Quartic NLS: Difference between revisions

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====Quartic NLS on <math>R</math>====
====Quartic NLS on <math>R</math>====


* Scaling is s<sub>c</sub> = -1/6.
* Scaling is <math>s_c = -1/6\,</math>.
* For any quartic non-linearity one can obtain LWP for s <font face="Symbol">³</font> 0 [[Bibliography#CaWe1990|CaWe1990]]
* For any quartic non-linearity one can obtain LWP for <math>s \ge 0\,</math> [[CaWe1990]]
** Below L^2 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 <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]].
* 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 |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.
* 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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====Quartic NLS on <math>T</math>====
====Quartic NLS on <math>T</math>====


* For any quartic non-linearity one has LWP for s>0 [[Bibliography#Bo1993|Bo1993]].
* For any quartic non-linearity one has LWP for <math>s>0\,</math> [[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 quartic non-linearity is of <math>\underline{uuuu}\,</math> type then one can obtain LWP for <math>s > -1/6\,</math>, [[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) [[Bibliography#Bo1995c|Bo1995c]]. Indeed one has an invariant measure.
* If the nonlinearity is of <math>|u|^3 u\,</math> type one has GWP for random data whose Fourier coefficients decay like <math>1/|k|\,</math> (times a Gaussian random variable) [[Bo1995c]]. Indeed one has an invariant measure.
 
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====Quartic NLS on <math>R^2</math>====
====Quartic NLS on <math>R^2</math>====


* Scaling is s<sub>c</sub> = 1/3.
* Scaling is <math>s_c = 1/3\,.</math>
* For any quartic non-linearity one can obtain LWP for s <font face="Symbol">³</font> s<sub>c</sub> [[Bibliography#CaWe1990|CaWe1990]].
* For any quartic non-linearity one can obtain LWP for <math>s \ge s_c\,</math> [[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.
** 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 |u|^3 u) we have GWP for s <font face="Symbol">³</font> 1 [[Bibliography#Ka1986|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 s > 1-<font face="Symbol">e</font> in [[Bibliography#CoKeStTkiTa2003c|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.
** Scattering in the energy space [[Bibliography#Na1999c|Na1999c]] in the defocusing Hamiltonian case.
** Scattering in the energy space [[Na1999c]] in the defocusing Hamiltonian case.
** One also has GWP and scattering for small H^{1/3} data for any quintic non-linearity.
** One also has GWP and scattering for small <math>H^{1/3}\,</math> data for any quintic non-linearity.


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Latest revision as of 06:25, 21 July 2007

Quartic NLS on

  • Scaling is .
  • For any quartic non-linearity one can obtain LWP for CaWe1990
    • Below 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 For one has LWP for , while for the other three types , , or one has LWP for Gr-p2.
  • In the Hamiltonian case (a non-linearity of type ) we have GWP for by 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

  • For any quartic non-linearity one has LWP for Bo1993.
  • If the quartic non-linearity is of type then one can obtain LWP for , Gr-p2.
  • If the nonlinearity is of type one has GWP for random data whose Fourier coefficients decay like (times a Gaussian random variable) Bo1995c. Indeed one has an invariant measure.

Quartic NLS on

  • Scaling is
  • For any quartic non-linearity one can obtain LWP for CaWe1990.
    • For we have ill-posedness, indeed the 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 ) we have GWP for Ka1986.
    • This has been improved to 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 data for any quintic non-linearity.