https://dispersivewiki.org/DispersiveWiki/api.php?action=feedcontributions&user=Oub&feedformat=atomDispersiveWiki - User contributions [en]2024-03-29T08:29:48ZUser contributionsMediaWiki 1.39.3https://dispersivewiki.org/DispersiveWiki/index.php?title=Well-posedness&diff=5554Well-posedness2008-10-03T11:09:54Z<p>Oub: Mark up fixed</p>
<hr />
<div><center>What is well-posedness?</center><br />
<br />
By well-posedness in <math>H^s</math> we generally mean that there exists a unique solution u for some time T for each set of initial data in <math>H^s</math>, which stays in <math>H^s</math> and depends continuously on the initial data as a map from <math>H^s</math> to <math>H^s</math>. However, there are a couple subtleties involved here.<br />
<br />
* Existence. For classical (smooth) solutions it is clear what it means for a solution to exist; for rough solutions one usually asks (as a bare minimum) for a solution to exist in the sense of distributions. (One may sometimes have to write the equation in conservation form before one can make sense of a distribution). It is possible for negative regularity solutions to exist if there is a sufficient amount of local smoothing available.<br />
* Uniqueness. There are many different notions of uniqueness. One common one is uniqueness in the class of limits of smooth solutions. Another is uniqueness assuming certain spacetime regularity assumptions on the solution. A stronger form of uniqueness is in the class of all <math>H^s</math> functions. Stronger still is uniqueness in the class of all distributions for which the equation makes sense.<br />
* Time of existence. In subcritical situations the time of existence typically depends only on the <math>H^s</math> norm of the initial data, or at a bare minimum one should get a fixed non-zero time of existence for data of sufficiently small norm. When combined with a conservation law this can often be extended to global existence. In critical situations one typically obtains global existence for data of small norm, and local existence for data of large norm but with a time of existence depending on the profile of the data (in particular, the frequencies where the norm is largest) and not just on the norm itself.<br />
* Continuity. There are many different ways the solution map can be continuous from <math>H^s</math> to <math>H^s</math>. One of the strongest is real analyticity (which is what is commonly obtained by iteration methods). Weaker than this are various types of <math>C^k </math> continuity ( <math>C^1 </math>, <math>C^2 </math> , <math>C^3 </math>, etc.). If the solution map is C^k, then this implies that the k^th derivative at the origin is in <math>H^s</math>, which roughly corresponds to some iterate (often the k^th iterate) lying in <math>H^s</math>. Weaker than this is Lipschitz continuity, and weaker than that is uniform continuity. Finally, there is just plain old continuity. Interestingly, several examples have emerged recently in which one form of continuity holds but not another; in particular we now have several examples (critical wave maps, low-regularity periodic [[KdV]] and [[mKdV]], [[Benjamin-Ono]], [[QNLW|quasilinear wave equations]], ...) where the solution map is continuous but not uniformly continuous.<br />
<br />
For a survey of LWP and GWP issues, see [[Ta2002]].<br />
<br />
[[Category:Concept]]</div>Oubhttps://dispersivewiki.org/DispersiveWiki/index.php?title=Well-posedness&diff=5553Well-posedness2008-10-03T11:01:27Z<p>Oub: Make up</p>
<hr />
<div><center>What is well-posedness?</center><br />
<br />
By well-posedness in <math>H^s</math> we generally mean that there exists a unique solution u for some time T for each set of initial data in <math>H^s</math>, which stays in <math>H^s</math> and depends continuously on the initial data as a map from <math>H^s</math> to <math>H^s</math>. However, there are a couple subtleties involved here.<br />
<br />
* Existence. For classical (smooth) solutions it is clear what it means for a solution to exist; for rough solutions one usually asks (as a bare minimum) for a solution to exist in the sense of distributions. (One may sometimes have to write the equation in conservation form before one can make sense of a distribution). It is possible for negative regularity solutions to exist if there is a sufficient amount of local smoothing available.<br />
* Uniqueness. There are many different notions of uniqueness. One common one is uniqueness in the class of limits of smooth solutions. Another is uniqueness assuming certain spacetime regularity assumptions on the solution. A stronger form of uniqueness is in the class of all <math>H^s</math> functions. Stronger still is uniqueness in the class of all distributions for which the equation makes sense.<br />
* Time of existence. In subcritical situations the time of existence typically depends only on the <math>H^s</math> norm of the initial data, or at a bare minimum one should get a fixed non-zero time of existence for data of sufficiently small norm. When combined with a conservation law this can often be extended to global existence. In critical situations one typically obtains global existence for data of small norm, and local existence for data of large norm but with a time of existence depending on the profile of the data (in particular, the frequencies where the norm is largest) and not just on the norm itself.<br />
* Continuity. There are many different ways the solution map can be continuous from <math>H^s</math> to <math>H^s</math>. One of the strongest is real analyticity (which is what is commonly obtained by iteration methods). Weaker than this are various types of C^k continuity (C^1, C^2, C^3, etc.). If the solution map is C^k, then this implies that the k^th derivative at the origin is in <math>H^s</math>, which roughly corresponds to some iterate (often the k^th iterate) lying in <math>H^s</math>. Weaker than this is Lipschitz continuity, and weaker than that is uniform continuity. Finally, there is just plain old continuity. Interestingly, several examples have emerged recently in which one form of continuity holds but not another; in particular we now have several examples (critical wave maps, low-regularity periodic [[KdV]] and [[mKdV]], [[Benjamin-Ono]], [[QNLW|quasilinear wave equations]], ...) where the solution map is continuous but not uniformly continuous.<br />
<br />
For a survey of LWP and GWP issues, see [[Ta2002]].<br />
<br />
[[Category:Concept]]</div>Oubhttps://dispersivewiki.org/DispersiveWiki/index.php?title=Semilinear_NLW&diff=4490Semilinear NLW2007-01-23T11:42:49Z<p>Oub: /* Semilinear wave equations */ format change</p>
<hr />
<div>===Semilinear wave equations===<br />
__TOC__<br />
[Note: Many references needed here!]<br />
<br />
Semilinear wave equations (NLW) and semi-linear Klein-Gordon equations (NLKG) take the form<br />
<br />
<center><math>\Box \phi = F( \phi ) , \Box \phi = \phi + F( \phi )</math></center><br />
<br />
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. <br />
<br />
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]]<br />
<br />
<center><math>\int \frac{ |\phi_t |^2}{ 2} + \frac{|\nabla \phi |^2}{2} + V( \phi )\ dx.</math></center><br />
<br />
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]].<br />
<br />
<br />
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>.<br />
<br />
The scaling regularity is <br />
