Semilinear NLW: Difference between revisions

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Typically <math>F</math> is a [[power type]] nonlinearity. If <math>F</math> is the gradient of some function <math>V</math>, then we have a [[conserved]] [[Hamiltonian]]
Typically <math>F</math> is a [[power type]] nonlinearity. If <math>F</math> is the gradient of some function <math>V</math>, then we have a [[conserved]] [[Hamiltonian]]


<center><math>\int | \frac{\phi_t |^2}{ 2} + | \frac{\nabla \phi |^2}{2} + V( \phi )\ dx.</math></center>
<center><math>\int \frac{ |\phi_t |^2}{ 2} + \frac{|\nabla \phi |^2}{2} + V( \phi )\ dx.</math></center>


For NLKG there is an additional term of <math>| \phi |^2 /2</math> in the integrand, which is useful for controlling the low frequencies of  <math>f</math> . If V is positive definite then we call the NLW [[defocusing]]; if <math>V</math> is negative definite we call the NLW [[focusing]].
For NLKG there is an additional term of <math>| \phi |^2 /2</math> in the integrand, which is useful for controlling the low frequencies of  <math>f</math> . If V is positive definite then we call the NLW [[defocusing]]; if <math>V</math> is negative definite we call the NLW [[focusing]].
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To analyze these equations in <math>H^s</math> we need the non-linearity to be sufficiently smooth. More precisely, we will always assume either that <math>F</math> is smooth, or that <math>F</math> is a p^th-[[power type]] non-linearity with <math>p > [s]+1</math>.
To analyze these equations in <math>H^s</math> we need the non-linearity to be sufficiently smooth. More precisely, we will always assume either that <math>F</math> is smooth, or that <math>F</math> is a p^th-[[power type]] non-linearity with <math>p > [s]+1</math>.


The scaling regularity is <math>s_c = d/2 - 2/(p-1)</math>. Notable powers of <math>p</math> include the <math>L^2</math>-critical power <math>p_{L^2} = 1 + 4/d</math>, the <math>H^{1/2}</math>-critical or [[conformal]] power p_{H^{1/2}} = 1 + 4/(d-1), and the <math>H^1</math>-critical'' power <math>p_{H^1} = 1 + 4/{d-2}</math>. <br />
The scaling regularity is  
<center>
<math>s_c = \frac{d}{2} - \frac{2}{(p-1)}</math>.  
</center>
Notable powers of <math>p</math> include the <math>L^2</math>-critical power <math>p_{L^2} = 1 + 4/d</math>, the <math>H^{1/2}</math>-critical or [[conformal]] power p_{H^{1/2}} = 1 + 4/(d-1), and the <math>H^1</math>-critical'' power <math>p_{H^1} = 1 + 4/{d-2}</math>. <br />


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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.  


<center><math>s_{conf} = (d+1)/4 - 1/(p-1)</math></center>
* 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> 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]]. 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]].


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 dimensions <math>d\leq3 </math> the above necessary conditions are also sufficient for LWP.
 
* 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.
 
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.
* For d>4 sufficiency is only known assuming the condition
* For d>4 sufficiency is only known assuming the condition
 
<center><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>
 
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 <center><math>p ((d+1)/4-s) \leq (d+1)/2d ( (d+3)/2 - s)</math>; (**)</center> 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 <math>H^{s_c}</math> norm when <math>p</math> is at or above the <math>H^{1/2}</math>-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 <math>H^1</math> in the defocussing case when p is at or below the <math>H^1</math>-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.
** 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>
 
