Semilinear NLW: Difference between revisions

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the significance of this factor is that it behaves well under conformal compactification. See [[Bibliography#Aa2002|Aa2002]], [[Bibliography#BcKkZz2002|BcKkZz2002]], [[Bibliography#Gue2003|Gue2003]] for some recent results.
the significance of this factor is that it behaves well under conformal compactification. See [[Bibliography#Aa2002|Aa2002]], [[Bibliography#BcKkZz2002|BcKkZz2002]], [[Bibliography#Gue2003|Gue2003]] for some recent results.
A substantial [[scattering for NLW/NLKG|scattering theory for NLW and NLKG]] is known.


The [[non-relativistic limit]] of NLKG has attracted a fair amount of research.
The [[non-relativistic limit]] of NLKG has attracted a fair amount of research.
[[Category:Equations]]
====Scattering theory for semilinear NLW====
[Thanks to Kenji Nakanishi for many helpful additions to this section - Ed.]
The ''Strauss exponent''
<center><math>p_0(d) = [d + 2 + \sqrt{d^2 + 12d + 4}]/2d</math></center>
plays a key role in the GWP and scattering theory. We have <math>p_0(1) = [3+\sqrt{17}]/2</math>; <math>p_0(2) = 1+sqrt(2); p_0(3) = 2</math>; note that <math>p_0(d-1)</math> is always between the <math>L^2</math> and <math>H^{1/2}</math> critical powers, and <math>p_0(d)</math> is always between the <math>H^{1/2}</math> and <math>H^1</math> critical powers.
Another key power is
<center><math>p_*(d) = [d+2 + sqrt(d^2 + 8d)]/2(d-1)</math></center>
which lies between the <math>L^2</math> critical power and <math>p_0(d-1)</math>.
'''Caveats''': the <math>d=1,2</math> 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 <math>H^1</math>-critical case. This is because (unlike NLS and NLKG) there is no a priori bound on the <math>L^2</math> norm (even with conservation of energy).
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]].
* For <math>p < p_0(d-1)</math> one has blow-up [[Bibliography#Si1984|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=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]]
Scattering for large <math>H^1</math> data for defocussing 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 = p_{H^{1/2}}</math>, <math>d=3</math> [[Bibliography#BaeSgZz1990|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=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.
Scattering for small smooth compactly supported data for arbitrary NLW:
* GWP and scattering when <math>p > p_0(d-1)</math> [[Bibliography#GeLbSo1997|GeLbSo1997]]
** For <math>d=3</math> this is in [[Bibliography#Jo1979|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]]
** For <math>d=4</math> this is in [[Bibliography#Gs1981b|Gs1981b]]
** For <math>d=3</math> this is in [[Bibliography#Jo1979|Jo1979]]
* 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 large data and arbitrary <math>d</math> this is in [[Bibliography#Lev1990|Lev1990]]
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]].
* 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.
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.
* Scattering is known for <math>p_{L^2} < p \leq p_{H^1}</math> [[Bibliography#Na1999c|Na1999c]], [[Bibliography#Na1999d|Na1999d]], [Na-p5]
** For <math>d>2</math> and <math>p</math> not <math>H^1</math>-critical this is in [[Bibliography#Br1985|Br1985]] [[Bibliography#GiVl1985b|GiVl1985b]]
** 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:
* GWP and scattering for <math>p > 1+2/d</math> when <math>d=1,2,3</math> [[Bibliography#LbSo1996|LbSo1996]]
** 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.
* 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.
** Failure of scattering for <math>p \leq 1+2/d</math> was shown in [[Bibliography#Gs1973|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]].
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]]


====Specific semilinear wave equations====
====Specific semilinear wave equations====
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[[Category:Wave]]
[[Category:Wave]]
[[Category:Equations]]

Revision as of 21:58, 30 July 2006

Semilinear wave equations

[Note: Many references needed here!]

Semilinear wave equations (NLW) and semi-linear Klein-Gordon equations (NLKG) take 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 \Box f = F( f ) , \Box f = f + F( f )}

respectively where is a function only 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 f} and not of its derivatives, which vanishes to more than first order.

Typically 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} grows 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 | f |^p} for some 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} . 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 F} is the gradient of some function 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 V} , then we have a conserved Hamiltonian

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 \int | f _t |^2 / 2 + | \nabla f |^2 / 2 + V( f )\ dx.}

For NLKG there is an additional term 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 | f |^2 /2} in the integrand, which is useful for controlling the low frequencies of . If V is positive definite then we call the NLW defocussing; 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 V} is negative definite we call the NLW focussing. The term "coercive" does not have a standard definition, but generally denotes a potential 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 V} which is positive for large values 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 f} .