<center><br />
<math>s_c = \frac{d}{2} - \frac{2}{(p-1)}</math>. <br />
</center><br />
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 /><br />
<br />
{| class="MsoNormalTable" style="width: 100.0%; mso-cellspacing: 1.5pt; mso-padding-alt: 0in 0in 0in 0in" width="100%" border="1"<br />
|- style="mso-yfti-irow: 0; mso-yfti-firstrow: yes"<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
Dimension d<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
Strauss exponent (NLKG)<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
<math>L^2</math>-critical exponent<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
Strauss exponent (NLW)<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
H^{1/2}-critical exponent<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
H^1-critical exponent<br />
|- style="mso-yfti-irow: 1"<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
1<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
3.56155...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
5<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
infinity<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
infinity<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
N/A<br />
|- style="mso-yfti-irow: 2"<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2.41421...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
3<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
3.56155...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
5<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
infinity<br />
|- style="mso-yfti-irow: 3"<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
3<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2.33333...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2.41421...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
3<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
5<br />
|- style="mso-yfti-irow: 4; mso-yfti-lastrow: yes"<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
4<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
1.78078...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2.33333...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
3<br />
|}<br />
<br />
====Necessary conditions for [[LWP]] ====<br />
The following necessary conditions for [[LWP]] are known. <br />
<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.<br />
** 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. <br />
* Finally, in three dimensions one has [[ill-posedness]] when <math>p=2</math> and <math>s = s_{conf} = 0</math> [[Lb1993]].<br />
<br />
* In dimensions <math>d\leq3 </math> the above necessary conditions are also sufficient for LWP.<br />
* For d>4 sufficiency is only known assuming the condition<br />
<center><math>p (d/4-s) \leq 1/2 ( (d+3)/2 - s)</math> (*)</center><br />
and excluding the double endpoint when (*) holds with equality and s=s_{conf} [[Ta1999]]. The main tool is two-scale [[Strichartz estimates]].<br />
* 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.<br />
For the defocussing NLKG, GWP in <math>H^s</math>, <math>s < 1</math>, is known in the following cases:<br />
* <math>d=3, p = 3, s > 3/4</math> [[KnPoVe-p2]]<br />
* <math>d=3, 3 \leq p < 5, s > [4(p-1) + (5-p)(3p-3-4)]/[2(p-1)(7-p)]</math> [[MiaZgFg-p]]<br />
* <math>d=3, 2 < p < 3, or n\geq4, (d+1)^2/((d-1)^2+4) \leq p <<br />
(d-1)/(d-3)</math>, and<br />
<br />
<center><math>s > [2(p-1)^2 - (d+2-p(d-2))(d+1-p(d-1))] /<br />
[2(p-1)(d+1-p(d-3))]</math></center><br />
<br />
[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 (**).<br />
* <math>d=2, 3 \leq p \leq 5, s > (p-2)/(p-1)</math> [[Fo-p]]; this is<br />
for the NLW instead of NLKG.<br />
* <math>d=2, p > 5, s > (p-1)/p</math> [[Fo-p]]; this is for the NLW<br />
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.<br />
<br />
====Specific semilinear wave equations====<br />
<br />
* [[Sine-Gordon]]<br />
* [[Quadratic NLW/NLKG]]<br />
* [[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]])<br />
* [[Quartic NLW/NLKG]]<br />
* [[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]])<br />
* [[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]])<br />
<br />
[[Category:Wave]]<br />
[[Category:Equations]]</div>Oubhttps://dispersivewiki.org/DispersiveWiki/index.php?title=Strichartz_estimates&diff=4481Strichartz estimates2007-01-17T10:22:01Z<p>Oub: Add Linear Strichartz inequality</p>
<hr />
<div>{{stub}}<br />
<br />
Strichartz estimates are spacetime estimates on homogeneous and<br />
inhomogeneous linear dispersive and wave equations. They are<br />
particularly useful for solving semilinear perturbations of such<br />
equations, in which no derivatives are present in the nonlinearity.<br />
<br />
Strichartz estimates can be derived abstractly as a consequence of a<br />
dispersive inequality and an energy inequality.<br />
<br />
==Linear Strichartz estimate ==<br />
Let <math> \dot H^{\alpha}(\Bbb R^n) </math> denote the homogeneous<br />
Sobolev space with norm <br />
<center><br />
<math><br />
\left\| u \right\|_{\dot H^{\alpha}(\Bbb R^n)}<br />
= \left\|(-\Delta^{\alpha/2}) u \right\|_{L^2(\Bbb R^n)}<br />
</math><br />
</center><br />
If <math> u </math> solves the ''linear wave equation''<br />
<center><br />
<math><br />
\Box u = F(t,x)<br />
</math><br />
</center><br />
with data<br />
<center><br />
<math><br />
u(0,\cdot)=f \qquad \partial_t u (0,\cdot )=g<br />
</math><br />
</center><br />
then the Strichartz estimates states that <br />
<br />
<center> <math> <br />
\left\| u \right\|_{L^{4}({{\Bbb R}^{3+1}}_+)} <br />
\leq C \left( <br />
\left\| f \right\|_{{\dot H^{1/2}}(\Bbb R^3)} <br />
+ \left\| g \right\|_{{\dot H^{-1/2}}(\Bbb R^3)}<br />
+ \int\limits_0^{\infty} \left\| F \right\|_{L^2(\Bbb R^{3})} <br />
\right)<br />
</math> </center><br />
<br />
<br />
[[Category:Estimates]] [[Category:Schrodinger]] [[Category:Wave]] <br />
[[Category:Airy]]</div>Oubhttps://dispersivewiki.org/DispersiveWiki/index.php?title=Semilinear_NLW&diff=4480Semilinear NLW2007-01-16T14:12:32Z<p>Oub: /* Semilinear wave equations */ Ditto</p>
<hr />
<div>===Semilinear wave equations===<br />
__TOC__<br />
[Note: Many references needed here!]<br />
<br />
Semilinear wave equations (NLW) and semi-linear Klein-Gordon equations (NLKG) take the form<br />
<br />
<center><math>\Box \phi = F( \phi ) , \Box \phi = \phi + F( \phi )</math></center><br />
<br />
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. <br />
<br />
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]]<br />
<br />
<center><math>\int \frac{ |\phi_t |^2}{ 2} + \frac{|\nabla \phi |^2}{2} + V( \phi )\ dx.</math></center><br />
<br />
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]].<br />
<br />
<br />
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>.<br />
<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 /><br />
<br />
{| class="MsoNormalTable" style="width: 100.0%; mso-cellspacing: 1.5pt; mso-padding-alt: 0in 0in 0in 0in" width="100%" border="1"<br />
|- style="mso-yfti-irow: 0; mso-yfti-firstrow: yes"<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
Dimension d<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
Strauss exponent (NLKG)<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
<math>L^2</math>-critical exponent<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
Strauss exponent (NLW)<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
