Scattering for NLW/NLKG: Difference between revisions
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The ''Strauss exponent'' | The ''Strauss exponent'' | ||
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* Known for <math>p_*(d) < p \leq p_{H^{1/2}}</math> [[Sr1981]]. | * Known for <math>p_*(d) < p \leq p_{H^{1/2}}</math> [[Sr1981]]. | ||
* For <math>p < p_0(d-1)</math> one has blow-up [[Si1984]]. | * For <math>p < p_0(d-1)</math> one has blow-up [[Si1984]]. | ||
* When <math>d=3</math> this is extended to <math>5/2 < p \leq p_{H^{1/2}}</math>, but scattering fails for <math>p<5/2</math> [Hi-p3] | * When <math>d=3</math> this is extended to <math>5/2 < p \leq p_{H^{1/2}}</math>, but scattering fails for <math>p<5/2</math> [[Hi-p3]] | ||
* When <math>d=4</math> this is extended to <math>p_0(d-1) = 2 < p < 5/2</math>, but scattering fails for <math>p<2</math> [Hi-p3] | * When <math>d=4</math> this is extended to <math>p_0(d-1) = 2 < p < 5/2</math>, but scattering fails for <math>p<2</math> [[Hi-p3]] | ||
* An alternate argument based on conformal compactification but giving slightly different results are in [[BcKkZz1999]] | * An alternate argument based on conformal compactification but giving slightly different results are in [[BcKkZz1999]] | ||
Scattering for large <math>H^1</math> data for | Scattering for large <math>H^1</math> data for [[defocusing]] NLW: | ||
* Known for <math>p_{H^{1/2}} < p \leq p_{H^1}</math> [[BaSa1998]], [[BaGd1997]] (GWP was established earlier in [[GiVl1987]]). | * Known for <math>p_{H^{1/2}} < p \leq p_{H^1}</math> [[BaSa1998]], [[BaGd1997]] ([[GWP]] was established earlier in [[GiVl1987]]). | ||
* Known for <math>p = p_{H^{1/2}}</math>, <math>d=3</math> [[BaeSgZz1990]] | * Known for <math>p = p_{H^{1/2}}</math>, <math>d=3</math> [[BaeSgZz1990]] | ||
* When <math>d=3</math> this is extended to <math>p_*(3) < p \leq p_{H^{1/2}}</math> [Hi-p3] | * When <math>d=3</math> this is extended to <math>p_*(3) < p \leq p_{H^{1/2}}</math> [[Hi-p3]] | ||
* When <math>d=4</math> this is extended to <math>p_*(4) < p < 5/2</math> [Hi-p3] | * When <math>d=4</math> this is extended to <math>p_*(4) < p < 5/2</math> [[Hi-p3]] | ||
* For <math>d>4</math> one expects scattering when <math>p_0(d-1) < p \leq p_{H^{1/2}}</math>, but this is not known. | * For <math>d>4</math> one expects scattering when <math>p_0(d-1) < p \leq p_{H^{1/2}}</math>, but this is not known. | ||
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** For <math>d=4</math> this is in [[Gs1981b]] | ** For <math>d=4</math> this is in [[Gs1981b]] | ||
** For <math>d=3</math> this is in [[Jo1979]] | ** For <math>d=3</math> this is in [[Jo1979]] | ||
* At the critical power <math>p = p_0(d-1)</math> there is blowup for non-negative non-trivial data [YoZgq-p2] | * At the critical power <math>p = p_0(d-1)</math> there is blowup for non-negative non-trivial data [[YoZgq-p2]] | ||
** For <math>d=2,3</math> and arbitrary nonzero data this is in [[Scf1985]] | ** For <math>d=2,3</math> and arbitrary nonzero data this is in [[Scf1985]] | ||
** For large data and arbitrary <math>d</math> this is in [[Lev1990]] | ** For large data and arbitrary <math>d</math> this is in [[Lev1990]] | ||
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* In this case one has an a priori <math>L^2</math> bound and one does not need decay at spatial infinity. | * In this case one has an a priori <math>L^2</math> bound and one does not need decay at spatial infinity. | ||
* Scattering is known for <math>p_{L^2} < p \leq p_{H^1}</math> [[Na1999c]], [[Na1999d]], [Na-p5] | * Scattering is known for <math>p_{L^2} < p \leq p_{H^1}</math> [[Na1999c]], [[Na1999d]], [[Na-p5]] | ||
** For <math>d>2</math> and <math>p</math> not <math>H^1</math>-critical this is in [[Br1985]] [[GiVl1985b]] | ** For <math>d>2</math> and <math>p</math> not <math>H^1</math>-critical this is in [[Br1985]] [[GiVl1985b]] | ||
** The <math>L^2</math>-critical case <math>p = p_{L^2}</math> is an interesting open problem. | ** The <math>L^2</math>-critical case <math>p = p_{L^2}</math> is an interesting open problem. | ||
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** Failure of scattering for <math>p \leq 1+2/d</math> was shown in [[Gs1973]]. | ** Failure of scattering for <math>p \leq 1+2/d</math> was shown in [[Gs1973]]. | ||
An interesting (and apparently under-explored) problem is what happens to these global existence and scattering results when there is an obstacle. For [ | An interesting (and apparently under-explored) problem is what happens to these global existence and scattering results when there is an obstacle. For [[Quintic_NLW/NLKG_on_R3|NLW-5 on <math>R^3</math>]] one has global regularity for convex obstacles [[SmhSo1995]], and for smooth non-linearities there is the [[QNLW|general quasilinear theory]]. If one adds a suitable damping term near the obstacle then one can recover some global existence results [[Nk2001]]. | ||
On the Schwarzschild manifold some scattering and decay results for NLW and NLWKG can be found in [[BchNic1993]], [[ | On the Schwarzschild manifold some scattering and decay results for NLW and NLWKG can be found in [[BchNic1993]], [[Nic1996]], [[BluSf2003]] | ||
[[Category:wave]] | [[Category:wave]] | ||
Revision as of 23:35, 14 August 2006
The Strauss exponent
![{\displaystyle p_{0}(d)=[d+2+{\sqrt {d^{2}+12d+4}}]/2d}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/935acc427d15d6b5a697ac085746874236db11c1)
plays a key role in the GWP and scattering theory. We have ; ; note that is always between the and critical powers, and is always between the and critical powers.
