Quadratic NLS: Difference between revisions

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* Scaling is <math>s_c=-3/2\,.</math>  
* Scaling is <math>s_c=-3/2\,.</math>  
* For any quadratic non-linearity one can obtain LWP for <math>s>=0\,</math> [[Bibliography#CaWe1990|CaWe1990]], [[Bibliography#Ts1987|Ts1987]].
* For any quadratic non-linearity one can obtain LWP for <math>s \ge 0\,</math> [[Bibliography#CaWe1990|CaWe1990]], [[Bibliography#Ts1987|Ts1987]].
* If the quadratic non-linearity is of <math>\underline{uu}\,</math> or <math>uu\,</math> type then one can push LWP to <math>s > -3/4.\,</math> [[Bibliography#KnPoVe1996b|KnPoVe1996b]].
* If the quadratic non-linearity is of <math>\underline{uu}\,</math> or <math>uu\,</math> type then one can push LWP to <math>s > -3/4.\,</math> [[Bibliography#KnPoVe1996b|KnPoVe1996b]].
** This can be improved to the Besov space <math>B^{-3/4}_{2,1}\,</math> [MurTao-p]. The <math>X^{s,b}\,</math> bilinear estimates fail for <math>H^{-3/4}\,</math> [[references:NaTkTs-p NaTkTs2001]].
** This can be improved to the Besov space <math>B^{-3/4}_{2,1}\,</math> [MurTao-p]. The <math>X^{s,b}\,</math> bilinear estimates fail for <math>H^{-3/4}\,</math> [[references:NaTkTs-p NaTkTs2001]].
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====Quadratic NLS on <math>T</math>====
====Quadratic NLS on <math>T</math \ge ===


* For any quadratic non-linearity one can obtain LWP for <math>s>=0\,</math> [[Bibliography#Bo1993|Bo1993]]. In the Hamiltonian case (<math>|u| u\,</math>) this is sharp by Gallilean invariance considerations [KnPoVe-p]
* For any quadratic non-linearity one can obtain LWP for <math>s \ge 0\,</math> [[Bibliography#Bo1993|Bo1993]]. In the Hamiltonian case (<math>|u| u\,</math>) this is sharp by Gallilean invariance considerations [KnPoVe-p]
* If the quadratic non-linearity is of <math>\underline{uu}\,</math> or <math>uu\,</math> type then one can push LWP to <math>s > -1/2.\,</math> [[Bibliography#KnPoVe1996b|KnPoVe1996b]].
* If the quadratic non-linearity is of <math>\underline{uu}\,</math> or <math>uu\,</math> type then one can push LWP to <math>s > -1/2.\,</math> [[Bibliography#KnPoVe1996b|KnPoVe1996b]].
* In the Hamiltonian case (a non-linearity of type <math>|u| u\,</math>) we have GWP for <math>s>=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| 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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[[Category:Equations]]
[[Category:Equations]]


====Quadratic NLS on <math>R^2</math>====
====Quadratic NLS on <math>R^2</math \ge ===


* Scaling <math>s_c = -1.\,</math>
* Scaling <math>s_c = -1.\,</math>
* For any quadratic non-linearity one can obtain LWP for <math>s>=0\,</math> [[Bibliography#CaWe1990|CaWe1990]], [[Bibliography#Ts1987|Ts1987]].
* For any quadratic non-linearity one can obtain LWP for <math>s \ge 0\,</math> [[Bibliography#CaWe1990|CaWe1990]], [[Bibliography#Ts1987|Ts1987]].
** In the Hamiltonian case (<math>|u| u\,</math>) this is sharp by Gallilean invariance considerations [KnPoVe-p]
** In the Hamiltonian case (<math>|u| u\,</math>) this is sharp by Gallilean invariance considerations [KnPoVe-p]
* If the quadratic non-linearity is of <math>\underline{uu}\,</math> or <math>u u\,</math> type then one can push LWP to <math>s > -3/4.\,</math> [[Bibliography#St1997|St1997]], [[references:CoDeKnSt-p CoDeKnSt-p]].
* If the quadratic non-linearity is of <math>\underline{uu}\,</math> or <math>u u\,</math> type then one can push LWP to <math>s > -3/4.\,</math> [[Bibliography#St1997|St1997]], [[references:CoDeKnSt-p CoDeKnSt-p]].
** This can be improved to the Besov space <math>B^{-3/4}_{2,1}\,</math> [MurTao-p].
** This can be improved to the Besov space <math>B^{-3/4}_{2,1}\,</math> [MurTao-p].
* If the quadratic non-linearity is of <math>u \underline{u}\,</math> type then one can push LWP to <math>s > -1/4.\,</math> [[references:Ta-p2 Ta-p2]].
* If the quadratic non-linearity is of <math>u \underline{u}\,</math> type then one can push LWP to <math>s > -1/4.\,</math> [[references:Ta-p2 Ta-p2]].
* In the Hamiltonian case (a non-linearity of type <math>|u| u\,</math>) we have GWP for <math>s>= 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| 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.
** 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.