H^{1/2}-critical exponent<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
H^1-critical exponent<br />
|- style="mso-yfti-irow: 1"<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
1<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
3.56155...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
5<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
infinity<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
infinity<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
N/A<br />
|- style="mso-yfti-irow: 2"<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2.41421...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
3<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
3.56155...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
5<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
infinity<br />
|- style="mso-yfti-irow: 3"<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
3<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2.33333...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2.41421...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
3<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
5<br />
|- style="mso-yfti-irow: 4; mso-yfti-lastrow: yes"<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
4<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
1.78078...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2.33333...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
3<br />
|}<br />
<br />
====Necessary conditions for [[LWP]] ====<br />
The following necessary conditions for [[LWP]] are known. <br />
<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.<br />
** 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. <br />
* Finally, in three dimensions one has [[ill-posedness]] when <math>p=2</math> and <math>s = s_{conf} = 0</math> [[Lb1993]].<br />
<br />
* In dimensions <math>d\leq3 </math> the above necessary conditions are also sufficient for LWP.<br />
* For d>4 sufficiency is only known assuming the condition<br />
<center><math>p (d/4-s) \leq 1/2 ( (d+3)/2 - s)</math> (*)</center><br />
and excluding the double endpoint when (*) holds with equality and s=s_{conf} [[Ta1999]]. The main tool is two-scale [[Strichartz estimates]].<br />
* 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.<br />
For the defocussing NLKG, GWP in <math>H^s</math>, <math>s < 1</math>, is known in the following cases:<br />
* <math>d=3, p = 3, s > 3/4</math> [[KnPoVe-p2]]<br />
* <math>d=3, 3 \leq p < 5, s > [4(p-1) + (5-p)(3p-3-4)]/[2(p-1)(7-p)]</math> [[MiaZgFg-p]]<br />
* <math>d=3, 2 < p < 3, or n\geq4, (d+1)^2/((d-1)^2+4) \leq p <<br />
(d-1)/(d-3)</math>, and<br />
<br />
<center><math>s > [2(p-1)^2 - (d+2-p(d-2))(d+1-p(d-1))] /<br />
[2(p-1)(d+1-p(d-3))]</math></center><br />
<br />
[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 (**).<br />
* <math>d=2, 3 \leq p \leq 5, s > (p-2)/(p-1)</math> [[Fo-p]]; this is<br />
for the NLW instead of NLKG.<br />
* <math>d=2, p > 5, s > (p-1)/p</math> [[Fo-p]]; this is for the NLW<br />
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.<br />
<br />
====Specific semilinear wave equations====<br />
<br />
* [[Sine-Gordon]]<br />
* [[Quadratic NLW/NLKG]]<br />
* [[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]])<br />
* [[Quartic NLW/NLKG]]<br />
* [[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]])<br />
* [[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]])<br />
<br />
[[Category:Wave]]<br />
[[Category:Equations]]</div>Oubhttps://dispersivewiki.org/DispersiveWiki/index.php?title=Semilinear_NLW&diff=4479Semilinear NLW2007-01-16T14:11:57Z<p>Oub: /* Semilinear wave equations */ Typo</p>
<hr />
<div>===Semilinear wave equations===<br />
__TOC__<br />
[Note: Many references needed here!]<br />
<br />
Semilinear wave equations (NLW) and semi-linear Klein-Gordon equations (NLKG) take the form<br />
<br />
<center><math>\Box \phi = F( \phi ) , \Box \phi = \phi + F( \phi )</math></center><br />
<br />
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. <br />
<br />
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]]<br />
<br />
<center><math>\int \frac{ <br />\phi_t |^2}{ 2} + \frac{|\nabla \phi |^2}{2} + V( \phi )\ dx.</math></center><br />
<br />
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]].<br />
<br />
<br />
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>.<br />
<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 /><br />
<br />
{| class="MsoNormalTable" style="width: 100.0%; mso-cellspacing: 1.5pt; mso-padding-alt: 0in 0in 0in 0in" width="100%" border="1"<br />
|- style="mso-yfti-irow: 0; mso-yfti-firstrow: yes"<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
Dimension d<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
Strauss exponent (NLKG)<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
<math>L^2</math>-critical exponent<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
Strauss exponent (NLW)<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
H^{1/2}-critical exponent<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
H^1-critical exponent<br />
|- style="mso-yfti-irow: 1"<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
1<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
3.56155...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
5<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
infinity<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
infinity<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
N/A<br />
|- style="mso-yfti-irow: 2"<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2.41421...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
3<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
3.56155...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
5<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
infinity<br />
|- style="mso-yfti-irow: 3"<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
3<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2.33333...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2.41421...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
3<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
5<br />
|- style="mso-yfti-irow: 4; mso-yfti-lastrow: yes"<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
4<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
1.78078...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2.33333...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
3<br />
|}<br />
<br />
====Necessary conditions for [[LWP]] ====<br />
The following necessary conditions for [[LWP]] are known. <br />
<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.<br />
** 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. <br />
* Finally, in three dimensions one has [[ill-posedness]] when <math>p=2</math> and <math>s = s_{conf} = 0</math> [[Lb1993]].<br />
<br />
* In dimensions <math>d\leq3 </math> the above necessary conditions are also sufficient for LWP.<br />
* For d>4 sufficiency is only known assuming the condition<br />
<center><math>p (d/4-s) \leq 1/2 ( (d+3)/2 - s)</math> (*)</center><br />
and excluding the double endpoint when (*) holds with equality and s=s_{conf} [[Ta1999]]. The main tool is two-scale [[Strichartz estimates]].<br />
* 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.<br />
For the defocussing NLKG, GWP in <math>H^s</math>, <math>s < 1</math>, is known in the following cases:<br />