![{\displaystyle p_{0}(1)=[3+{\sqrt {17}}]/2}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/298897d74823be5dc049b8ce7ff054fa2647b9a3)
Another key power is
![{\displaystyle p_{*}(d)=[d+2+sqrt(d^{2}+8d)]/2(d-1)}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/ed0197c0038937de43ad980ce9fac1044881285d)
which lies between the critical power and .
Caveats: the cases may be somewhat different from what is stated here (partly because some of the powers here are not well-defined). Also, in many of the NLW results one needs some additional decay at spatial infinity (e.g. finiteness of the conformal energy), except in the special -critical case. This is because (unlike NLS and NLKG) there is no a priori bound on the norm (even with conservation of energy).
Scattering for small data for arbitrary NLW:
- Known for Sr1981.
- For one has blow-up Si1984.
- When this is extended to , but scattering fails for Hi-p3
- When this is extended to , but scattering fails for Hi-p3
- An alternate argument based on conformal compactification but giving slightly different results are in BcKkZz1999
Scattering for large data for defocusing NLW:
- Known for BaSa1998, BaGd1997 (GWP was established earlier in GiVl1987).
- 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 d=3} BaeSgZz1990
- When 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 d=3} this is extended 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 p_*(3) < p \leq p_{H^{1/2}}} Hi-p3
- When 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 d=4} this is extended 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 p_*(4) < p < 5/2} Hi-p3
- 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 d>4} one expects scattering when 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 p_0(d-1) < p \leq p_{H^{1/2}}} , but this is not known.
Scattering for small smooth compactly supported data for arbitrary NLW:
- GWP and scattering when 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 p > p_0(d-1)}
GeLbSo1997
- 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 d=3} this is in Jo1979
- Blow-up for arbitrary nonzero data when 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 p < p_0(d-1)}
Si1984 (see also Rm1987, JiZz2003
- 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 d=4} this is in Gs1981b
- 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 d=3} this is in Jo1979
- At the critical power 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 p = p_0(d-1)}
there is blowup for non-negative non-trivial data YoZgq-p2
- 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 d=2,3} and arbitrary nonzero data this is in Scf1985
- For large data and arbitrary 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 d} this is in Lev1990
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} data for arbitrary NLKG:
- Decay estimates are known when 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 p_0(d) < p \leq p_{L^2}} MsSrWa1980, Br1984, Sr1981, Pe1985.
- Known when 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 p_{L^2} \leq p \leq p_{H^1}} Na1999c, Na1999d, [Na-p5]. Indeed, one has existence of wave operators and asymptotic completeness in these cases.
Scattering 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 H^1} data for defocussing NLKG:
- In this case one has an 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 L^2} bound and one does not need decay at spatial infinity.
- Scattering 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 p_{L^2} < p \leq p_{H^1}}
Na1999c, Na1999d, Na-p5
- 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 d>2} and 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 p} not -critical this is in Br1985 GiVl1985b
- The -critical case is an interesting open problem.
Scattering for small smooth compactly supported data for arbitrary NLKG:
- GWP and scattering for when LbSo1996
- When this can be obtained by energy estimates and decay estimates.
- In principle this extends to higher dimensions but there is a difficulty with lack of smoothness in the nonlinearity.
- Blowup in the non-Hamiltonian case when KeTa1999. The endpoint remains open but one probably also has blow-up here.
- Failure of scattering for was shown in Gs1973.
An interesting (and apparently under-explored) problem is what happens to these global existence and scattering results when there is an obstacle. For NLW-5 on one has global regularity for convex obstacles SmhSo1995, and for smooth non-linearities there is the general quasilinear theory. If one adds a suitable damping term near the obstacle then one can recover some global existence results Nk2001.
On the Schwarzschild manifold some scattering and decay results for NLW and NLWKG can be found in BchNic1993, Nic1996, BluSf2003