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[[Category:Equations]]
[[Category:Equations]]


====Quadratic NLS on <math>R^3</math>====
====Quadratic NLS on <math>R^3</math \ge ===


* Scaling is <math>s_c = -1/2.\,</math>
* Scaling is <math>s_c = -1/2.\,</math>
* For any quadratic non-linearity one can obtain LWP for <math>s>= 0\,</math> [[Bibliography#CaWe1990|CaWe1990]], [[Bibliography#Ts1987|Ts1987]].
* For any quadratic non-linearity one can obtain LWP for <math>s \ge  0\,</math> [[Bibliography#CaWe1990|CaWe1990]], [[Bibliography#Ts1987|Ts1987]].
* If the quadratic non-linearity is of <math>\underline{uu}\,</math> or <math>u u\,</math> type then one can push LWP to <math>s > -1/2.\,</math> [[Bibliography#St1997|St1997]], [[references:CoDeKnSt-p CoDeKnSt-p]].
* If the quadratic non-linearity is of <math>\underline{uu}\,</math> or <math>u u\,</math> type then one can push LWP to <math>s > -1/2.\,</math> [[Bibliography#St1997|St1997]], [[references:CoDeKnSt-p CoDeKnSt-p]].
* If the quadratic non-linearity is of <math>u \underline{u}\,</math> type then one can push LWP to <math>s > -1/4.\,</math> [[references:Ta-p2 Ta-p2]].
* If the quadratic non-linearity is of <math>u \underline{u}\,</math> type then one can push LWP to <math>s > -1/4.\,</math> [[references:Ta-p2 Ta-p2]].
* In the Hamiltonian case (a non-linearity of type <math>|u| u\,</math>) we have GWP for <math>s >= 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| 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.
** 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.


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[[Category:Equations]]
[[Category:Equations]]


====Quadratic NLS on <math>T^3</math>====
====Quadratic NLS on <math>T^3</math \ge ===


* If the quadratic non-linearity is of <math>\underline{uu}\,</math> type then one can obtain LWP for <math>s > -3/10\,</math> [[references#Gr-p2 Gr-p2]]
* If the quadratic non-linearity is of <math>\underline{uu}\,</math> type then one can obtain LWP for <math>s > -3/10\,</math> [[references#Gr-p2 Gr-p2]]

Revision as of 22:36, 29 July 2006

Quadratic NLS

Equations of the form

which a quadratic function of its arguments are quadratic nonlinear Schrodinger equations.


Quadratic NLS on R

  • Scaling is
  • For any quadratic non-linearity one can obtain LWP for CaWe1990, Ts1987.
  • If the quadratic non-linearity is of or type then one can push LWP to KnPoVe1996b.
    • This can be improved to the Besov space [MurTao-p]. The bilinear estimates fail for references:NaTkTs-p NaTkTs2001.
  • If the quadratic non-linearity is of type then one can push LWP to KnPoVe1996b.
  • Since these equations do not have conservation it is not clear whether there is any reasonable GWP result, except possibly for very small data.
  • If the non-linearity is then there is GWP in thanks to conservation, and ill-posedness below by Gallilean invariance considerations in both the focusing [KnPoVe-p] and defocusing [CtCoTa-p2] cases.

====Quadratic NLS on Bo1993. In the Hamiltonian case () this is sharp by Gallilean invariance considerations [KnPoVe-p]

  • If the quadratic non-linearity is of or type then one can push LWP to KnPoVe1996b.
  • 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.

====Quadratic NLS on

  • For any quadratic non-linearity one can obtain LWP for CaWe1990, Ts1987.
    • In the Hamiltonian case () this is sharp by Gallilean invariance considerations [KnPoVe-p]
  • If the quadratic non-linearity is of or type then one can push LWP to St1997, references:CoDeKnSt-p CoDeKnSt-p.
    • This can be improved to the Besov space [MurTao-p].
  • If the quadratic non-linearity is of type then one can push LWP to references:Ta-p2 Ta-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.
    • Below we have ill-posedness by Gallilean invariance considerations in both the focusing [KnPoVe-p] and defocusing [CtCoTa-p2] cases.

Quadratic NLS on T^2

  • If the quadratic non-linearity is of type then one can obtain LWP for references#Gr-p2 Gr-p2

====Quadratic NLS on

  • For any quadratic non-linearity one can obtain LWP for CaWe1990, Ts1987.
  • If the quadratic non-linearity is of or type then one can push LWP to St1997, references:CoDeKnSt-p CoDeKnSt-p.
  • If the quadratic non-linearity is of type then one can push LWP to references:Ta-p2 Ta-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.
    • Below we have ill-posedness by Gallilean invariance considerations in both the focusing [KnPoVe-p] and defocusing [CtCoTa-p2] cases.

====Quadratic NLS on type then one can obtain LWP for references#Gr-p2 Gr-p2