* <math>d=3, p = 3, s > 3/4</math> [[KnPoVe-p2]]<br />
* <math>d=3, 3 \leq p < 5, s > [4(p-1) + (5-p)(3p-3-4)]/[2(p-1)(7-p)]</math> [[MiaZgFg-p]]<br />
* <math>d=3, 2 < p < 3, or n\geq4, (d+1)^2/((d-1)^2+4) \leq p <<br />
(d-1)/(d-3)</math>, and<br />
<br />
<center><math>s > [2(p-1)^2 - (d+2-p(d-2))(d+1-p(d-1))] /<br />
[2(p-1)(d+1-p(d-3))]</math></center><br />
<br />
[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 (**).<br />
* <math>d=2, 3 \leq p \leq 5, s > (p-2)/(p-1)</math> [[Fo-p]]; this is<br />
for the NLW instead of NLKG.<br />
* <math>d=2, p > 5, s > (p-1)/p</math> [[Fo-p]]; this is for the NLW<br />
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.<br />
<br />
====Specific semilinear wave equations====<br />
<br />
* [[Sine-Gordon]]<br />
* [[Quadratic NLW/NLKG]]<br />
* [[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]])<br />
* [[Quartic NLW/NLKG]]<br />
* [[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]])<br />
* [[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]])<br />
<br />
[[Category:Wave]]<br />
[[Category:Equations]]</div>Oubhttps://dispersivewiki.org/DispersiveWiki/index.php?title=Semilinear_NLW&diff=4478Semilinear NLW2007-01-16T14:11:17Z<p>Oub: /* Necessary conditions for LWP */ Structure</p>
<hr />
<div>===Semilinear wave equations===<br />
__TOC__<br />
[Note: Many references needed here!]<br />
<br />
Semilinear wave equations (NLW) and semi-linear Klein-Gordon equations (NLKG) take the form<br />
<br />
<center><math>\Box \phi = F( \phi ) , \Box \phi = \phi + F( \phi )</math></center><br />
<br />
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. <br />
<br />
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]]<br />
<br />
<center><math>\int | \frac{\phi_t |^2}{ 2} + | \frac{\nabla \phi |^2}{2} + V( \phi )\ dx.</math></center><br />
<br />
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]].<br />
<br />
<br />
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>.<br />
<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 /><br />
<br />
{| class="MsoNormalTable" style="width: 100.0%; mso-cellspacing: 1.5pt; mso-padding-alt: 0in 0in 0in 0in" width="100%" border="1"<br />
|- style="mso-yfti-irow: 0; mso-yfti-firstrow: yes"<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
Dimension d<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
Strauss exponent (NLKG)<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
<math>L^2</math>-critical exponent<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
Strauss exponent (NLW)<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
H^{1/2}-critical exponent<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
H^1-critical exponent<br />
|- style="mso-yfti-irow: 1"<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
1<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
3.56155...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
5<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
infinity<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
infinity<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
N/A<br />
|- style="mso-yfti-irow: 2"<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2.41421...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
3<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
3.56155...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
5<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
infinity<br />
|- style="mso-yfti-irow: 3"<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
3<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2.33333...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2.41421...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
3<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
5<br />
|- style="mso-yfti-irow: 4; mso-yfti-lastrow: yes"<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
4<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
1.78078...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2.33333...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
3<br />
|}<br />
<br />
====Necessary conditions for [[LWP]] ====<br />
The following necessary conditions for [[LWP]] are known. <br />
<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.<br />
** 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. <br />
* Finally, in three dimensions one has [[ill-posedness]] when <math>p=2</math> and <math>s = s_{conf} = 0</math> [[Lb1993]].<br />
<br />
* In dimensions <math>d\leq3 </math> the above necessary conditions are also sufficient for LWP.<br />
* For d>4 sufficiency is only known assuming the condition<br />
<center><math>p (d/4-s) \leq 1/2 ( (d+3)/2 - s)</math> (*)</center><br />
and excluding the double endpoint when (*) holds with equality and s=s_{conf} [[Ta1999]]. The main tool is two-scale [[Strichartz estimates]].<br />
* 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.<br />
For the defocussing NLKG, GWP in <math>H^s</math>, <math>s < 1</math>, is known in the following cases:<br />
* <math>d=3, p = 3, s > 3/4</math> [[KnPoVe-p2]]<br />
* <math>d=3, 3 \leq p < 5, s > [4(p-1) + (5-p)(3p-3-4)]/[2(p-1)(7-p)]</math> [[MiaZgFg-p]]<br />
* <math>d=3, 2 < p < 3, or n\geq4, (d+1)^2/((d-1)^2+4) \leq p <<br />
(d-1)/(d-3)</math>, and<br />
<br />
<center><math>s > [2(p-1)^2 - (d+2-p(d-2))(d+1-p(d-1))] /<br />
[2(p-1)(d+1-p(d-3))]</math></center><br />
<br />
[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 (**).<br />
* <math>d=2, 3 \leq p \leq 5, s > (p-2)/(p-1)</math> [[Fo-p]]; this is<br />
for the NLW instead of NLKG.<br />
* <math>d=2, p > 5, s > (p-1)/p</math> [[Fo-p]]; this is for the NLW<br />
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.<br />
<br />
====Specific semilinear wave equations====<br />
<br />
* [[Sine-Gordon]]<br />
* [[Quadratic NLW/NLKG]]<br />
* [[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]])<br />
* [[Quartic NLW/NLKG]]<br />
* [[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]])<br />
* [[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]])<br />
<br />
[[Category:Wave]]<br />
[[Category:Equations]]</div>Oubhttps://dispersivewiki.org/DispersiveWiki/index.php?title=Semilinear_NLW&diff=4477Semilinear NLW2007-01-16T13:59:34Z<p>Oub: /* Necessary conditions for LWP */ Typos</p>
<hr />
<div>===Semilinear wave equations===<br />
__TOC__<br />
[Note: Many references needed here!]<br />
<br />
Semilinear wave equations (NLW) and semi-linear Klein-Gordon equations (NLKG) take the form<br />
<br />
<center><math>\Box \phi = F( \phi ) , \Box \phi = \phi + F( \phi )</math></center><br />
<br />
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. <br />
<br />
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]]<br />
<br />
<center><math>\int | \frac{\phi_t |^2}{ 2} + | \frac{\nabla \phi |^2}{2} + V( \phi )\ dx.</math></center><br />
<br />
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]].<br />
<br />
<br />
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>.<br />
<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 /><br />
<br />
{| class="MsoNormalTable" style="width: 100.0%; mso-cellspacing: 1.5pt; mso-padding-alt: 0in 0in 0in 0in" width="100%" border="1"<br />
|- style="mso-yfti-irow: 0; mso-yfti-firstrow: yes"<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
Dimension d<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
Strauss exponent (NLKG)<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
