Semilinear NLW: Difference between revisions

From DispersiveWiki
Jump to navigationJump to search
No edit summary
 
(15 intermediate revisions by 4 users not shown)
Line 1: Line 1:
===Semilinear wave equations===
===Semilinear wave equations===
 
__TOC__
[Note: Many references needed here!]
[Note: Many references needed here!]


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


<center><math>\Box = F( f ) , \Box = f + F( f )</math></center>
<center><math>\Box \phi = F( \phi ) , \Box \phi = \phi + F( \phi )</math></center>


respectively where <math>F</math> is a function only of  <math>f</math>  and not of its derivatives, which vanishes to more than first order.  
respectively where <math>F</math> is a function only of  <math>f</math>  and not of its derivatives, which vanishes to more than first order.  


Typically <math>F</math> grows like <math>| f |^p</math> for some power <math>p</math>. 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 | f _t |^2 / 2 + | \nabla f |^2 / 2 + V( f )\ 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>| f |^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 defocussing; if <math>V</math> is negative definite we call the NLW focussing. The term "coercive" does not have a standard definition, but generally denotes a potential <math>V</math> which is positive for large values of  <math>f</math> .
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]].


[[semilinear NLW]]


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  
 
<center>
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 />
<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 />


{| class="MsoNormalTable" style="width: 100.0%; mso-cellspacing: 1.5pt; mso-padding-alt: 0in 0in 0in 0in" width="100%" border="1"
{| class="MsoNormalTable" style="width: 100.0%; mso-cellspacing: 1.5pt; mso-padding-alt: 0in 0in 0in 0in" width="100%" border="1"
Line 90: Line 92:
|}
|}


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
====Necessary conditions for [[LWP]] ====
 
The following necessary conditions for [[LWP]] are known.  
<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 [[Bibliography#MeZaa2003|MeZaa2003]]; earlier results are in [[Bibliography#AntMe2001|AntMe2001]], [[Bibliography#CafFri1986|CafFri1986]], [[Bibliography#Al1995|Al1995]], [[Bibliography#KiLit1993|KiLit1993]], [[Bibliography#KiLit1993b|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> [[Bibliography#Lb1993|Lb1993]]. <br />
* 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 dimensions d\leq3 the above necessary conditions are also sufficient for LWP.
* In dimensions <math>d\leq3 </math> 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]].
 
* 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.
and excluding the double endpoint when (*) holds with equality and s=s_{conf} [[Bibliography#Ta1999|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 [[Bibliography#KeTa1998|KeTa1998]] for the double endpoint when (**) holds with equality and s=s_{conf}, and [[Bibliography#LbSo1995|LbSo1995]] for all other cases. A slightly weaker result also appears in [[Bibliography#Kp1994|Kp1994]].
 
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 [[Bibliography#Gl1992|Gl1992]]; see also [[Bibliography#SaSw1994|SaSw1994]]. For radial data this was shown in [[Bibliography#Sw1988|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, 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, p = 3, s > 3/4</math> [[references:KnPoVe-p2 KnPoVe-p2]]
<center><math>s > [2(p-1)^2 - (d+2-p(d-2))(d+1-p(d-1))] /
* <math>d=3, 3 \leq p < 5, s > [4(p-1) + (5-p)(3p-3-4)]/[2(p-1)(7-p)]</math> [MiaZgFg-p]
[2(p-1)(d+1-p(d-3))]</math></center>
* <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>


[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 [[Bibliography#Aa2002|Aa2002]], [[Bibliography#BcKkZz2002|BcKkZz2002]], [[Bibliography#Gue2003|Gue2003]] for some recent results.
 
