Schrodinger:quadratic NLS: Difference between revisions

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===Quadratic NLS===
#REDIRECT [[Quadratic NLS]]
 
Equations of the form
 
<math> i \partial_t u + \Delta u = Q(u, \overline{u})</math>
 
which <math>Q(u, \overline{u})</math> a quadratic function of its arguments are [[Schrodinger:quadratic NLS|quadratic nonlinear Schrodinger equations]].
 
 
====Quadratic NLS on R====
 
* 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]].
* 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]].
* If the quadratic non-linearity is of  <math>\underline{u}u\,</math> type then one can push LWP to <math>s > -1/4.\,</math> [[Bibliography#KnPoVe1996b|KnPoVe1996b]].
* Since these equations do not have <math>L^2\,</math> conservation it is not clear whether there is any reasonable GWP result, except possibly for very small data.
* If the non-linearity is <math>|u|u\,</math> then there is GWP in <math>L^2\,</math>  thanks to <math>L^2\,</math> conservation, and ill-posedness below <math>L^2\,</math> by Gallilean invariance considerations in both the focusing [KnPoVe-p] and defocusing [CtCoTa-p2] cases.
 
[[Category:Equations]]
 
<div class="MsoNormal" style="text-align: center"><center>
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====Quadratic NLS on <math>T</math>====
 
* 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]
* 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.
 
<div class="MsoNormal" style="text-align: center"><center>
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</center></div>
 
[[Category:Equations]]
 
====Quadratic NLS on <math>R^2</math>====
 
* Scaling is s<sub>c</sub> = -1.
* For any quadratic non-linearity one can obtain LWP for s <font face="Symbol">³</font> 0 [[Bibliography#CaWe1990|CaWe1990]], [[Bibliography#Ts1987|Ts1987]].
** In the Hamiltonian case (|u| u) this is sharp by Gallilean invariance considerations [KnPoVe-p]
* If the quadratic non-linearity is of <u>u</u> <u>u</u> or u u type then one can push LWP to s > -3/4. [[Bibliography#St1997|St1997]], [[references:CoDeKnSt-p CoDeKnSt-p]].
** This can be improved to the Besov space B^{-3/4}_{2,1} [MurTao-p].
* If the quadratic non-linearity is of u <u>u</u> type then one can push LWP to s > -1/4. [[references:Ta-p2 Ta-p2]].
* In the Hamiltonian case (a non-linearity of type |u| 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.
** Below L^2 we have ill-posedness by Gallilean invariance considerations in both the focusing [KnPoVe-p] and defocusing [CtCoTa-p2] cases.
 
<div class="MsoNormal" style="text-align: center"><center>
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</center></div>
[[Category:Equations]]
 
 
====Quadratic NLS on T^2====
 
* If the quadratic non-linearity is of <u>u</u> <u>u</u> type then one can obtain LWP for s > -1/2 [[references#Gr-p2 Gr-p2]]
 
<div class="MsoNormal" style="text-align: center"><center>
----
</center></div>
 
[[Category:Equations]]
 
====Quadratic NLS on <math>R^3</math>====
 
* Scaling is s<sub>c</sub> = -1/2.
* For any quadratic non-linearity one can obtain LWP for s <font face="Symbol">³</font> 0 [[Bibliography#CaWe1990|CaWe1990]], [[Bibliography#Ts1987|Ts1987]].
* If the quadratic non-linearity is of <u>u</u> <u>u</u> or u u type then one can push LWP to s > -1/2. [[Bibliography#St1997|St1997]], [[references:CoDeKnSt-p CoDeKnSt-p]].
* If the quadratic non-linearity is of u <u>u</u> type then one can push LWP to s > -1/4. [[references:Ta-p2 Ta-p2]].
* In the Hamiltonian case (a non-linearity of type |u| 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.
** Below L^2 we have ill-posedness by Gallilean invariance considerations in both the focusing [KnPoVe-p] and defocusing [CtCoTa-p2] cases.
 
<div class="MsoNormal" style="text-align: center"><center>
----
</center></div>
[[Category:Equations]]
 
====Quadratic NLS on <math>T^3</math>====
 
* If the quadratic non-linearity is of <u>u</u> <u>u</u> type then one can obtain LWP for s > -3/10 [[references#Gr-p2 Gr-p2]]
 
<div class="MsoNormal" style="text-align: center"><center>
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</center></div>
[[Category:Equations]]

Latest revision as of 16:53, 30 July 2006

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