<math>L^2</math>-critical exponent<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
Strauss exponent (NLW)<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
H^{1/2}-critical exponent<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
H^1-critical exponent<br />
|- style="mso-yfti-irow: 1"<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
1<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
3.56155...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
5<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
infinity<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
infinity<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
N/A<br />
|- style="mso-yfti-irow: 2"<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2.41421...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
3<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
3.56155...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
5<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
infinity<br />
|- style="mso-yfti-irow: 3"<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
3<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2.33333...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2.41421...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
3<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
5<br />
|- style="mso-yfti-irow: 4; mso-yfti-lastrow: yes"<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
4<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
1.78078...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2.33333...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
3<br />
|}<br />
<br />
====Necessary conditions for [[LWP]] ====<br />
The following necessary conditions for [[LWP]] are known. <br />
<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<br />
<br />
<center><math>s_{conf} = (d+1)/4 - 1/(p-1)</math></center><br />
<br />
in the focusing case; the defocusing case is still open. In the<br />
<math>H^{1/2}</math>-critical power or below, this condition is<br />
stronger than the scaling requirement.<br />
<br />
** 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]].<br />
<br />
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.<br />
<br />
#Finally, in three dimensions one has [[ill-posedness]] when <math>p=2</math> and <math>s = s_{conf} = 0</math> [[Lb1993]]. <br /><br />
<br />
** In dimensions d\leq3 the above necessary conditions are also sufficient for LWP.<br />
** For d>4 sufficiency is only known assuming the condition<br />
<br />
<math>p (d/4-s) \leq 1/2 ( (d+3)/2 - s)</math> (*)</center><br />
<br />
and excluding the double endpoint when (*) holds with equality and s=s_{conf} [[Ta1999]]. The main tool is two-scale [[Strichartz estimates]].<br />
<br />
*** By using standard Strichartz estimates this was proven with (*) replaced by<br />
<br />
<center><math>p ((d+1)/4-s) \leq (d+1)/2d ( (d+3)/2 - s)</math>;<br />
(**)</center><br />
<br />
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]].<br />
<br />
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.<br />
<br />
For the defocussing NLKG, GWP in <math>H^s</math>, <math>s < 1</math>, is known in the following cases:<br />
<br />
* <math>d=3, p = 3, s > 3/4</math> [[KnPoVe-p2]]<br />
* <math>d=3, 3 \leq p < 5, s > [4(p-1) + (5-p)(3p-3-4)]/[2(p-1)(7-p)]</math> [[MiaZgFg-p]]<br />
* <math>d=3, 2 < p < 3, or n\geq4, (d+1)^2/((d-1)^2+4) \leq p <<br />
(d-1)/(d-3)</math>, and<br />
<br />
<center><math>s > [2(p-1)^2 - (d+2-p(d-2))(d+1-p(d-1))] /<br />
[2(p-1)(d+1-p(d-3))]</math></center><br />
<br />
[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 (**).<br />
<br />
* <math>d=2, 3 \leq p \leq 5, s > (p-2)/(p-1)</math> [[Fo-p]]; this is<br />
for the NLW instead of NLKG.<br />
* <math>d=2, p > 5, s > (p-1)/p</math> [[Fo-p]]; this is for the NLW<br />
instead of NLKG.<br />
<br />
GWP and blowup has also been studied for the NLW with a conformal factor<br />
<br />
<center><math>\Box u = (t^2 + (1 - (t^2-x^2)/4)^2)^{-(d-1)p/4 +<br />
(d+3)/4} |u|^p</math>;</center><br />
<br />
the significance of this factor is that it behaves well under conformal compactification. See [[Aa2002]], [[BcKkZz2002]], [[Gue2003]] for some recent results.<br />
<br />
A substantial [[scattering for NLW/NLKG|scattering theory for NLW and NLKG]] is known.<br />
<br />
The [[non-relativistic limit]] of NLKG has attracted a fair amount of research.<br />
<br />
====Specific semilinear wave equations====<br />
<br />
* [[Sine-Gordon]]<br />
* [[Quadratic NLW/NLKG]]<br />
* [[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]])<br />
* [[Quartic NLW/NLKG]]<br />
* [[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]])<br />
* [[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]])<br />
<br />
[[Category:Wave]]<br />
[[Category:Equations]]</div>Oubhttps://dispersivewiki.org/DispersiveWiki/index.php?title=Semilinear_NLW&diff=4476Semilinear NLW2007-01-16T13:54:43Z<p>Oub: /* Necessary conditions for LWP */ restructured</p>
<hr />
<div>===Semilinear wave equations===<br />
__TOC__<br />
[Note: Many references needed here!]<br />
<br />
Semilinear wave equations (NLW) and semi-linear Klein-Gordon equations (NLKG) take the form<br />
<br />
<center><math>\Box \phi = F( \phi ) , \Box \phi = \phi + F( \phi )</math></center><br />
<br />
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. <br />
<br />
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]]<br />
<br />
<center><math>\int | \frac{\phi_t |^2}{ 2} + | \frac{\nabla \phi |^2}{2} + V( \phi )\ dx.</math></center><br />
<br />
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]].<br />
<br />
<br />
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>.<br />
<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 /><br />
<br />
{| class="MsoNormalTable" style="width: 100.0%; mso-cellspacing: 1.5pt; mso-padding-alt: 0in 0in 0in 0in" width="100%" border="1"<br />
|- style="mso-yfti-irow: 0; mso-yfti-firstrow: yes"<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
Dimension d<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
Strauss exponent (NLKG)<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
<math>L^2</math>-critical exponent<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
Strauss exponent (NLW)<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
H^{1/2}-critical exponent<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
H^1-critical exponent<br />
|- style="mso-yfti-irow: 1"<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
1<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
3.56155...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
5<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
infinity<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
infinity<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
N/A<br />
|- style="mso-yfti-irow: 2"<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2.41421...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
3<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
3.56155...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
5<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
infinity<br />
|- style="mso-yfti-irow: 3"<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
3<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2.33333...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2.41421...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