To analyze these equations 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} we need the non-linearity to be sufficiently smooth. More precisely, we will always assume either 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 F} is smooth, or 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 F} is a p^th-power type non-linearity with 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 > [s]+1} .

The scaling regularity 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 = d/2 - 2/(p-1)} . Notable powers 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 p} include 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 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} = 1 + 4/d} , 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}} -critical or conformal power p_{H^{1/2}} = 1 + 4/(d-1), and 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} -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_{H^1} = 1 + 4/{d-2}} .

Dimension d

Strauss exponent (NLKG)

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 exponent

Strauss exponent (NLW)

H^{1/2}-critical exponent

H^1-critical exponent

1

3.56155...

5

infinity

infinity

N/A

2

2.41421...

3

3.56155...

5

infinity

3

2

2.33333...

2.41421...

3

5

4

1.78078...

2

2

2.33333...

3

The following necessary conditions for LWP are known. Firstly, for focussing NLW/NLKG one has blowup in finite time for large data, as can be seen by the ODE method. One can scale this and obtain ill-posedness for any focussing NLW/NLKG in the supercritical regime s < s_c; this has been extended to the defocusing case in [CtCoTa-p2]. By using Lorentz scaling instead of isotropic scaling one can also obtain ill-posedness whenever s is below the conformal 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 s_{conf} = (d+1)/4 - 1/(p-1)}

in the focusing case; the defocusing case is still open. In 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}} -critical power or below, this condition is stronger than the scaling requirement.

  • 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 \geq 2} and 1 < p < p_{H^{1/2}} with the focusing sign, blowup is known to occur when a certain Lyapunov functional is negative, and the rate of blowup is self-similar MeZaa2003; earlier results are in AntMe2001, CafFri1986, Al1995, KiLit1993, KiLit1993b.

To make sense of the non-linearity in the sense of distributions we need s \geq 0 (indeed we have illposedness below this regularity by a high-to-low cascade, see [CtCoTa-p2]). In the one-dimensional case one also needs the condition 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 1/2 - s < 1/p} to keep the non-linearity integrable, because there is no Strichartz smoothing to exploit.

Finally, in three dimensions one has ill-posedness 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=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 s = s_{conf} = 0} Lb1993.

  • In dimensions d\leq3 the above necessary conditions are also sufficient for LWP.
  • For d>4 sufficiency is only known assuming the condition

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/4-s) \leq 1/2 ( (d+3)/2 - s)} (*)

and excluding the double endpoint when (*) holds with equality and s=s_{conf} Ta1999. The main tool is two-scale Strichartz estimates.

    • By using standard Strichartz estimates this was proven with (*) replaced by
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+1)/4-s) \leq (d+1)/2d ( (d+3)/2 - s)} ; (**)

see KeTa1998 for the double endpoint when (**) holds with equality and s=s_{conf}, and LbSo1995 for all other cases. A slightly weaker result also appears in Kp1994.

GWP and scattering for NLW is known for data 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 H^{s_c}} norm 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} is at or above 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}} -critical power (and this has been extended to Besov spaces; see [Pl-p4]. This can be used to obtain self-similar solutions, see [MiaZg-p2]). One also has GWP 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} in the defocussing case when p is at or below 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} -critical power. (At the critical power this result is due to Gl1992; see also SaSw1994. For radial data this was shown in Sw1988). For more scattering results, see below.

For the defocussing NLKG, GWP 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} , 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} , is known in the following cases:

  • 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, p = 3, s > 3/4} references:KnPoVe-p2 KnPoVe-p2
  • 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, 3 \leq p < 5, s > [4(p-1) + (5-p)(3p-3-4)]/[2(p-1)(7-p)]} [MiaZgFg-p]
  • 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, 2 < p < 3, or n\geq4, (d+1)^2/((d-1)^2+4) \leq p < (d-1)/(d-3)} , 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 s > [2(p-1)^2 - (d+2-p(d-2))(d+1-p(d-1))] / [2(p-1)(d+1-p(d-3))]}

[MiaZgFg-p]. Note that this is the range of p for which s_conf obeys both the scaling condition 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_{conf} > s_c} and the condition (**).

  • 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 \leq p \leq 5, s > (p-2)/(p-1)} [Fo-p]; this is for the NLW instead of NLKG.
  • 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, p > 5, s > (p-1)/p} [Fo-p]; this is for the NLW instead of NLKG.

GWP and blowup has also been studied for the NLW with a conformal factor

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 \Box u = (t^2 + (1 - (t^2-x^2)/4)^2)^{-(d-1)p/4 + (d+3)/4} |u|^p} ;

the significance of this factor is that it behaves well under conformal compactification. See Aa2002, BcKkZz2002, Gue2003 for some recent results.

A substantial scattering theory for NLW and NLKG is known.

The non-relativistic limit of NLKG has attracted a fair amount of research.

Specific semilinear wave equations

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