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 <math>H^{s_c}</math> norm when <math>p</math> is at or above the <math>H^{1/2}</math>-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 <math>H^1</math> in the defocussing case when p is at or below the <math>H^1</math>-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 <math>H^s</math>, <math>s < 1</math>, is known in the following cases:
For the defocussing NLKG, GWP in <math>H^s</math>, <math>s < 1</math>, is known in the following cases:
* <math>d=3, p = 3, s > 3/4</math> [[KnPoVe-p2]]
* <math>d=3, p = 3, s > 3/4</math> [[KnPoVe-p2]]
* <math>d=3, 3 \leq p < 5, s > [4(p-1) + (5-p)(3p-3-4)]/[2(p-1)(7-p)]</math> [[MiaZgFg-p]]
* <math>d=3, 3 \leq p < 5, s > [4(p-1) + (5-p)(3p-3-4)]/[2(p-1)(7-p)]</math> [[MiaZgFg-p]]
* <math>d=3, 2 < p < 3, or n\geq4, (d+1)^2/((d-1)^2+4) \leq p < (d-1)/(d-3)</math>, and
* <math>d=3, 2 < p < 3, or n\geq4, (d+1)^2/((d-1)^2+4) \leq p <
(d-1)/(d-3)</math>, and


<center><math>s > [2(p-1)^2 - (d+2-p(d-2))(d+1-p(d-1))] / [2(p-1)(d+1-p(d-3))]</math></center>
<center><math>s > [2(p-1)^2 - (d+2-p(d-2))(d+1-p(d-1))] /
[2(p-1)(d+1-p(d-3))]</math></center>


[MiaZgFg-p]. Note that this is the range of p for which s_conf obeys both the scaling condition <math>s_{conf} > s_c</math> and the condition (**).
[MiaZgFg-p]. Note that this is the range of p for which s_conf obeys both the scaling condition <math>s_{conf} > s_c</math> and the condition (**).
 
* <math>d=2, 3 \leq p \leq 5, s > (p-2)/(p-1)</math> [[Fo-p]]; this is
* <math>d=2, 3 \leq p \leq 5, s > (p-2)/(p-1)</math> [[Fo-p]]; this is for the NLW instead of NLKG.
for the NLW instead of NLKG.
* <math>d=2, p > 5, s > (p-1)/p</math> [[Fo-p]]; this is for the NLW instead of NLKG.
* <math>d=2, p > 5, s > (p-1)/p</math> [[Fo-p]]; this is for the NLW
 
instead of NLKG. GWP and blowup has also been studied for the NLW with a conformal factor <center><math>\Box u = (t^2 + (1 - (t^2-x^2)/4)^2)^{-(d-1)p/4 + (d+3)/4} |u|^p</math>;</center> the significance of this factor is that it behaves well under conformal compactification. See [[Aa2002]], [[BcKkZz2002]], [[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.
GWP and blowup has also been studied for the NLW with a conformal factor
 
<center><math>\Box u = (t^2 + (1 - (t^2-x^2)/4)^2)^{-(d-1)p/4 + (d+3)/4} |u|^p</math>;</center>
 
the significance of this factor is that it behaves well under conformal compactification. See [[Aa2002]], [[BcKkZz2002]], [[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.


====Specific semilinear wave equations====
====Specific semilinear wave equations====


* [[Sine-Gordon]]
* [[Sine-Gordon]]
* [[Liouville's equation]]
* [[Quadratic NLW/NLKG]]
* [[Quadratic NLW/NLKG]]
* [[Cubic NLW/NLKG]] ([[Cubic NLW/NLKG on R|on R]], [[Cubic NLW/NLKG on R2|on R^2]], [[Cubic NLW/NLKG on R3|on R^3]], and [[Cubic NLW/NLKG on R4|on R^4]])
* [[Cubic NLW/NLKG]] ([[Cubic NLW/NLKG on R|on R]], [[Cubic NLW/NLKG on R2|on R^2]], [[Cubic NLW/NLKG on R3|on R^3]], and [[Cubic NLW/NLKG on R4|on R^4]])

Latest revision as of 23:37, 22 January 2009

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.

  • 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.
    • When 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 to keep the non-linearity integrable, because there is no Strichartz smoothing to exploit.
  • Finally, in three dimensions one has ill-posedness when and Lb1993.
  • In dimensions 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