3<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
5<br />
|- style="mso-yfti-irow: 4; mso-yfti-lastrow: yes"<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
4<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
1.78078...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2.33333...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
3<br />
|}<br />
<br />
====Necessary conditions for [[LWP]] ====<br />
The following necessary conditions for [[LWP]] are known. <br />
<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<br />
<br />
<center><math>s_{conf} = (d+1)/4 - 1/(p-1)</math></center><br />
<br />
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.<br />
<br />
** 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]].<br />
<br />
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.<br />
<br />
#Finally, in three dimensions one has [[ill-posedness]] when <math>p=2</math> and <math>s = s_{conf} = 0</math> [[Lb1993]]. <br /><br />
<br />
** In dimensions d\leq3 the above necessary conditions are also sufficient for LWP.<br />
** For d>4 sufficiency is only known assuming the condition<br />
<br />
<math>p (d/4-s) \leq 1/2 ( (d+3)/2 - s)</math> (*)</center><br />
<br />
and excluding the double endpoint when (*) holds with equality and s=s_{conf} [[Ta1999]]. The main tool is two-scale [[Strichartz estimates]].<br />
<br />
*** By using standard Strichartz estimates this was proven with (*) replaced by<br />
<br />
<center><math>p ((d+1)/4-s) \leq (d+1)/2d ( (d+3)/2 - s)</math>; (**)</center><br />
<br />
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]].<br />
<br />
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.<br />
<br />
For the defocussing NLKG, GWP in <math>H^s</math>, <math>s < 1</math>, is known in the following cases:<br />
<br />
* <math>d=3, p = 3, s > 3/4</math> [[KnPoVe-p2]]<br />
* <math>d=3, 3 \leq p < 5, s > [4(p-1) + (5-p)(3p-3-4)]/[2(p-1)(7-p)]</math> [[MiaZgFg-p]]<br />
* <math>d=3, 2 < p < 3, or n\geq4, (d+1)^2/((d-1)^2+4) \leq p < (d-1)/(d-3)</math>, and<br />
<br />
<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><br />
<br />
[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 (**).<br />
<br />
* <math>d=2, 3 \leq p \leq 5, s > (p-2)/(p-1)</math> [[Fo-p]]; this is for the NLW instead of NLKG.<br />
* <math>d=2, p > 5, s > (p-1)/p</math> [[Fo-p]]; this is for the NLW instead of NLKG.<br />
<br />
GWP and blowup has also been studied for the NLW with a conformal factor<br />
<br />
<center><math>\Box u = (t^2 + (1 - (t^2-x^2)/4)^2)^{-(d-1)p/4 + (d+3)/4} |u|^p</math>;</center><br />
<br />
the significance of this factor is that it behaves well under conformal compactification. See [[Aa2002]], [[BcKkZz2002]], [[Gue2003]] for some recent results.<br />
<br />
A substantial [[scattering for NLW/NLKG|scattering theory for NLW and NLKG]] is known.<br />
<br />
The [[non-relativistic limit]] of NLKG has attracted a fair amount of research.<br />
<br />
====Specific semilinear wave equations====<br />
<br />
* [[Sine-Gordon]]<br />
* [[Quadratic NLW/NLKG]]<br />
* [[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]])<br />
* [[Quartic NLW/NLKG]]<br />
* [[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]])<br />
* [[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]])<br />
<br />
[[Category:Wave]]<br />
[[Category:Equations]]</div>Oubhttps://dispersivewiki.org/DispersiveWiki/index.php?title=Semilinear_NLW&diff=4475Semilinear NLW2007-01-16T13:52:54Z<p>Oub: New subsection introduced energy expression changed. May be more structure needed?</p>
<hr />
<div>===Semilinear wave equations===<br />
__TOC__<br />
[Note: Many references needed here!]<br />
<br />
Semilinear wave equations (NLW) and semi-linear Klein-Gordon equations (NLKG) take the form<br />
<br />
<center><math>\Box \phi = F( \phi ) , \Box \phi = \phi + F( \phi )</math></center><br />
<br />
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. <br />
<br />
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]]<br />
<br />
<center><math>\int | \frac{\phi_t |^2}{ 2} + | \frac{\nabla \phi |^2}{2} + V( \phi )\ dx.</math></center><br />
<br />
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]].<br />
<br />
<br />
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>.<br />
<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 /><br />
<br />
{| class="MsoNormalTable" style="width: 100.0%; mso-cellspacing: 1.5pt; mso-padding-alt: 0in 0in 0in 0in" width="100%" border="1"<br />
|- style="mso-yfti-irow: 0; mso-yfti-firstrow: yes"<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
Dimension d<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
Strauss exponent (NLKG)<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
<math>L^2</math>-critical exponent<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
Strauss exponent (NLW)<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
H^{1/2}-critical exponent<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
H^1-critical exponent<br />
|- style="mso-yfti-irow: 1"<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
1<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
3.56155...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
5<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
infinity<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
infinity<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
N/A<br />
|- style="mso-yfti-irow: 2"<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2.41421...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
3<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
3.56155...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
5<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
infinity<br />
|- style="mso-yfti-irow: 3"<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
3<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2.33333...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2.41421...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
3<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
5<br />
|- style="mso-yfti-irow: 4; mso-yfti-lastrow: yes"<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
4<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
1.78078...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
2.33333...<br />
| style="padding: .75pt .75pt .75pt .75pt" |<br />
3<br />
|}<br />
<br />
====Necessary conditions for [[LWP]] ====<br />
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<br />
<br />
<center><math>s_{conf} = (d+1)/4 - 1/(p-1)</math></center><br />
<br />
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.<br />
<br />
* 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]].<br />
<br />
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.<br />
<br />
Finally, in three dimensions one has [[ill-posedness]] when <math>p=2</math> and <math>s = s_{conf} = 0</math> [[Lb1993]]. <br /><br />
<br />
* In dimensions d\leq3 the above necessary conditions are also sufficient for LWP.<br />
* For d>4 sufficiency is only known assuming the condition<br />
<br />
<math>p (d/4-s) \leq 1/2 ( (d+3)/2 - s)</math> (*)</center><br />
<br />
and excluding the double endpoint when (*) holds with equality and s=s_{conf} [[Ta1999]]. The main tool is two-scale [[Strichartz estimates]].<br />
<br />
** By using standard Strichartz estimates this was proven with (*) replaced by<br />
<br />
