Difference between revisions of "Schrodinger:quintic NLS"

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====Quintic NLS on <math>R</math>====
 
====Quintic NLS on <math>R</math>====
  
* This equation may be viewed as a simpler version of [#dnls-3_on_R cubic DNLS], and is always at least as well-behaved. It has been proposed as a modifiation of the Gross-Pitaevski approximation for low-dimesional Bose liquids [[references:KolNewStrQi2000 KolNewStrQi2000]]
+
* This equation may be viewed as a simpler version of [#dnls-3_on_R cubic DNLS], and is always at least as well-behaved. It has been proposed as a modifiation of the Gross-Pitaevski approximation for low-dimesional Bose liquids [[Bibliography#KolNewStrQi2000|KolNewStrQi2000]]
 
* Scaling is s<sub>c</sub> = 0, thus this is an [#L^2-critical_NLS L^2 critical NLS].
 
* Scaling is s<sub>c</sub> = 0, thus this is an [#L^2-critical_NLS L^2 critical NLS].
* LWP is known for s <font face="Symbol">³</font> 0 [[references:CaWe1990 CaWe1990]], [[references:Ts1987 Ts1987]].
+
* LWP is known for s <font face="Symbol">³</font> 0 [[Bibliography#CaWe1990|CaWe1990]], [[Bibliography#Ts1987|Ts1987]].
 
** For s=0 the time of existence depends on the profile of the data as well as the norm.
 
** For s=0 the time of existence depends on the profile of the data as well as the norm.
 
** Below L^2 we have ill-posedness by Gallilean invariance considerations in both the focusing [KnPoVe-p] and defocusing [CtCoTa-p2] cases.
 
** 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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** For s<font face="Symbol">³</font> 1 this follows from LWP and Hamiltonian conservation.
 
** For s<font face="Symbol">³</font> 1 this follows from LWP and Hamiltonian conservation.
 
** One has GWP and scattering for small L<sup>2</sup> data for any quintic non-linearity. The corresponding problem for large L<sup>2</sup> data and defocussing nonlinearity is very interesting, but probably very difficult, perhaps only marginally easier than the corresponding problem for the [#Cubic_NLS_on_R^2 2D cubic NLS]. It would suffice to show that the solution has a bounded L<sup>6</sup> norm in spacetime.
 
** One has GWP and scattering for small L<sup>2</sup> data for any quintic non-linearity. The corresponding problem for large L<sup>2</sup> data and defocussing nonlinearity is very interesting, but probably very difficult, perhaps only marginally easier than the corresponding problem for the [#Cubic_NLS_on_R^2 2D cubic NLS]. It would suffice to show that the solution has a bounded L<sup>6</sup> norm in spacetime.
** Explicit blowup solutions (with large L<sup>2</sup> norm) are known in the focussing case [[references:BirKnPoSvVe1996 BirKnPoSvVe1996]]. The blowup rate in H<sup>1</sup> is t<sup>-1</sup> in these solutions. This is not the optimal blowup rate; in fact an example has been constructed where the blowup rate is |t|^{-1/2} (log log|t|)^{1/2}[Per-p]. Furthermore, one always this blowup behavior (or possibly slower, though one must still blow up by at least |t|^{-1/2}) whenever the energy is negative [MeRap-p], [MeRap-p2], and one either assumes that the mass is close to the critical mass or that xu is in L^2.
+
** Explicit blowup solutions (with large L<sup>2</sup> norm) are known in the focussing case [[Bibliography#BirKnPoSvVe1996|BirKnPoSvVe1996]]. The blowup rate in H<sup>1</sup> is t<sup>-1</sup> in these solutions. This is not the optimal blowup rate; in fact an example has been constructed where the blowup rate is |t|^{-1/2} (log log|t|)^{1/2}[Per-p]. Furthermore, one always this blowup behavior (or possibly slower, though one must still blow up by at least |t|^{-1/2}) whenever the energy is negative [MeRap-p], [MeRap-p2], and one either assumes that the mass is close to the critical mass or that xu is in L^2.
*** One can modify the explicit solutions from [[references:BirKnPoSvVe1996 BirKnPoSvVe1996]] and in fact create solutions which blow up at any collection of specified points in spacetime [[references:BoWg1997 BoWg1997]], [[references:Nw1998 Nw1998]].
+
*** One can modify the explicit solutions from [[Bibliography#BirKnPoSvVe1996|BirKnPoSvVe1996]] and in fact create solutions which blow up at any collection of specified points in spacetime [[Bibliography#BoWg1997|BoWg1997]], [[Bibliography#Nw1998|Nw1998]].
 
* ''Remark''<nowiki>: This equation is pseudo-conformally invariant. GWP results in H</nowiki><sup>s</sup> automatically transfer to GWP and scattering results in L<sup>2</sup>(|x|<sup>s</sup>) thanks to the pseudo-conformal transformation.
 
