Semilinear NLW: Difference between revisions

From DispersiveWiki
Jump to navigationJump to search
m (bibliography test)
Line 5: Line 5:
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( f ) , \Box f  = 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 | \phi_t |^2 / 2 + | \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]].




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 <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 />


{| 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 88: Line 88:
|}
|}


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 94: Line 94:
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.
Line 105: Line 105:
<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
** By using standard Strichartz estimates this was proven with (*) replaced by
Line 111: Line 111:
<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>


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]].
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.
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


Line 125: Line 125:
[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 for the NLW instead of NLKG.
* <math>d=2, 3 \leq p \leq 5, s > (p-2)/(p-1)</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.
* <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
GWP and blowup has also been studied for the NLW with a conformal factor
Line 142: Line 142:
* [[Sine-Gordon]]
* [[Sine-Gordon]]
* [[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]])
* [[Quartic NLW/NLKG]]
* [[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]])
* [[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]])
* [[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]]
[[Category:Wave]]
[[Category:Equations]]
[[Category:Equations]]

Revision as of 00:00, 15 August 2006

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 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 .

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 Kp1993.

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} 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

Retrieved from "https://dispersivewiki.org/DispersiveWiki/index.php?title=Semilinear_NLW&oldid=4119"