Quintic NLS: Difference between revisions
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** Explicit blowup solutions (with large <math>L^2\,</math> norm) are known in the focussing case [[Bibliography#BirKnPoSvVe1996|BirKnPoSvVe1996]]. The blowup rate in <math>H^1\,</math> is <math>t^{-1}\,</math> in these solutions. This is not the optimal blowup rate; in fact an example has been constructed where the blowup rate is <math>|t|^{-1/2} (log log|t|)^{1/2}\,</math>[Per-p]. Furthermore, one always this blowup behavior (or possibly slower, though one must still blow up by at least <math>|t|^{-1/2}\,</math>) whenever the energy is negative [MeRap-p], [MeRap-p2], and one either assumes that the mass is close to the critical mass or that <math>xu\,</math> is in <math>L^2\,</math>. | ** Explicit blowup solutions (with large <math>L^2\,</math> norm) are known in the focussing case [[Bibliography#BirKnPoSvVe1996|BirKnPoSvVe1996]]. The blowup rate in <math>H^1\,</math> is <math>t^{-1}\,</math> in these solutions. This is not the optimal blowup rate; in fact an example has been constructed where the blowup rate is <math>|t|^{-1/2} (log log|t|)^{1/2}\,</math>[Per-p]. Furthermore, one always this blowup behavior (or possibly slower, though one must still blow up by at least <math>|t|^{-1/2}\,</math>) whenever the energy is negative [MeRap-p], [MeRap-p2], and one either assumes that the mass is close to the critical mass or that <math>xu\,</math> is in <math>L^2\,</math>. | ||
*** 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]]. | *** 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'' | * ''Remark'': This equation is pseudo-conformally invariant. GWP results in <math>H^s\,</math> automatically transfer to GWP and scattering results in <math>L^2(|x|^s)\,</math> thanks to the pseudo-conformal transformation. | ||
* Solitons are <math>H^1\,</math>-unstable. | * Solitons are <math>H^1\,</math>-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 | * Scaling is <math>s_c = 0\,</math>. | ||
* LWP is known for s > 0 [[Bibliography#Bo1993|Bo1993]]. | * LWP is known for <math>s > 0\,</math> [[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 <math>s < 0\,</math> the solution map is not uniformly continuous from <math>C^k\,</math> to <math>C^{-k}\,</math> for any <math>k\,</math> [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 <math>s > 4/9\,</math> (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 <math>s > 2/3\,</math> this is commented upon in [Bo-p2] and is a minor modification of [CoKeStTkTa-p]. | ||
** For s > | ** For <math>s \ge 1\,</math> one has GWP in the defocusing case, or in the focusing case with small <math>L^2\,</math> 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) [[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 | *** In the defocusing case one has GWP for random data whose Fourier coefficients decay like <math>1/|k|\,</math> (times a Gaussian random variable) [[Bibliography#Bo1995c|Bo1995c]]; this is roughly of the regularity of <math>H^{1/2}\,</math>. Indeed one has an invariant measure. In the focusing case the same result holds assuming the <math>L^2\,</math> norm is sufficiently small. | ||
<div class="MsoNormal" style="text-align: center"><center> | <div class="MsoNormal" style="text-align: center"><center> | ||
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====Quintic NLS on <math>R^2</math>==== | ====Quintic NLS on <math>R^2</math>==== | ||
* Scaling is | * Scaling is <math>s_c = 1/2\,</math>. | ||
* LWP is known for | * LWP is known for <math>s \ge 1/2\,</math> [[Bibliography#CaWe1990|CaWe1990]]. | ||
** For s=1/2 the time of existence depends on the profile of the data as well as the norm. | ** For <math>s=1/2\,</math> 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 <math>s<s_c\,</math> 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 | * GWP for <math>s \ge 1\,</math> by Hamiltonian conservation. | ||
** This has been improved to s > 1- | ** This has been improved to <math>s > 1-e\,</math>- in [[Bibliography#CoKeStTkTa2003b|CoKeStTkTa2003b]]. This result can of course be improved further. | ||
** Scattering in the energy space [[Bibliography#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 <math>H^{1/2}\,</math> data for any quintic non-linearity. | ||
<div class="MsoNormal" style="text-align: center"><center> | <div class="MsoNormal" style="text-align: center"><center> | ||
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====Quintic NLS on <math>R^3</math>==== | ====Quintic NLS on <math>R^3</math>==== | ||
* Scaling is | * Scaling is <math>s_c = 1\,</math>. | ||
* LWP is known for | * LWP is known for <math>s \ge 1\,</math> [[Bibliography#CaWe1990|CaWe1990]]. | ||
** For s=1 the time of existence depends on the profile of the data as well as the norm. | ** For <math>s=1\,</math> 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 <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. | ||
* GWP and scattering for | * GWP and scattering for <math>s\ge 1\,</math> in the defocusing case [CoKeStTkTa-p] | ||
** For radial data this is in [Bo-p], [[Bibliography#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 13:59, 3 August 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 , thus this is an [#L^2-critical_NLS critical NLS].
- LWP is known for CaWe1990, Ts1987.
- For the time of existence depends on the profile of the data as well as the norm.
- Below we have ill-posedness by Gallilean invariance considerations in both the focusing [KnPoVe-p] and defocusing [CtCoTa-p2] cases.
- GWP for in the defocussing case [Tzi-p]
- For this is in CoKeStTkTa-p6
- For this is in CoKeStTkTa-p4.
- For this is implicit in Tk-p.
- For this follows from LWP and Hamiltonian conservation.
- One has GWP and scattering for small data for any quintic non-linearity. The corresponding problem for large 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 norm in spacetime.
- Explicit blowup solutions (with large norm) are known in the focussing case BirKnPoSvVe1996. The blowup rate in is in these solutions. This is not the optimal blowup rate; in fact an example has been constructed where the blowup rate is [Per-p]. Furthermore, one always this blowup behavior (or possibly slower, though one must still blow up by at least ) whenever the energy is negative [MeRap-p], [MeRap-p2], and one either assumes that the mass is close to the critical mass or that is in .
- 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 automatically transfer to GWP and scattering results in thanks to the pseudo-conformal transformation.
- Solitons are -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 .
- LWP is known for Bo1993.
- For the solution map is not uniformly continuous from to for any [CtCoTa-p3].
- GWP is known in the defocusing case for (De Silva, Pavlovic, Staffilani, Tzirakis)
- For this is commented upon in [Bo-p2] and is a minor modification of [CoKeStTkTa-p].
- For one has GWP in the defocusing case, or in the focusing case with small norm, by Hamiltonian conservation.
- In the defocusing case one has GWP for random data whose Fourier coefficients decay like (times a Gaussian random variable) Bo1995c; this is roughly of the regularity of . Indeed one has an invariant measure. In the focusing case the same result holds assuming the norm is sufficiently small.
Quintic NLS on
- Scaling is .
- LWP is known for CaWe1990.
- For the time of existence depends on the profile of the data as well as the norm.
- For 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 by Hamiltonian conservation.
- This has been improved to - in CoKeStTkTa2003b. This result can of course be improved further.
- Scattering in the energy space Na1999c
- One also has GWP and scattering for small data for any quintic non-linearity.
Quintic NLS on
- Scaling is .
- LWP is known for CaWe1990.
- For the time of existence depends on the profile of the data as well as the norm.
- 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.
- GWP and scattering for 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.