identity and scaling.
- In the Hamiltonian case (a non-linearity of type 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 |u|^3 u\,}
) we have 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 \ge 1\,}
Ka1986.
- This has been improved to 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 > 1-e\,} in CoKeStTkTa2003c in the defocusing Hamiltonian case. This result can of course be improved further.
- Scattering in the energy space Na1999c in the defocusing Hamiltonian case.
- One also has GWP and scattering for small 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^{1/3}\,} data for any quintic non-linearity.
