Cubic NLS on R2: Difference between revisions

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{{equation
| name = Cubic NLS on <math>\R^2</math>
| equation = <math>iu_t + \Delta u = \pm |u|^2 u</math>
| fields = <math>u: \R \times \R^2 \to \mathbb{C}</math>
| data = <math>u(0) \in H^s(\R^2)</math>
| hamiltonian = [[Hamiltonian]]
| linear = [[free Schrodinger equation|Schrodinger]]
| nonlinear = [[semilinear]]
| critical = <math>L^2(\R^2)</math>
| criticality = [[mass critical NLS|mass-critical]];<br> energy-subcritical;<br> scattering-subcritical
| covariance = [[Galilean]] [[pseudoconformal]]
| lwp = <math>H^s(\R^2)</math> for <math>s \geq 0</math>
| gwp = <math>H^s(\R^2)</math> for <math>s \geq 1/2</math>
| parent = [[cubic NLS]]
| special = -
| related = -
}}
The theory of the [[cubic NLS]] on R^2 is as follows.
The theory of the [[cubic NLS]] on R^2 is as follows.


* Scaling is <math>s_c = 0\,</math>, thus this is a [mass critical NLS].
* LWP for <math>s \ge 0\,</math> [[CaWe1990]].
* LWP for <math>s \ge 0\,</math> [[CaWe1990]].
** For <math>s=0\,</math> the time of existence depends on the profile of the data as well as the norm.
** For <math>s=0\,</math> the time of existence depends on the profile of the data as well as the norm.
** LWP has also been obtained in Besov spaces [[Pl2000]], [[Pl-p]] and Fourier-Lorentz spaces [[CaVeVi-p]] at the scaling of <math>L^2\,</math>. This is also connected with the construction of self-similar solutions to NLS (which are generally not in the usual Sobolev spaces globally in space).
** LWP has also been obtained in Besov spaces [[Pl2000]], [[Pl-p]] and Fourier-Lorentz spaces [[CaVeVi-p]] at the scaling of <math>L^2\,</math>. This is also connected with the construction of self-similar solutions to NLS (which are generally not in the usual Sobolev spaces globally in space).
** Below <math>L^2\,</math> we have ill-posedness by Gallilean invariance considerations in both the focusing [[KnPoVe-p]] and defocusing [[CtCoTa-p2]] cases.
** Below <math>L^2\,</math> we have ill-posedness by Gallilean invariance considerations in both the focusing [[KnPoVe2001]] and defocusing [[CtCoTa-p2]] cases.
* GWP for <math>s>4/7\,</math> in the defocussing case [[CoKeStTkTa2002]]
* GWP for <math>s\geq 1/2\,</math> in the defocussing case [[FgGl2006]]
** For <math>s>4/7\,</math> in the defocussing case, this was shown in [[CoKeStTkTa2002]]
** For <math>s>3/5\,</math> this was shown in [[Bo1998]].
** For <math>s>3/5\,</math> this was shown in [[Bo1998]].
** For <math>s>2/3\,</math> this was shown in [[Bo1998]], [[Bo1999]].
** For <math>s>2/3\,</math> this was shown in [[Bo1998]], [[Bo1999]].
** For <math>s\ge 1\,</math> this follows from Hamiltonian conservation.
** For <math>s\ge 1\,</math> this follows from Hamiltonian conservation.
** For small <math>L^2\,</math> data one has GWP and scattering for any cubic nonlinearity (not necessarily defocussing or Hamiltonian). More precisely, one has global well-posedness whenever the data has an <math>L^2\,</math> norm strictly smaller than the ground state Q [[Me1993]]. If the <math>L^2\,</math> norm is exactly equal to that of Q then one has blow-up if and only if the data is a pseudo-conformal transformation of the ground state [[Me1993]], [[Me1992]]. In particular, the ground state is unstable.
** For small <math>L^2\,</math> data one has GWP and scattering for any cubic nonlinearity (not necessarily defocussing or Hamiltonian). More precisely, one has global well-posedness whenever the data has an <math>L^2\,</math> norm strictly smaller than the ground state Q [[Me1993]]. If the <math>L^2\,</math> norm is exactly equal to that of Q then one has blow-up if and only if the data is a pseudo-conformal transformation of the ground state [[Me1993]], [[Me1992]]. In particular, the ground state is unstable.
*** Scattering is known whenever the solution is sufficiently small in <math>L^2\,</math> norm, or more generally whenever the solution is <math>L^4\,</math> in spacetime.Presumably one in fact has scattering whenever the mass is strictly smaller than the ground state, though this has not yet been established.
*** Scattering is known whenever the solution is sufficiently small in <math>L^2\,</math> norm, or more generally whenever the solution is <math>L^4\,</math> in spacetime.
** The <math>s>4/7\,</math> result is probably improvable by correction term methods.
** The <math>s>4/7\,</math> result is probably improvable by correction term methods.
** Remark: <math>s=1/2\,</math> is the least regularity for which the non-linear part of the solution has finite energy (Bourgain, private communication).
** Remark: <math>s=1/2\,</math> is the least regularity for which the non-linear part of the solution has finite energy (Bourgain, private communication).
** For powers slightly higher than cubic, one has scattering for large mass data [[Na1999c]], and indeed we have bounded <math>H^k\,</math> norms in this case [Bourgain?].
** For powers slightly higher than cubic, one has scattering for large mass data [[Na1999c]], and indeed we have bounded <math>H^k\,</math> norms in this case [Bourgain?].
** If the data has sufficient decay then one has scattering. For instance if <math>xu(0)\,</math> is in <math>L^2\,</math> [[Bibliography#Ts1985|Ts1985]]. This was improved to <math>x^{2/3+} u(0) \in L^2\,</math> in [[Bo1998]], [[Bibliography#Bo1999|Bo1999]]; the above results on GWP will probably also extend to scattering.
** If the data has sufficient decay then one has scattering. For instance if <math>xu(0)\,</math> is in <math>L^2\,</math> [[Ts1985]]. This was improved to <math>x^{2/3+} u(0) \in L^2\,</math> in [[Bo1998]], [[Bo1999]]; the above results on GWP will probably also extend to scattering.
* ''Remark'': This equation is pseudo-conformally invariant. Heuristically, GWP results in <math>H^s\,</math> transfer to GWP and scattering results in <math>L^2(|x|^{2s})\,</math> thanks to the pseudo-conformal transformation. Thus for instance GWP and scattering occurs this weighted space for <math>s>2/3\,</math> (the corresponding statement for, say, <math>s > 4/7\,</math> has not yet been checked).
* ''Remark'': This equation is pseudo-conformally invariant. Heuristically, GWP results in <math>H^s\,</math> transfer to GWP and scattering results in <math>L^2(|x|^{2s})\,</math> thanks to the pseudo-conformal transformation. Thus for instance GWP and scattering occurs this weighted space for <math>s>2/3\,</math> (the corresponding statement for, say, <math>s > 4/7\,</math> has not yet been checked).
* The H^k norms grow like <math>O(t^{(k-1)+})\,</math> as long as the H^1 norm stays bounded [[St1997]], [[St1997b]]; this was improved to <math>t^{2/3 (k-1)+}\,</math> in [[CoDeKnSt2001]], and also generalized to higher order multilinearity. A preliminary analysis suggests that the I-method can push the growth bounds down to <math>t^{(k-1)+/2}.</math>
== Open question: large mass behavior ==


