Schrodinger:quartic NLS: Difference between revisions

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#REDIRECT [[Quartic NLS]]
====Quartic NLS on <math>R</math>====
 
* Scaling is s<sub>c</sub> = -1/6.
* For any quartic non-linearity one can obtain LWP for s <font face="Symbol">³</font> 0 [[references:CaWe1990 CaWe1990]]
** Below 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 <u>u</u> <u>u</u> <u>u</u> <u>u</u> type then one can obtain LWP for s > -1/6. For |u|<sup>4</sup> one has LWP for s > -1/8, while for the other three types u<sup>4</sup>, u u u <u>u</u>, or u <u>uuu</u> one has LWP for s > -1/6 [[references#Gr-p2 Gr-p2]].
* In the Hamiltonian case (a non-linearity of type |u|^3 u) we have GWP for s <font face="Symbol">³</font> 0 by L<sup>2</sup> 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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====Quartic NLS on <math>T</math>====
 
* For any quartic non-linearity one has LWP for s>0 [[references:Bo1993 Bo1993]].
* If the quartic non-linearity is of <u>u</u> <u>u</u> <u>u</u> <u>u</u> type then one can obtain LWP for s > -1/6 [[references#Gr-p2 Gr-p2]].
* If the nonlinearity is of |u|<sup>3</sup> u type one has GWP for random data whose Fourier coefficients decay like 1/|k| (times a Gaussian random variable) [[references:Bo1995c Bo1995c]]. Indeed one has an invariant measure.
 
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====Quartic NLS on <math>R^2</math>====
 
* Scaling is s<sub>c</sub> = 1/3.
* For any quartic non-linearity one can obtain LWP for s <font face="Symbol">³</font> s<sub>c</sub> [[references:CaWe1990 CaWe1990]].
** For s<s_c we have ill-posedness, indeed the 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 |u|^3 u) we have GWP for s <font face="Symbol">³</font> 1 [[references:Ka1986 Ka1986]].
** This has been improved to s > 1-<font face="Symbol">e</font> in [[references:CoKeStTkiTa2003c CoKeStTkTa2003c]] in the defocusing Hamiltonian case. This result can of course be improved further.
** Scattering in the energy space [[references:Na1999c Na1999c]] in the defocusing Hamiltonian case.
** One also has GWP and scattering for small H^{1/3} data for any quintic non-linearity.
 
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Latest revision as of 16:53, 30 July 2006

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