Cubic DNLS on R: Difference between revisions
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Suppose the non-linearity has the form f = i (u <u>u</u> u)<sub>x</sub>. Then: | Suppose the non-linearity has the form f = i (u <u>u</u> u)<sub>x</sub>. Then: | ||
* Scaling is | * Scaling is <math>s_c = 0</math>. | ||
* LWP for s | * LWP for <math>s = 1/2</math> [[Tk2001]]. | ||
** This is sharp in the | ** This is sharp in the uniform sense [[BiLi2001]] (see also [[Tk2001]] for failure of analytic well-posedness below 1/2). | ||
** For s | ** For <math>s > 1</math> this was proven in [[HaOz1994]]. | ||
* GWP for s>1/2 and small | * GWP for <math>s>1/2</math> and small <math>L^2</math> norm [[CoKeStTkTa2002b]]. The <math>s=1/2</math> case remains open. The existence of finite time blowup solutions emerging from smooth data with large <math>L^2</math> size is also unknown. | ||
** for s>2/3 and small | ** for <math> s>2/3</math> and small <math>L^2</math> norm this was proven in [[CoKeStTkTa2001b]]. | ||
** For s > 32/33 with small | ** For <math>s > 32/33</math> with small <math>L^2</math> norm this was proven in [[Tk-p]]. | ||
** For s | ** For <math>s >1</math> and small norm this was proven in [[HaOz1994]]. One can also handle certain pure power additional terms [[Oz1996]]. | ||
** The small | ** The small <math>L^2</math> norm condition is used to obtain a priori <math>H^1</math> control on the gauge transformed solution; see [[HaOz1992]], [[Oz1996]]. | ||
* Solutions do not scatter to free Schrodinger solutions. In the | * Solutions do not scatter to free Schrodinger solutions. In the focusing case this can be easily seen from the existence of solitons. But even in the defocussing case wave operators do not exist, and must be replaced by modified wave operators (constructed in [[HaOz1994]] for small data). | ||
This equation has the same scaling as the [ | This equation has the same scaling as the [[Quintic NLS on R|quintic NLS]], and there is a certain gauge invariance which unifies the two (together with an additional nonlinear term u <u>u</u><sub>x</sub> u). | ||
For non-linearities of the form f = a (u | For non-linearities of the form <math>f = a (u {\overline{u}})_x u + b (u \overline{u})_x u_x</math> one can obtain GWP for small data [[KyTs1994]] for arbitrary complex constants <math>a, b</math>. See also [[Ts1994]]. | ||
==Problem: Maximal-in-time behavior for large <math>L^2</math> data?== | |||
For smooth (say <math>H^1</math>) initial data with large <math>L^2</math>, what happens? Do there exist finite time blowup solutions? This problem is gauge transforms into quintic focusing NLS on <math>\mathbb{R}</math> which is known to have blowup solutions. However, the extra term in the equation <math>u {u} {\overline{u}}_x </math> scales the same way so could conceivably counteract the focusing quintic nonlinearity. Also, the equation is completely integrable so blowup may possibly be ruled out by looking at other conservation properties. | |||
</ | |||
==Problem: Global well-posedness in <math>H^{1/2}</math> under the mass constraint?== | |||
For <math>s=1/2</math>, we have LWP [[Tk2001]]. For <math>s>1/2</math> and under a mass upper bound, we have GWP [[CoKeStTkTa2002b]]. Does GWP hold also for the endpoint <math>s = 1/2</math>? | |||
==Problem: LWP for <math>s<1/2 </math>?== | |||
The fixed-point approach to proving LWP [[Tk2001]], [[BiLi2001]] fails for <math>s<1/2</math>. Does LWP with merely continuous dependence upon the initial data hold for <math>0 \leq s < 1/2</math>? | |||
[[Category:Open problems]] | |||
[[Category:Schrodinger]] | |||
[[Category:Equations]] | [[Category:Equations]] | ||
Latest revision as of 12:30, 18 May 2007
Cubic DNLS 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}
Suppose the non-linearity has the form f = i (u u u)x. Then:
- 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 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}
Tk2001.
- This is sharp in the uniform sense BiLi2001 (see also Tk2001 for failure of analytic well-posedness below 1/2).
- 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 > 1} this was proven in HaOz1994.
- 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>1/2}
and 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 CoKeStTkTa2002b. 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=1/2}
case remains open. The existence of finite time blowup solutions emerging from smooth data 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}
size is also unknown.
- 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} and 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 this was proven in CoKeStTkTa2001b.
- 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} 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 this was proven 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 >1} and small norm this was proven in HaOz1994. One can also handle certain pure power additional terms Oz1996.
- The 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 condition is used to obtain a priori 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} control on the gauge transformed solution; see HaOz1992, Oz1996.
- Solutions do not scatter to free Schrodinger solutions. In the focusing case this can be easily seen from the existence of solitons. But even in the defocussing case wave operators do not exist, and must be replaced by modified wave operators (constructed in HaOz1994 for small data).
This equation has the same scaling as the quintic NLS, and there is a certain gauge invariance which unifies the two (together with an additional nonlinear term u ux u).
For non-linearities of the form 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 f = a (u {\overline{u}})_x u + b (u \overline{u})_x u_x} one can obtain GWP for small data KyTs1994 for arbitrary complex constants 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 a, b} . See also Ts1994.
Problem: Maximal-in-time behavior 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?
For smooth (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 H^1} ) initial data 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} , what happens? Do there exist finite time blowup solutions? This problem is gauge transforms into quintic focusing 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 \mathbb{R}} which is known to have blowup solutions. However, the extra term in the 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 u {u} {\overline{u}}_x } scales the same way so could conceivably counteract the focusing quintic nonlinearity. Also, the equation is completely integrable so blowup may possibly be ruled out by looking at other conservation properties.
Problem: Global well-posedness 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^{1/2}} under the mass constraint?
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} , we have LWP Tk2001. 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} and under a mass upper bound, we have GWP CoKeStTkTa2002b. Does GWP hold also for the endpoint 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} ?
Problem: 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<1/2 } ?
The fixed-point approach to proving LWP Tk2001, BiLi2001 fails 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} . Does LWP with merely continuous dependence upon the initial data hold 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 0 \leq s < 1/2} ?
