Difference between revisions of "Schrodinger:quadratic NLS"

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(Cleaning bibliographic references)
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* Scaling is s<sub>c</sub> = -3/2.
 
* Scaling is s<sub>c</sub> = -3/2.
* For any quadratic non-linearity one can obtain LWP for s <font face="Symbol">³</font> 0 [[references:CaWe1990 CaWe1990]], [[references:Ts1987 Ts1987]].
+
* 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 > -3/4. [[references:KnPoVe1996b KnPoVe1996b]].
+
* 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#KnPoVe1996b|KnPoVe1996b]].
 
** This can be improved to the Besov space B^{-3/4}_{2,1} [MurTao-p]. The X^{s,b} bilinear estimates fail for H^{-3/4} [[references:NaTkTs-p NaTkTs2001]].
 
** This can be improved to the Besov space B^{-3/4}_{2,1} [MurTao-p]. The X^{s,b} bilinear estimates fail for H^{-3/4} [[references:NaTkTs-p NaTkTs2001]].
* If the quadratic non-linearity is of u <u>u</u> type then one can push LWP to s > -1/4. [[references:KnPoVe1996b KnPoVe1996b]].
+
* If the quadratic non-linearity is of u <u>u</u> type then one can push LWP to s > -1/4. [[Bibliography#KnPoVe1996b|KnPoVe1996b]].
 
* Since these equations do not have L<sup>2</sup> conservation it is not clear whether there is any reasonable GWP result, except possibly for very small data.
 
* Since these equations do not have L<sup>2</sup> conservation it is not clear whether there is any reasonable GWP result, except possibly for very small data.
 
* If the non-linearity is |u|u then there is GWP in L<sup>2</sup> thanks to L<sup>2</sup> conservation, and ill-posedness below L<sup>2</sup> by Gallilean invariance considerations in both the focusing [KnPoVe-p] and defocusing [CtCoTa-p2] cases.
 
* If the non-linearity is |u|u then there is GWP in L<sup>2</sup> thanks to L<sup>2</sup> conservation, and ill-posedness below L<sup>2</sup> by Gallilean invariance considerations in both the focusing [KnPoVe-p] and defocusing [CtCoTa-p2] cases.
Line 26: Line 26:
 
====Quadratic NLS on <math>T</math>====
 
====Quadratic NLS on <math>T</math>====
  
* For any quadratic non-linearity one can obtain LWP for s <font face="Symbol">³</font> 0 [[references:Bo1993 Bo1993]]. In the Hamiltonian case (|u| u) this is sharp by Gallilean invariance considerations [KnPoVe-p]
+
* For any quadratic non-linearity one can obtain LWP for s <font face="Symbol">³</font> 0 [[Bibliography#Bo1993|Bo1993]]. 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 > -1/2. [[references:KnPoVe1996b KnPoVe1996b]].
+
* 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#KnPoVe1996b|KnPoVe1996b]].
 
* 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.
 
* 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.
  
Line 39: Line 39:
  
 
* Scaling is s<sub>c</sub> = -1.
 
* Scaling is s<sub>c</sub> = -1.
* For any quadratic non-linearity one can obtain LWP for s <font face="Symbol">³</font> 0 [[references:CaWe1990 CaWe1990]], [[references:Ts1987 Ts1987]].
+
* 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]
 
** 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. [[references:St1997 St1997]], [[references:CoDeKnSt-p CoDeKnSt-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].
 
** 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]].
 
* 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]].
Line 66: Line 66:
  
 
* Scaling is s<sub>c</sub> = -1/2.
 
* 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 [[references:CaWe1990 CaWe1990]], [[references:Ts1987 Ts1987]].
+
* 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. [[references:St1997 St1997]], [[references:CoDeKnSt-p CoDeKnSt-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 > -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]].
 
* 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.
 
* 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.

Revision as of 20:03, 28 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 sc = -3/2.
  • For any quadratic non-linearity one can obtain LWP for s ³ 0 CaWe1990, Ts1987.
  • If the quadratic non-linearity is of u u or u u type then one can push LWP to s > -3/4. KnPoVe1996b.
  • If the quadratic non-linearity is of u u type then one can push LWP to s > -1/4. KnPoVe1996b.
  • Since these equations do not have L2 conservation it is not clear whether there is any reasonable GWP result, except possibly for very small data.
  • If the non-linearity is |u|u then there is GWP in L2 thanks to L2 conservation, and ill-posedness below L2 by Gallilean invariance considerations in both the focusing [KnPoVe-p] and defocusing [CtCoTa-p2] cases.

Quadratic NLS on

  • For any quadratic non-linearity one can obtain LWP for s ³ 0 Bo1993. In the Hamiltonian case (|u| u) this is sharp by Gallilean invariance considerations [KnPoVe-p]
  • If the quadratic non-linearity is of u u or u u type then one can push LWP to s > -1/2. KnPoVe1996b.
  • In the Hamiltonian case (a non-linearity of type |u| u) we have GWP for s ³ 0 by L2 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

  • Scaling is sc = -1.
  • For any quadratic non-linearity one can obtain LWP for s ³ 0 CaWe1990, 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 or u u type then one can push LWP to s > -3/4. 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 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 ³ 0 by L2 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.


Quadratic NLS on T^2


Quadratic NLS on

  • Scaling is sc = -1/2.
  • For any quadratic non-linearity one can obtain LWP for s ³ 0 CaWe1990, Ts1987.
  • If the quadratic non-linearity is of u u or u u type then one can push LWP to s > -1/2. St1997, references:CoDeKnSt-p CoDeKnSt-p.
  • If the quadratic non-linearity is of 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 ³ 0 by L2 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.

Quadratic NLS on