<center><math>p ((d+1)/4-s) \leq (d+1)/2d ( (d+3)/2 - s)</math>; (**)</center><br />
<br />
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]].<br />
<br />
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.<br />
<br />
For the defocussing NLKG, GWP in <math>H^s</math>, <math>s < 1</math>, is known in the following cases:<br />
<br />
* <math>d=3, p = 3, s > 3/4</math> [[KnPoVe-p2]]<br />
* <math>d=3, 3 \leq p < 5, s > [4(p-1) + (5-p)(3p-3-4)]/[2(p-1)(7-p)]</math> [[MiaZgFg-p]]<br />
* <math>d=3, 2 < p < 3, or n\geq4, (d+1)^2/((d-1)^2+4) \leq p < (d-1)/(d-3)</math>, and<br />
<br />
<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><br />
<br />
[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 (**).<br />
<br />
* <math>d=2, 3 \leq p \leq 5, s > (p-2)/(p-1)</math> [[Fo-p]]; this is for the NLW instead of NLKG.<br />
* <math>d=2, p > 5, s > (p-1)/p</math> [[Fo-p]]; this is for the NLW instead of NLKG.<br />
<br />
GWP and blowup has also been studied for the NLW with a conformal factor<br />
<br />
<center><math>\Box u = (t^2 + (1 - (t^2-x^2)/4)^2)^{-(d-1)p/4 + (d+3)/4} |u|^p</math>;</center><br />
<br />
the significance of this factor is that it behaves well under conformal compactification. See [[Aa2002]], [[BcKkZz2002]], [[Gue2003]] for some recent results.<br />
<br />
A substantial [[scattering for NLW/NLKG|scattering theory for NLW and NLKG]] is known.<br />
<br />
The [[non-relativistic limit]] of NLKG has attracted a fair amount of research.<br />
<br />
====Specific semilinear wave equations====<br />
<br />
* [[Sine-Gordon]]<br />
* [[Quadratic NLW/NLKG]]<br />
* [[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]])<br />
* [[Quartic NLW/NLKG]]<br />
* [[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]])<br />
* [[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]])<br />
<br />
[[Category:Wave]]<br />
[[Category:Equations]]</div>Oubhttps://dispersivewiki.org/DispersiveWiki/index.php?title=Quadratic_NLW/NLKG&diff=4474Quadratic NLW/NLKG2007-01-16T13:48:00Z<p>Oub: Typo</p>
<hr />
<div>* Scaling is <math>s_c = \frac{d}{2} - 2</math>.<br />
* For <math>d>4</math> LWP is known for <math>s \geq \frac{d}{2} - 2</math> by Strichartz estimates ([[LbSo1995]]). This is sharp by scaling arguments.<br />
* For <math>d=4</math> LWP is known for <math>s \geq \frac{1}{4}</math> by Strichartz estimates ([[LbSo1995]]).This is sharp from Lorentz invariance (concentration) considerations.<br />
* For <math>d=3</math> LWP is known for <math>s > 0</math> by Strichartz estimates ([[LbSo1995]]).<br />
** One has ill-posedness for <math>s=0</math> ([[Lb1996]]). This is related to the failure of endpoint Strichartz when <math>d=3</math>.<br />
* For <math>d=1,2</math> LWP is known for <math>s\geq 0</math> by Strichartz estimates (or energy estimates and Sobolev in the <math>d=1</math> case).<br />
** For s<0 one has rather severe ill-posedness generically, indeed cannot even interpret the non-linearity <math>f^2</math> as a distribution ([[CtCoTa-p2]]).<br />
** In the [[two-speed wave equations|two-speed case]] one can improve this to <math>s>-1/4</math> for non-linearities of the form <math>F = uv</math> and <math>G = uv</math> ([[Tg-p]]).<br />
<br />
[[Category:Wave]]<br />
[[Category:Equations]]</div>Oubhttps://dispersivewiki.org/DispersiveWiki/index.php?title=Quadratic_NLW/NLKG&diff=4473Quadratic NLW/NLKG2007-01-16T13:47:14Z<p>Oub: Formating the equations</p>
<hr />
<div>* Scaling is <math>s_c = d/2 - 2</math>.<br />
* For <math>d>4</math> LWP is known for <math>s \geq \frac{d}{2} d/2 - 2</math> by Strichartz estimates ([[LbSo1995]]). This is sharp by scaling arguments.<br />
* For <math>d=4</math> LWP is known for <math>s \geq \frac{1}{4}</math> by Strichartz estimates ([[LbSo1995]]).This is sharp from Lorentz invariance (concentration) considerations.<br />
* For <math>d=3</math> LWP is known for <math>s > 0</math> by Strichartz estimates ([[LbSo1995]]).<br />
** One has ill-posedness for <math>s=0</math> ([[Lb1996]]). This is related to the failure of endpoint Strichartz when <math>d=3</math>.<br />
* For <math>d=1,2</math> LWP is known for <math>s\geq 0</math> by Strichartz estimates (or energy estimates and Sobolev in the <math>d=1</math> case).<br />
** For s<0 one has rather severe ill-posedness generically, indeed cannot even interpret the non-linearity <math>f^2</math> as a distribution ([[CtCoTa-p2]]).<br />
** In the [[two-speed wave equations|two-speed case]] one can improve this to <math>s>-1/4</math> for non-linearities of the form <math>F = uv</math> and <math>G = uv</math> ([[Tg-p]]).<br />
<br />
[[Category:Wave]]<br />
[[Category:Equations]]</div>Oubhttps://dispersivewiki.org/DispersiveWiki/index.php?title=DispersiveWiki:Community_Portal&diff=4326DispersiveWiki:Community Portal2006-09-26T16:31:32Z<p>Oub: /* Enumerate equations */</p>
<hr />
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Older discussion [[DispersiveWiki:Community Portal/Archive|has been archived here]].<br />
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Would you like to help out? Suggested projects are posted on our [[Current events| current projects]] page.<br />
<br />
== Connections to other Wikis ==<br />
<br />
<br />
There is a [http://www.wikiindex.com/Category:Math list of Math Wiki's] on Wikipedia. <br />
See for example the [http://en.wikipedia.org/wiki/Category:Partial_differential_equations collection] of PDE related pages on wikipedia. For basic concepts in PDE, perhaps we should further develop<br />
those pages. For more specialized resources like a bibliography, it may be better to use this space. [[User:Colliand|Colliand]] 11:28, 15 September 2006 (EDT)<br />
<br />
==Enumerate equations ==<br />
<br />
Hello <br />
<br />
I am currently searching the net for a possibility to enumerate equations automatically, and to have the possiblty to add labels and references. It seems that there are 2 enhancement of texcv which are capable of doing it (with a little hacking as the authors say): <br />
#http://www.blahtex.org<br />
#http://en.wikipedia.org/wiki/Wikipedia:WikiTeX<br />
Do I understand correctly from earlier discussions that such a functionality would be most welcome?<br />
Regards<br />
[[User:Oub|Uwe Brauer]] 10:36, 26 September 2006 (EDT):<br />
<br />
:Improvements to texvc and referencing are indeed welcome. Please feel free to make edits and suggestions to improve the DispersiveWiki. [[User:Colliand|Colliand]] 11:50, 26 September 2006 (EDT)<br />
<br />
::'''Re: [[User:Colliand|Colliand]]''' ok, sadly I just learnt that blahtex is '''not''' able to enumerate equations automatically (misunderstanding from my side), wikitex however seems to be able. When I know more I come back. <br />
::[[User:Oub|Uwe Brauer]] 12:30, 26 September 2006 (EDT):</div>Oubhttps://dispersivewiki.org/DispersiveWiki/index.php?title=DispersiveWiki:Community_Portal&diff=4325DispersiveWiki:Community Portal2006-09-26T16:30:36Z<p>Oub: /* Enumerate equations */</p>
<hr />
<div>This is the portal for discussing the general Dispersive Wiki project and for making announcements. You can sign your name and timestamp by writing three or four tildes <nowiki>~~~</nowiki> at the end of your message, and use : at the start of a message to indent. Use == (title) == to start a new topic. <br />
<br />
Older discussion [[DispersiveWiki:Community Portal/Archive|has been archived here]].<br />