* ''Remark''<nowiki>: This equation is pseudo-conformally invariant. GWP results in H</nowiki><sup>s</sup> automatically transfer to GWP and scattering results in L<sup>2</sup>(|x|<sup>s</sup>) thanks to the pseudo-conformal transformation.
 
* Solitons are H<sup>1</sup>-unstable.
 
* Solitons are H<sup>1</sup>-unstable.
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* This equation may be viewed as a simpler version of cubic DNLS, and is always at least as well-behaved.
 
* This equation may be viewed as a simpler version of cubic DNLS, and is always at least as well-behaved.
 
* Scaling is s<sub>c</sub> = 0.
 
* Scaling is s<sub>c</sub> = 0.
* LWP is known for s > 0 [[references:Bo1993 Bo1993]].
+
* LWP is known for s > 0 [[Bibliography#Bo1993|Bo1993]].
 
** For s < 0 the solution map is not uniformly continuous from C^k to C^{-k} for any k [CtCoTa-p3].
 
** For s < 0 the solution map is not uniformly continuous from C^k to C^{-k} for any k [CtCoTa-p3].
 
* GWP is known in the defocusing case for s > 4/9 (De Silva, Pavlovic, Staffilani, Tzirakis)
 
* GWP is known in the defocusing case for s > 4/9 (De Silva, Pavlovic, Staffilani, Tzirakis)
 
** For s > 2/3 this is commented upon in [Bo-p2] and is a minor modification of [CoKeStTkTa-p].
 
** For s > 2/3 this is commented upon in [Bo-p2] and is a minor modification of [CoKeStTkTa-p].
 
** For s >= 1 one has GWP in the defocusing case, or in the focusing case with small L^2 norm, by Hamiltonian conservation.
 
** For s >= 1 one has GWP in the defocusing case, or in the focusing case with small L^2 norm, by Hamiltonian conservation.
*** In the defocusing case one has GWP for random data whose Fourier coefficients decay like 1/|k| (times a Gaussian random variable) [[references:Bo1995c Bo1995c]]; this is roughly of the regularity of H^{1/2}. Indeed one has an invariant measure. In the focusing case the same result holds assuming the L<sup>2</sup> norm is sufficiently small.
+
*** In the defocusing case one has GWP for random data whose Fourier coefficients decay like 1/|k| (times a Gaussian random variable) [[Bibliography#Bo1995c|Bo1995c]]; this is roughly of the regularity of H^{1/2}. Indeed one has an invariant measure. In the focusing case the same result holds assuming the L<sup>2</sup> norm is sufficiently small.
  
 
<div class="MsoNormal" style="text-align: center"><center>
 
<div class="MsoNormal" style="text-align: center"><center>
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* Scaling is s<sub>c</sub> = 1/2.
 
* Scaling is s<sub>c</sub> = 1/2.
* LWP is known for s <font face="Symbol">³</font> 1/2 [[references:CaWe1990 CaWe1990]].
+
* LWP is known for s <font face="Symbol">³</font> 1/2 [[Bibliography#CaWe1990|CaWe1990]].
 
** For s=1/2 the time of existence depends on the profile of the data as well as the norm.
 
** For s=1/2 the time of existence depends on the profile of the data as well as the norm.
 
** 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 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.
 
* GWP for s <font face="Symbol">³</font> 1 by Hamiltonian conservation.
 
* GWP for s <font face="Symbol">³</font> 1 by Hamiltonian conservation.
** This has been improved to s > 1-<font face="Symbol">e</font> in [[references:CoKeStTkTa2003b CoKeStTkTa2003b]]. This result can of course be improved further.
+
** This has been improved to s > 1-<font face="Symbol">e</font> in [[Bibliography#CoKeStTkTa2003b|CoKeStTkTa2003b]]. This result can of course be improved further.
** Scattering in the energy space [[references:Na1999c Na1999c]]
+
** Scattering in the energy space [[Bibliography#Na1999c|Na1999c]]
 
** One also has GWP and scattering for small H^{1/2} data for any quintic non-linearity.
 
** One also has GWP and scattering for small H^{1/2} data for any quintic non-linearity.
  
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* Scaling is s<sub>c</sub> = 1.
 
* Scaling is s<sub>c</sub> = 1.
* LWP is known for s <font face="Symbol">³</font> 1 [[references:CaWe1990 CaWe1990]].
+
* LWP is known for s <font face="Symbol">³</font> 1 [[Bibliography#CaWe1990|CaWe1990]].
 
