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| ====Quartic NLS on <math>R</math>====
| | #REDIRECT [[Quartic NLS]] |
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| * Scaling is s<sub>c</sub> = -1/6.
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| * For any quartic non-linearity one can obtain LWP for s <font face="Symbol">³</font> 0 [[Bibliography#CaWe1990|CaWe1990]]
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| ** Below L^2 we have ill-posedness by Gallilean invariance considerations in both the focusing [KnPoVe-p] and defocusing [CtCoTa-p2] cases.
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| * 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]].
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| * 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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| ----
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| </center></div>
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| [[Category:Equations]] | |
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| ====Quartic NLS on <math>T</math>====
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| * For any quartic non-linearity one has LWP for s>0 [[Bibliography#Bo1993|Bo1993]].
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| * 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]].
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| * 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) [[Bibliography#Bo1995c|Bo1995c]]. Indeed one has an invariant measure.
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| ----
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| </center></div>
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| [[Category:Equations]]
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| ====Quartic NLS on <math>R^2</math>====
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| * Scaling is s<sub>c</sub> = 1/3.
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| * For any quartic non-linearity one can obtain LWP for s <font face="Symbol">³</font> s<sub>c</sub> [[Bibliography#CaWe1990|CaWe1990]].
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| ** 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.
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| * In the Hamiltonian case (a non-linearity of type |u|^3 u) we have GWP for s <font face="Symbol">³</font> 1 [[Bibliography#Ka1986|Ka1986]].
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| ** This has been improved to s > 1-<font face="Symbol">e</font> in [[Bibliography#CoKeStTkiTa2003c|CoKeStTkTa2003c]] in the defocusing Hamiltonian case. This result can of course be improved further.
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| ** Scattering in the energy space [[Bibliography#Na1999c|Na1999c]] in the defocusing Hamiltonian case.
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| ** One also has GWP and scattering for small H^{1/3} data for any quintic non-linearity.
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| ----
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| </center></div>
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| [[Category:Equations]]
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