Semilinear NLW: Difference between revisions

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====Necessary conditions for [[LWP]] ====
====Necessary conditions for [[LWP]] ====
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
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


<center><math>s_{conf} = (d+1)/4 - 1/(p-1)</math></center>
<center><math>s_{conf} = (d+1)/4 - 1/(p-1)</math></center>
Line 95: Line 97:
in the focusing case; the defocusing case is still open. In the <math>H^{1/2}</math>-critical power or below, this condition is stronger than the scaling requirement.
in the focusing case; the defocusing case is still open. In the <math>H^{1/2}</math>-critical power or below, this condition is stronger than the scaling requirement.


* When <math>d \geq 2</math> 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]].
** When <math>d \geq 2</math> 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 <math>1/2 - s < 1/p</math> to keep the non-linearity integrable, because there is no Strichartz smoothing to exploit.
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 <math>1/2 - s < 1/p</math> to keep the non-linearity integrable, because there is no Strichartz smoothing to exploit.


Finally, in three dimensions one has [[ill-posedness]] when <math>p=2</math> and <math>s = s_{conf} = 0</math> [[Lb1993]]. <br />
#Finally, in three dimensions one has [[ill-posedness]] when <math>p=2</math> and <math>s = s_{conf} = 0</math> [[Lb1993]]. <br />


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


<math>p (d/4-s) \leq 1/2 ( (d+3)/2 - s)</math> (*)</center>
<math>p (d/4-s) \leq 1/2 ( (d+3)/2 - s)</math> (*)</center>
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and excluding the double endpoint when (*) holds with equality and s=s_{conf} [[Ta1999]]. The main tool is two-scale [[Strichartz estimates]].
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
*** By using standard Strichartz estimates this was proven with (*) replaced by


<center><math>p ((d+1)/4-s) \leq (d+1)/2d ( (d+3)/2 - s)</math>; (**)</center>
<center><math>p ((d+1)/4-s) \leq (d+1)/2d ( (d+3)/2 - s)</math>; (**)</center>

Revision as of 13:54, 16 January 2007

Semilinear wave equations

[Note: Many references needed here!]

Semilinear wave equations (NLW) and semi-linear Klein-Gordon equations (NLKG) take the form

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

Typically is a power type nonlinearity. If is the gradient of some function , then we have a conserved Hamiltonian

For NLKG there is an additional term of in the integrand, which is useful for controlling the low frequencies of . If V is positive definite then we call the NLW defocusing; if is negative definite we call the NLW focusing.


To analyze these equations in we need the non-linearity to be sufficiently smooth. More precisely, we will always assume either that is smooth, or that is a p^th-power type non-linearity with .

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

Dimension d

Strauss exponent (NLKG)

-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

Necessary conditions for LWP

The following necessary conditions for LWP are known.

  1. 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

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

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 to keep the non-linearity integrable, because there is no Strichartz smoothing to exploit.

  1. Finally, in three dimensions one has ill-posedness when and Lb1993.
    • In dimensions d\leq3 the above necessary conditions are also sufficient for LWP.
    • For d>4 sufficiency is only known assuming the condition

(*)

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
; (**)

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 Kp1993.

GWP and scattering for NLW is known for data with small norm when is at or above the -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 in the defocussing case when p is at or below the -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 , , is known in the following cases:

  • KnPoVe-p2
  • MiaZgFg-p
  • , and

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

  • Fo-p; this is for the NLW instead of NLKG.
  • Fo-p; this is for the NLW instead of NLKG.

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

;

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