== Open question: large mass scattering ==
It is conjectured that global wellposedness, regularity, scattering and spacetime bounds occur for all large mass initial data in the defocusing case, and all data of mass less than that of the ground state in the focusing case.


What happens for large <math>L^2\,</math> data? It is known that the only way GWP can fail at <math>L^2\,</math> is if the <math>L^2\,</math> norm concentrates [[Bo1998]]. Blowup examples with multiple blowup points are known, either simultaneously [[Me1992]] or non-simultaneously [[BoWg1997]]. It is conjectured that the amount of energy which can go into blowup points is quantized. The <math>H^1\,</math> norm in these examples blows up like <math>|t|^{-1}\,.</math> It is conjectured that slower blow-up examples exist, in particular numerics suggest a blowup rate of <math>|t|^{-1/2} (log log|t|)^{1/2}\,</math> [[LanPapSucSup1988]]; interestingly, however, if we perturb NLS to the [[Zakharov system]] then one can only have blowup rates of <math>|t|^{-1}\,.</math>
* It is known that the only way GWP can fail at <math>L^2\,</math> is if the <math>L^2\,</math> norm concentrates [[Bo1998]].  
* Blowup examples with multiple blowup points are known, either simultaneously [[Me1992]] or non-simultaneously [[BoWg1997]]. In all known examples the mass has to be larger than that of the ground state.
** It is conjectured that the amount of energy which can go into blowup points is quantized.  
** The <math>H^1\,</math> norm in these examples blows up like <math>|t|^{-1}\,.</math> It is conjectured that slower blow-up examples exist, in particular numerics suggest a blowup rate of <math>|t|^{-1/2} (log log|t|)^{1/2}\,</math> [[LanPapSucSup1988]]. (This conjecture has been established by G. Perelman and in a remarkable series of papers by F. Merle and P. Raphael.) Interestingly, however, if we perturb NLS to the [[Zakharov system]] then one can only have blowup rates of <math>|t|^{-1}\,.</math>


