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''<nowiki>: 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.
* ''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 s<sub>c</sub> = 0.
* 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 >= 1 one has GWP in the defocusing case, or in the focusing case with small L^2 norm, by Hamiltonian conservation.
** 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 L<sup>2</sup> norm is sufficiently small.
*** 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.


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====Quintic NLS on <math>R^2</math>====
====Quintic NLS on <math>R^2</math>====


* Scaling is s<sub>c</sub> = 1/2.
* Scaling is <math>s_c = 1/2\,</math>.
* LWP is known for s <font face="Symbol">³</font> 1/2 [[Bibliography#CaWe1990|CaWe1990]].
* 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 s <font face="Symbol">³</font> 1 by Hamiltonian conservation.
* GWP for <math>s \ge 1\,</math> by Hamiltonian conservation.
** This has been improved to s > 1-<font face="Symbol">e</font> in [[Bibliography#CoKeStTkTa2003b|CoKeStTkTa2003b]]. This result can of course be improved further.
** 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.


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====Quintic NLS on <math>R^3</math>====
====Quintic NLS on <math>R^3</math>====


* Scaling is s<sub>c</sub> = 1.
* Scaling is <math>s_c = 1\,</math>.
* LWP is known for s <font face="Symbol">³</font> 1 [[Bibliography#CaWe1990|CaWe1990]].
* 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 s<font face="Symbol">³</font>1 in the defocusing case [CoKeStTkTa-p]
* 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R}

  • 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s_c = 0\,} , thus this is an [#L^2-critical_NLS Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^2} critical NLS].
  • LWP is known for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s \ge 0\,} CaWe1990, Ts1987.
    • For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s=0\,} the time of existence depends on the profile of the data as well as the norm.
    • Below Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^2\,} we have ill-posedness by Gallilean invariance considerations in both the focusing [KnPoVe-p] and defocusing [CtCoTa-p2] cases.
  • GWP for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s>4/9\,} in the defocussing case [Tzi-p]
    • For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s>1/2\,} this is in CoKeStTkTa-p6
    • For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s>2/3\,} this is in CoKeStTkTa-p4.
    • For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s > 32/33\,} this is implicit in Tk-p.
    • For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s\ge 1\,} this follows from LWP and Hamiltonian conservation.
    • One has GWP and scattering for small Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^2\,} data for any quintic non-linearity. The corresponding problem for large Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^2\,} 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^6\,} norm in spacetime.
    • Explicit blowup solutions (with large Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^2\,} norm) are known in the focussing case BirKnPoSvVe1996. The blowup rate in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^1\,} is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t^{-1}\,} in these solutions. This is not the optimal blowup rate; in fact an example has been constructed where the blowup rate is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle |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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle |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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle xu\,} is in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^s\,} automatically transfer to GWP and scattering results in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^2(|x|^s)\,} thanks to the pseudo-conformal transformation.
  • Solitons are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^1\,} -unstable.

Quintic NLS on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle T}

  • This equation may be viewed as a simpler version of cubic DNLS, and is always at least as well-behaved.
  • Scaling is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s_c = 0\,} .
  • LWP is known for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s > 0\,} Bo1993.
    • For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s < 0\,} the solution map is not uniformly continuous from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle C^k\,} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle C^{-k}\,} for any Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k\,} [CtCoTa-p3].
  • GWP is known in the defocusing case for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s > 4/9\,} (De Silva, Pavlovic, Staffilani, Tzirakis)
    • For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s > 2/3\,} this is commented upon in [Bo-p2] and is a minor modification of [CoKeStTkTa-p].
    • For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s \ge 1\,} one has GWP in the defocusing case, or in the focusing case with small Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^2\,} norm, by Hamiltonian conservation.
      • In the defocusing case one has GWP for random data whose Fourier coefficients decay like Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 1/|k|\,} (times a Gaussian random variable) Bo1995c; this is roughly of the regularity of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^{1/2}\,} . Indeed one has an invariant measure. In the focusing case the same result holds assuming the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^2\,} norm is sufficiently small.


Quintic NLS on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R^2}

  • Scaling is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s_c = 1/2\,} .
  • LWP is known for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s \ge 1/2\,} CaWe1990.
    • For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s=1/2\,} the time of existence depends on the profile of the data as well as the norm.
    • For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s \ge 1\,} by Hamiltonian conservation.
    • This has been improved to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^{1/2}\,} data for any quintic non-linearity.

Quintic NLS on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R^3}

  • Scaling is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s_c = 1\,} .
  • LWP is known for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s \ge 1\,} CaWe1990.
    • For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s=1\,} 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.
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