<br />
Would you like to help out? Suggested projects are posted on our [[Current events| current projects]] page.<br />
<br />
== Connections to other Wikis ==<br />
<br />
<br />
There is a [http://www.wikiindex.com/Category:Math list of Math Wiki's] on Wikipedia. <br />
See for example the [http://en.wikipedia.org/wiki/Category:Partial_differential_equations collection] of PDE related pages on wikipedia. For basic concepts in PDE, perhaps we should further develop<br />
those pages. For more specialized resources like a bibliography, it may be better to use this space. [[User:Colliand|Colliand]] 11:28, 15 September 2006 (EDT)<br />
<br />
==Enumerate equations ==<br />
<br />
Hello <br />
<br />
I am currently searching the net for a possibility to enumerate equations automatically, and to have the possiblty to add labels and references. It seems that there are 2 enhancement of texcv which are capable of doing it (with a little hacking as the authors say): <br />
#http://www.blahtex.org<br />
#http://en.wikipedia.org/wiki/Wikipedia:WikiTeX<br />
Do I understand correctly from earlier discussions that such a functionality would be most welcome?<br />
Regards<br />
[[User:Oub|Uwe Brauer]] 10:36, 26 September 2006 (EDT):<br />
<br />
:Improvements to texvc and referencing are indeed welcome. Please feel free to make edits and suggestions to improve the DispersiveWiki. [[User:Colliand|Colliand]] 11:50, 26 September 2006 (EDT)<br />
<br />
::'''Re: [[User:Colliand|Colliand]]''' ok, sadly I just learnt that blahtex is '''not''' able to enumerate equations automatically, wikitex however seem to be able. When I know more I come back. <br />
::[[User:Oub|Uwe Brauer]] 12:30, 26 September 2006 (EDT):</div>Oubhttps://dispersivewiki.org/DispersiveWiki/index.php?title=DispersiveWiki:Community_Portal&diff=4317DispersiveWiki:Community Portal2006-09-26T14:41:34Z<p>Oub: /* Enumerate equations */</p>
<hr />
<div>This is the portal for discussing the general Dispersive Wiki project and for making announcements. You can sign your name and timestamp by writing three or four tildes <nowiki>~~~</nowiki> at the end of your message, and use : at the start of a message to indent. Use == (title) == to start a new topic. <br />
<br />
Older discussion [[DispersiveWiki:Community Portal/Archive|has been archived here]].<br />
<br />
Would you like to help out? Suggested projects are posted on our [[Current events| current projects]] page.<br />
<br />
== Connections to other Wikis ==<br />
<br />
An [http://tosio.math.toronto.edu/pdewiki/index.php/Main_Page emerging PDEwiki] covering basic concepts of PDEs has been set up. [[User:Colliand|Colliand]] 11:52, 13 September 2006 (EDT)<br />
<br />
<br />
:It turns out that there is already a [http://en.wikipedia.org/wiki/Category:Partial_differential_equations collection] of PDE related pages on wikipedia. For basic concepts in PDE, perhaps we should further develop<br />
:those pages. For more specialized resources like a bibliography, it may be better to use this space. [[User:Colliand|Colliand]] 11:28, 15 September 2006 (EDT)<br />
<br />
==Enumerate equations ==<br />
<br />
Hello <br />
<br />
I am currently searching the net for a possibility to enumerate equations automatically, and to have the possiblty to add labels and references. It seems that there are 2 enhancement of texcv which are capable of doing it (with a little hacking as the authors say): <br />
#http://www.blahtex.org<br />
#http://en.wikipedia.org/wiki/Wikipedia:WikiTeX<br />
Do I understand correctly from earlier discussions that such a functionality would be most welcome?<br />
Regards<br />
[[User:Oub|Uwe Brauer]] 10:36, 26 September 2006 (EDT):</div>Oubhttps://dispersivewiki.org/DispersiveWiki/index.php?title=DispersiveWiki:Community_Portal&diff=4316DispersiveWiki:Community Portal2006-09-26T14:37:19Z<p>Oub: /* Enumerate equations = */</p>
<hr />
<div>This is the portal for discussing the general Dispersive Wiki project and for making announcements. You can sign your name and timestamp by writing three or four tildes <nowiki>~~~</nowiki> at the end of your message, and use : at the start of a message to indent. Use == (title) == to start a new topic. <br />
<br />
Older discussion [[DispersiveWiki:Community Portal/Archive|has been archived here]].<br />
<br />
Would you like to help out? Suggested projects are posted on our [[Current events| current projects]] page.<br />
<br />
== Connections to other Wikis ==<br />
<br />
An [http://tosio.math.toronto.edu/pdewiki/index.php/Main_Page emerging PDEwiki] covering basic concepts of PDEs has been set up. [[User:Colliand|Colliand]] 11:52, 13 September 2006 (EDT)<br />
<br />
<br />
:It turns out that there is already a [http://en.wikipedia.org/wiki/Category:Partial_differential_equations collection] of PDE related pages on wikipedia. For basic concepts in PDE, perhaps we should further develop<br />
:those pages. For more specialized resources like a bibliography, it may be better to use this space. [[User:Colliand|Colliand]] 11:28, 15 September 2006 (EDT)<br />
<br />
==Enumerate equations ==<br />
<br />
Hello <br />
<br />
I am currently searching for a possibility to enumerate equations automatically, and to have the possiblty to add labels and references. It seems that there are 2 enhancement of texcv which are capable of doing it (with a little hacking as the authors say): <br />
#http://www.blahtex.org<br />
#http://en.wikipedia.org/wiki/Wikipedia:WikiTeX<br />
Do I understand correctly from earlier discussions that such a functionality would be most welcome?<br />
Regards<br />
Uwe Brauer [[User:Oub|Oub]] 10:36, 26 September 2006 (EDT):</div>Oubhttps://dispersivewiki.org/DispersiveWiki/index.php?title=DispersiveWiki:Community_Portal&diff=4315DispersiveWiki:Community Portal2006-09-26T14:36:59Z<p>Oub: Enumerate equations</p>
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== Connections to other Wikis ==<br />
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An [http://tosio.math.toronto.edu/pdewiki/index.php/Main_Page emerging PDEwiki] covering basic concepts of PDEs has been set up. [[User:Colliand|Colliand]] 11:52, 13 September 2006 (EDT)<br />
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:It turns out that there is already a [http://en.wikipedia.org/wiki/Category:Partial_differential_equations collection] of PDE related pages on wikipedia. For basic concepts in PDE, perhaps we should further develop<br />
:those pages. For more specialized resources like a bibliography, it may be better to use this space. [[User:Colliand|Colliand]] 11:28, 15 September 2006 (EDT)<br />
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==Enumerate equations ===<br />
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Hello <br />
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I am currently searching for a possibility to enumerate equations automatically, and to have the possiblty to add labels and references. It seems that there are 2 enhancement of texcv which are capable of doing it (with a little hacking as the authors say): <br />
#http://www.blahtex.org<br />
#http://en.wikipedia.org/wiki/Wikipedia:WikiTeX<br />
Do I understand correctly from earlier discussions that such a functionality would be most welcome?<br />
Regards<br />
Uwe Brauer [[User:Oub|Oub]] 10:36, 26 September 2006 (EDT):</div>Oub