[[Category:Open problems]]
[[Category:Open problems]]
[[Category:Schrodinger]]
[[Category:Schrodinger]]
[[Category:Equations]]
[[Category:Equations]]

Latest revision as of 18:16, 11 May 2007

Cubic 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}
Description
Equation 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 iu_t + \Delta u = \pm |u|^2 u}
Fields 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 u: \R \times \R^2 \to \mathbb{C}}
Data class 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 u(0) \in H^s(\R^2)}
Basic characteristics
Structure Hamiltonian
Nonlinearity semilinear
Linear component Schrodinger
Critical regularity 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(\R^2)}
Criticality mass-critical;
energy-subcritical;
scattering-subcritical
Covariance Galilean pseudoconformal
Theoretical results
LWP 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(\R^2)} 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 \geq 0}
GWP 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(\R^2)} 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 \geq 1/2}
Related equations
Parent class cubic NLS
Special cases -
Other related -


The theory of the cubic NLS on R^2 is as follows.

  • LWP 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.
    • 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.
    • LWP has also been obtained in Besov spaces Pl2000, Pl-p and Fourier-Lorentz spaces CaVeVi-p at the scaling 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 L^2\,} . This is also connected with the construction of self-similar solutions to NLS (which are generally not in the usual Sobolev spaces globally in space).
    • 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 KnPoVe2001 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\geq 1/2\,} in the defocussing case FgGl2006
    • 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>4/7\,} in the defocussing case, this was shown in CoKeStTkTa2002
    • 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>3/5\,} this was shown in Bo1998.
    • 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 was shown in Bo1998, Bo1999.
    • 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 Hamiltonian conservation.
    • 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 one has GWP and scattering for any cubic nonlinearity (not necessarily defocussing or Hamiltonian). More precisely, one has global well-posedness whenever the data has an 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 strictly smaller than the ground state Q Me1993. If 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 exactly equal to that of Q then one has blow-up if and only if the data is a pseudo-conformal transformation of the ground state Me1993, Me1992. In particular, the ground state is unstable.
      • Scattering is known whenever the solution is sufficiently small 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\,} norm, or more generally whenever the solution 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 L^4\,} in spacetime.
    • 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 s>4/7\,} result is probably improvable by correction term methods.
    • Remark: 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\,} is the least regularity for which the non-linear part of the solution has finite energy (Bourgain, private communication).
    • For powers slightly higher than cubic, one has scattering for large mass data Na1999c, and indeed we have 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 H^k\,} norms in this case [Bourgain?].
    • If the data has sufficient decay then one has scattering. For instance if 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(0)\,} 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\,} Ts1985. This was 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 x^{2/3+} u(0) \in L^2\,} in Bo1998, Bo1999; the above results on GWP will probably also extend to scattering.
  • Remark: This equation is pseudo-conformally invariant. Heuristically, 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\,} 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|^{2s})\,} thanks to the pseudo-conformal transformation. Thus for instance GWP and scattering occurs this weighted space 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\,} (the corresponding statement for, say, 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 > 4/7\,} has not yet been checked).
  • The H^k norms grow 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 O(t^{(k-1)+})\,} as long as the H^1 norm stays bounded St1997, St1997b; this was 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 t^{2/3 (k-1)+}\,} in CoDeKnSt2001, and also generalized to higher order multilinearity. A preliminary analysis suggests that the I-method can push the growth bounds down 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 t^{(k-1)+/2}.}


Open question: large mass behavior

It is conjectured that global wellposedness, regularity, scattering and spacetime bounds occur for all large mass initial data in the defocusing case, and all data of mass less than that of the ground state in the focusing case.

  • It is known that the only way GWP can fail at 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\,} is if 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 concentrates Bo1998.
  • Blowup examples with multiple blowup points are known, either simultaneously Me1992 or non-simultaneously BoWg1997. In all known examples the mass has to be larger than that of the ground state.
    • It is conjectured that the amount of energy which can go into blowup points is quantized.
    • 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 H^1\,} norm in these examples blows up 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 |t|^{-1}\,.} It is conjectured that slower blow-up examples exist, in particular numerics suggest a blowup rate 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 |t|^{-1/2} (log log|t|)^{1/2}\,} LanPapSucSup1988. (This conjecture has been established by G. Perelman and in a remarkable series of papers by F. Merle and P. Raphael.) Interestingly, however, if we perturb NLS to the Zakharov system then one can only have blowup rates 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 |t|^{-1}\,.}
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