** For s=1 the time of existence depends on the profile of the data as well as the norm.
 
** For s=1 the time of existence depends on the profile of the data as well as the norm.
 
** 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 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.
 
* GWP and scattering for s<font face="Symbol">³</font>1 in the defocusing case [CoKeStTkTa-p]
 
* GWP and scattering for s<font face="Symbol">³</font>1 in the defocusing case [CoKeStTkTa-p]
** For radial data this is in [Bo-p], [[references:Bo1999 Bo1999]].
+
** For radial data this is in [Bo-p], [[Bibliography#Bo1999|Bo1999]].
 
** Blowup can occur in the focussing case from Glassey's virial identity.
 
** Blowup can occur in the focussing case from Glassey's virial identity.
  

Revision as of 20:11, 28 July 2006

Quintic NLS on

  • This equation may be viewed as a simpler version of [#dnls-3_on_R cubic DNLS], and is always at least as well-behaved. It has been proposed as a modifiation of the Gross-Pitaevski approximation for low-dimesional Bose liquids KolNewStrQi2000
  • Scaling is sc = 0, thus this is an [#L^2-critical_NLS L^2 critical NLS].
  • LWP is known for s ³ 0 CaWe1990, Ts1987.
    • For s=0 the time of existence depends on the profile of the data as well as the norm.
    • Below L^2 we have ill-posedness by Gallilean invariance considerations in both the focusing [KnPoVe-p] and defocusing [CtCoTa-p2] cases.
  • GWP for s>4/9 in the defocussing case [Tzi-p]
    • For s>1/2 this is in references:CoKeStTkTa-p6 CoKeStTkTa-p6
    • For s>2/3 this is in references:CoKeStTkTa-p4 CoKeStTkTa-p4.
    • For s > 32/33 this is implicit in references:Tk-p Tk-p.
    • For s³ 1 this follows from LWP and Hamiltonian conservation.
    • One has GWP and scattering for small L2 data for any quintic non-linearity. The corresponding problem for large L2 data and defocussing nonlinearity is very interesting, but probably very difficult, perhaps only marginally easier than the corresponding problem for the [#Cubic_NLS_on_R^2 2D cubic NLS]. It would suffice to show that the solution has a bounded L6 norm in spacetime.
    • Explicit blowup solutions (with large L2 norm) are known in the focussing case BirKnPoSvVe1996. The blowup rate in H1 is t-1 in these solutions. This is not the optimal blowup rate; in fact an example has been constructed where the blowup rate is |t|^{-1/2} (log log|t|)^{1/2}[Per-p]. Furthermore, one always this blowup behavior (or possibly slower, though one must still blow up by at least |t|^{-1/2}) whenever the energy is negative [MeRap-p], [MeRap-p2], and one either assumes that the mass is close to the critical mass or that xu is in L^2.
      • One can modify the explicit solutions from BirKnPoSvVe1996 and in fact create solutions which blow up at any collection of specified points in spacetime BoWg1997, Nw1998.
  • Remark: This equation is pseudo-conformally invariant. GWP results in Hs automatically transfer to GWP and scattering results in L2(|x|s) thanks to the pseudo-conformal transformation.
  • Solitons are H1-unstable.

Quintic NLS on

  • This equation may be viewed as a simpler version of cubic DNLS, and is always at least as well-behaved.
  • Scaling is sc = 0.
  • LWP is known for s > 0 Bo1993.
    • For s < 0 the solution map is not uniformly continuous from C^k to C^{-k} for any k [CtCoTa-p3].
  • GWP is known in the defocusing case for s > 4/9 (De Silva, Pavlovic, Staffilani, Tzirakis)
    • For s > 2/3 this is commented upon in [Bo-p2] and is a minor modification of [CoKeStTkTa-p].
    • For s >= 1 one has GWP in the defocusing case, or in the focusing case with small L^2 norm, by Hamiltonian conservation.
      • In the defocusing case one has GWP for random data whose Fourier coefficients decay like 1/|k| (times a Gaussian random variable) Bo1995c; this is roughly of the regularity of H^{1/2}. Indeed one has an invariant measure. In the focusing case the same result holds assuming the L2 norm is sufficiently small.


Quintic NLS on

  • Scaling is sc = 1/2.
  • LWP is known for s ³ 1/2 CaWe1990.
    • For s=1/2 the time of existence depends on the profile of the data as well as the norm.
    • 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.
  • GWP for s ³ 1 by Hamiltonian conservation.
    • This has been improved to s > 1-e in CoKeStTkTa2003b. This result can of course be improved further.
    • Scattering in the energy space Na1999c
    • One also has GWP and scattering for small H^{1/2} data for any quintic non-linearity.

Quintic NLS on

  • Scaling is sc = 1.
  • LWP is known for s ³ 1 CaWe1990.
    • For s=1 the time of existence depends on the profile of the data as well as the norm.
    • 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.
  • GWP and scattering for s³1 in the defocusing case [CoKeStTkTa-p]
    • For radial data this is in [Bo-p], Bo1999.
    • Blowup can occur in the focussing case from Glassey's virial identity.