Scattering for NLW/NLKG: Difference between revisions

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[Thanks to Kenji Nakanishi for many helpful additions to this section - Ed.]
The ''Strauss exponent''
The ''Strauss exponent''


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Scattering for small <math>H^1</math> data for arbitrary NLW:
Scattering for small <math>H^1</math> data for arbitrary NLW:


* Known for <math>p_*(d) < p \leq p_{H^{1/2}}</math> [[Bibliography#Sr1981|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 [[Bibliography#Si1984|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 [[Bibliography#BcKkZz1999|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 defocussing NLW:
Scattering for large <math>H^1</math> data for [[defocusing]] NLW:


* Known for <math>p_{H^{1/2}} < p \leq p_{H^1}</math> [[Bibliography#BaSa1998|BaSa1998]], [[Bibliography#BaGd1997|BaGd1997]] (GWP was established earlier in [[Bibliography#GiVl1987|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> [[Bibliography#BaeSgZz1990|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.


Scattering for small smooth compactly supported data for arbitrary NLW:
Scattering for small smooth compactly supported data for arbitrary NLW:


* GWP and scattering when <math>p > p_0(d-1)</math> [[Bibliography#GeLbSo1997|GeLbSo1997]]
* GWP and scattering when <math>p > p_0(d-1)</math> [[GeLbSo1997]]
** For <math>d=3</math> this is in [[Bibliography#Jo1979|Jo1979]]
** For <math>d=3</math> this is in [[Jo1979]]
* Blow-up for arbitrary nonzero data when <math>p < p_0(d-1)</math> [[Bibliography#Si1984|Si1984]] (see also [[Bibliography#Rm1987|Rm1987]], [[Bibliography#JiZz2003|JiZz2003]]
* Blow-up for arbitrary nonzero data when <math>p < p_0(d-1)</math> [[Si1984]] (see also [[Rm1987]], [[JiZz2003]]
** For <math>d=4</math> this is in [[Bibliography#Gs1981b|Gs1981b]]
** For <math>d=4</math> this is in [[Gs1981b]]
** For <math>d=3</math> this is in [[Bibliography#Jo1979|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 [[Bibliography#Scf1985|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 [[Bibliography#Lev1990|Lev1990]]
** For large data and arbitrary <math>d</math> this is in [[Lev1990]]


Scattering for small <math>H^1</math> data for arbitrary NLKG:
Scattering for small <math>H^1</math> data for arbitrary NLKG:


* Decay estimates are known when <math>p_0(d) < p \leq p_{L^2}</math>[[Bibliography#MsSrWa1980|MsSrWa1980]], [[Bibliography#Br1984|Br1984]], [[Bibliography#Sr1981|Sr1981]], [[Bibliography#Pe1985|Pe1985]].
* Decay estimates are known when <math>p_0(d) < p \leq p_{L^2}</math>[[MsSrWa1980]], [[Br1984]], [[Sr1981]], [[Pe1985]].
* Known when <math>p_{L^2} \leq p \leq p_{H^1}</math> [[Bibliography#Na1999c|Na1999c]], [[Bibliography#Na1999d|Na1999d]], [Na-p5]. Indeed, one has existence of wave operators and asymptotic completeness in these cases.
* Known when <math>p_{L^2} \leq p \leq p_{H^1}</math> [[Na1999c]], [[Na1999d]], [[Na2001]]. Indeed, one has existence of wave operators and asymptotic completeness in these cases.


Scattering for large <math>H^1</math> data for defocussing NLKG:
Scattering for large <math>H^1</math> data for defocussing NLKG:


* 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> [[Bibliography#Na1999c|Na1999c]], [[Bibliography#Na1999d|Na1999d]], [Na-p5]
* Scattering is known for <math>p_{L^2} < p \leq p_{H^1}</math> [[Na1999c]], [[Na1999d]], [[Na2001]]
** For <math>d>2</math> and <math>p</math> not <math>H^1</math>-critical this is in [[Bibliography#Br1985|Br1985]] [[Bibliography#GiVl1985b|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.


Scattering for small smooth compactly supported data for arbitrary NLKG:
Scattering for small smooth compactly supported data for arbitrary NLKG:


* GWP and scattering for <math>p > 1+2/d</math> when <math>d=1,2,3</math> [[Bibliography#LbSo1996|LbSo1996]]
* GWP and scattering for <math>p > 1+2/d</math> when <math>d=1,2,3</math> [[LbSo1996]]
** When <math>d=1,2</math> this can be obtained by energy estimates and decay estimates.
** When <math>d=1,2</math> 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.
** 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 <math>p < 1+2/d</math> [[Bibliography#KeTa1999|KeTa1999]]. The endpoint <math>p=1+2/d</math> remains open but one probably also has blow-up here.
* Blowup in the non-Hamiltonian case when <math>p < 1+2/d</math> [[KeTa1999]]. The endpoint <math>p=1+2/d</math> remains open but one probably also has blow-up here.
** Failure of scattering for <math>p \leq 1+2/d</math> was shown in [[Bibliography#Gs1973|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 [#nlw-5_on_R^3 NLW-5 on <math>R^3</math>] one has global regularity for convex obstacles [[Bibliography#SmhSo1995|SmhSo1995]], and for smooth non-linearities there is the [#gwp_qnlw general quasilinear theory]. If one adds a suitable damping term near the obstacle then one can recover some global existence results [[Bibliography#Nk2001|Nk2001]].
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 [[Bibliography#BchNic1993|BchNic1993]], [[Bibliography#Nic1995|Nic1995]], [[Bibliography#BluSf2003|BluSf2003]]
On the Schwarzschild manifold some scattering and decay results for NLW and NLWKG can be found in [[BchNic1993]], [[Nic1996]], [[BluSf2003]]


[[Category:wave]]
[[Category:wave]]

Latest revision as of 22:40, 24 July 2007

The Strauss exponent

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) = [d + 2 + \sqrt{d^2 + 12d + 4}]/2d}

plays a key role in the GWP and scattering theory. We have 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(1) = [3+\sqrt{17}]/2} ; 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(2) = 1+sqrt(2); p_0(3) = 2} ; note 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 p_0(d-1)} is always between 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} 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 H^{1/2}} critical powers, 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_0(d)} is always between 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^{1/2}} 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 H^1} critical powers.

Another key power 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 p_*(d) = [d+2 + sqrt(d^2 + 8d)]/2(d-1)}

which lies between 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} critical power 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_0(d-1)} .

Caveats: 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 d=1,2} 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 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} -critical case. This is because (unlike NLS and NLKG) there is no a priori bound on 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 (even with conservation of energy).

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 NLW:

  • 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_*(d) < p \leq p_{H^{1/2}}} Sr1981.
  • 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 < p_0(d-1)} one has blow-up Si1984.
  • 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 5/2 < p \leq p_{H^{1/2}}} , but scattering 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 p<5/2} 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_0(d-1) = 2 < p < 5/2} , but scattering 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 p<2} Hi-p3
  • An alternate argument based on conformal compactification but giving slightly different results are in BcKkZz1999

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 defocusing NLW:

  • 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_{H^{1/2}} < p \leq p_{H^1}} 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 p = p_{H^{1/2}}} , 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, Na2001. 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, Na2001
    • 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

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