 
----  [[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]]
 
----  [[Category:Equations]]
 
====Non-relativistic limit of NLKG====
 
By inserting a parameter <math>c</math> (the speed of light), one can rewrite NLKG as
 
<center><math>u_{tt}/c^2 -  D  u + c^2 u + f(u) = 0</math>.</center>
 
One can then ask for what happens in the non-relativistic limit <math>c \rightarrow \infty</math> (keeping the initial position fixed, and dealing with the initial velocity appropriately). In Fourier space, <math>u</math> should be localized near the double hyperboloid
 
<center><math>t  = \pm c \sqrt{c^2 +  x^2}</math>.</center>
 
In the non-relativistic limit this becomes two paraboloids
 
<center><math>t  = \pm (c^2 +  x^2/2)</math></center>
 
and so one expects <math>u</math> to resolve as
 
<center><math> u = exp(i c^2 t) v_+ + exp(-i c^2 t) v_- </math></center>
<center><math> u_t = ic^2 exp(ic^2 t) v_+ - ic^2 exp(ic^2 t) v_- </math></center>
 
where <math>v_+</math>, <math>v_-</math> solve some suitable NLS.
 
A special case arises if one assumes <math>(u_t - ic^2 u)</math> to be small at time zero (say <math>o(c)</math> in some Sobolev norm). Then one expects <math>v_-</math> to vanish and to get a scalar NLS. Many results of this nature exist, see [Mac-p], [[Bibliography#Nj1990|Nj1990]], [[Bibliography#Ts1984|Ts1984]], [MacNaOz-p], [Na-p]. In more general situations one expects <math>v_+</math> and <math>v_-</math> to evolve by a coupled NLS; see [[Bibliography#MasNa2002|MasNa2002]].
 
Heuristically, the frequency <math>\ll c</math> portion of the evolution should evolve in a Schrodinger-type manner, while the frequency <math>\gg c</math> portion of the evolution should evolve in a wave-type manner. (This is consistent with physical intuition, since frequency is proportional to momentum, and hence (in the nonrelativistic regime) to velocity).
 
A similar non-relativistic limit result holds for the [#mkg Maxwell-Klein-Gordon] system (in the Coulomb gauge), where the limiting equation is the coupled <br /> Schrodinger-Poisson system
 
<center><math>i v^+_t +  D  v/2 = u v^+ </math></center>
<center><math>i v^-_t -  D  v/2 = u v^- </math></center>
<center><math>D  u = - |v^+|^2 + |v^-|^2</math></center>
 
under reasonable <math>H^1</math> hypotheses on the initial data [BecMauSb-p]. The asymptotic relation between the MKG-CG fields  <math>f</math> , <math>A</math>, <math>A_0</math> and the Schrodinger-Poisson fields u, v^+, v^- are
 
<center><math>A_0 \sim u </math></center>
<center><math>f  \sim exp(ic^2 t) v^+ + exp(-ic^2 t) v^- </math></center>
<center><math>f _t \sim i M exp(ic^2)v^+ - i M exp(-ic^2 t) v^-</math></center>
 
where <math>M = sqrt{c^4 - c^2 D}</math> (a variant of <math>c^2</math>).
 
 
----  [[Category:Equations]]


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


* [[Sine-Gordon]]
* [[Liouville's equation]]
* [[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]])
* [[Quartic NLW/NLKG]]
* [[Quintic NLW/NLKG]] ([[Quintic NLW/NLKG on R|on R]], [[Quintic NLW/NLKG on R2|on R^2]], and [[Quintic NLW/NLKG on R3|on R^3]])
* [[Septic NLW/NLKG]] ([[Septic NLW/NLKG on R|on R]], [[Septic NLW/NLKG on R2|on R^2]], and [[Septic NLW/NLKG on R3|on R^3]])


 
[[Category:Wave]]
[[Sine-Gordon]]
[[Category:Equations]]
 
[[Quadratic NLW/NLKG]]
 
[[Cubic NLW/NLKG on R]]
 
[[Cubic NLW/NLKG on R2]]
[[Cubic NLW/NLKG on R3]]
 
[[Cubic NLW/NLKG on R4]]
 
[[Quartic NLW/NLKG]]
 
[[Quintic NLW/NLKG on R]]
 
[[Quintic NLW/NLKG on R2]]
[[Quintic NLW/NLKG on R3]]
 
[[Septic NLW/NLKG on R]]
 
[[Septic NLW/NLKG on R2]]
 
[[Septic NLW/NLKG on R3]]

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