Cubic NLS on R3: Difference between revisions
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==== | {{equation | ||
| name = Cubic NLS on <math>\R^3</math> | |||
| equation = <math>iu_t + \Delta u = \pm |u|^2 u</math> | |||
| fields = <math>u: \R \times \R^3 \to \mathbb{C}</math> | |||
| data = <math>u(0) \in H^s(\R^3)</math> | |||
| hamiltonian = [[Hamiltonian]] | |||
| linear = [[free Schrodinger equation|Schrodinger]] | |||
| nonlinear = [[semilinear]] | |||
| critical = <math>\dot H^{1/2}(\R^3)</math> | |||
| criticality = mass-supercritical;<br> energy-subcritical;<br> scattering-subcritical | |||
| covariance = [[Galilean]] | |||
| lwp = <math>H^s(\R)</math> for <math>s \geq 1/2</math> | |||
| gwp = <math>H^s(\R)</math> for <math>s \geq 4/5</math> (+)<br>or for <math>s \geq 0</math>, small norm | |||
| parent = [[cubic NLS]] | |||
| special = - | |||
| related = - | |||
}} | |||
The theory of the [[cubic NLS]] on <math>\R^3</math> is as follows. | |||
* LWP for <math>s \ge 1/2\,</math> [[CaWe1990]]. | * LWP for <math>s \ge 1/2\,</math> [[CaWe1990]]. | ||
** For <math>s=1/2\,</math> 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 <math>s<1/2\,</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. | ** For <math>s<1/2\,</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. | ||
** For <math>s > 1/2\,</math> there is unconditional well-posedness [[FurPlTer2001]] | ** For <math>s > 1/2\,</math> there is unconditional well-posedness [[FurPlTer2001]] | ||
*** For <math>s >= 2/3\,</math> this is in [[Ka1995]]. | *** For <math>s >= 2/3\,</math> this is in [[Ka1995]]. | ||
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** For <math>s\ge 1\,</math> this follows from Hamiltonian conservation. One also has scattering in this case [[GiVl1985]]. | ** For <math>s\ge 1\,</math> this follows from Hamiltonian conservation. One also has scattering in this case [[GiVl1985]]. | ||
** For small <math>H^{1/2}\,</math> data one has GWP and scattering for any cubic nonlinearity (not necessarily defocussing or Hamiltonian). More generally one has scattering whenever the solution is <math>L^5\,</math> in spacetime. | ** For small <math>H^{1/2}\,</math> data one has GWP and scattering for any cubic nonlinearity (not necessarily defocussing or Hamiltonian). More generally one has scattering whenever the solution is <math>L^5\,</math> in spacetime. | ||
** In the focusing case one has blowup whenever the energy is negative [[Gs1977]], [[OgTs1991]], and in particular one has blowup arbitrarily close to the ground state [[BerCa1981]], [[SaSr1985]].If however the energy remains bounded (which is automatic in the defocusing case) then one has at most polynomial growth of high Sobolev norms, with the local higher Sobolev norms <math>H^s_{loc}\,</math> remaining bounded for all time [[ | ** In the focusing case one has blowup whenever the energy is negative [[Gs1977]], [[OgTs1991]], and in particular one has blowup arbitrarily close to the ground state [[BerCa1981]], [[SaSr1985]].If however the energy remains bounded (which is automatic in the defocusing case) then one has at most polynomial growth of high Sobolev norms, with the local higher Sobolev norms <math>H^s_{loc}\,</math> remaining bounded for all time [[Bo1996c]], [[Bo1998b]]. Also in the focusing radial case with bounded energy, the solution becomes asymptotically smooth and spatially decaying away from the origin, once one strips out the radiation component [[Ta2004b]] | ||
[[Category:Schrodinger]] | [[Category:Schrodinger]] | ||
[[Category:Equations]] | [[Category:Equations]] | ||
Latest revision as of 06:22, 21 July 2007
| Description | |
|---|---|
| Equation | |
| Fields | |
| Data class | |
| Basic characteristics | |
| Structure | Hamiltonian |
| Nonlinearity | semilinear |
| Linear component | Schrodinger |
| Critical regularity | |
| Criticality | mass-supercritical; energy-subcritical; scattering-subcritical |
| Covariance | Galilean |
| Theoretical results | |
| LWP | for |
| GWP | for (+) or for , small norm |
| Related equations | |
| Parent class | cubic NLS |
| Special cases | - |
| Other related | - |
The theory of the cubic NLS on is as follows.
- LWP 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.
- For there is unconditional well-posedness FurPlTer2001
- For this is in Ka1995.
- GWP and scattering for in the defocussing case CoKeStTkTa-p8
- For GWP is in CoKeStTkTa2002
- For GWP is in Bo1999
- For radial data and GWP and scattering is in Bo1998b, Bo1999.
- For this follows from Hamiltonian conservation. One also has scattering in this case GiVl1985.
- For small data one has GWP and scattering for any cubic nonlinearity (not necessarily defocussing or Hamiltonian). More generally one has scattering whenever the solution is in spacetime.
- In the focusing case one has blowup whenever the energy is negative Gs1977, OgTs1991, and in particular one has blowup arbitrarily close to the ground state BerCa1981, SaSr1985.If however the energy remains bounded (which is automatic in the defocusing case) then one has at most polynomial growth of high Sobolev norms, with the local higher Sobolev norms remaining bounded for all time Bo1996c, Bo1998b. Also in the focusing radial case with bounded energy, the solution becomes asymptotically smooth and spatially decaying away from the origin, once one strips out the radiation component Ta2004b
