DDNLW: Difference between revisions

From DispersiveWiki
Jump to navigationJump to search
No edit summary
No edit summary
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
By '''quadratic-derivative nonlinear wave equation''' (DDNLW), we mean a semi-linear wave equation whose non-linear term is quadratic in the derivatives, i.e.
By '''quadratic-derivative nonlinear wave equation''' (DDNLW), we mean a semi-linear wave equation whose non-linear term is quadratic in the derivatives, i.e.


<center>\Box  f  =  G ( f ) D f  D f .</center>
<center><math>\Box  f  =  G ( f ) D f  D f</math> .</center>


A fairly trivial example of such equations arise by considering fields of the form  f  = f(u), where f is a given smooth function (e.g. f(x) = exp(ix)) and u is a scalar solution to the free wave equation. In this case  f  solves the equation
A fairly trivial example of such equations arise by considering fields of the form  f  = f(u), where f is a given smooth function (e.g. f(x) = exp(ix)) and u is a scalar solution to the free wave equation. In this case  f  solves the equation


<center>\Box  f <nowiki> = f''(</nowiki> f ) Q_0( f , f )</center>
<center><math>\Box  f = f''( f ) Q_0( f , f )</math></center>


where Q_0 is the [[null form]]
where <math>Q_0</math> is the [[null form]]


<center>Q_0( f y ) := \partial_ af \partial^ a  y Ñf .  Ñy f _t y _t.</center>
<center><math>Q_0( \phi \psi ) := \partial_\alpha \phi \partial^\alpha \psi \nabla \phi \cdot \nabla \psi \phi_t \psi_t.</math></center>


The above equation can be viewed as the wave equation on the one-dimensional manifold f(R), with the induced metric from R. The higher-dimensional version of this equation is known as the [[wave maps|wave map equation]].
The above equation can be viewed as the wave equation on the one-dimensional manifold <math>f(R)</math>, with the induced metric from <math>R</math>. The higher-dimensional version of this equation is known as the [[wave maps|wave map equation]].


Quadratic-derivative nonlinear wave equations behave like their [[linear-derivative nonlinear wave equations|linear-derivative counterparts]], with all fields requiring one more derivative of regularity. One explicit way to make this connection is to differentiate the DDNLW equation and view the resulting as an instance of a linear-derivative NLW for the system of fields ( f , D f ). The reader should compare the results below with the [[linear-derivative nonlinear wave equations|linear-derivative counterparts]].
Quadratic-derivative nonlinear wave equations behave like their [[linear-derivative nonlinear wave equations|linear-derivative counterparts]], with all fields requiring one more derivative of regularity. One explicit way to make this connection is to differentiate the DDNLW equation and view the resulting as an instance of a linear-derivative NLW for the system of fields ( <math>f</math> , <math>D f</math> ). The reader should compare the results below with the [[linear-derivative nonlinear wave equations|linear-derivative counterparts]].


The critical regularity is s_c = d/2. For subcritical regularities s > s_c,  f  has some Holder continuity, and so one heuristically expects the  G ( f ) terms <br /> to be negligible. However, this term must play a crucial role in the critical case s=s_c. For instance, Nirenberg [ref?] observed that the real scalar equation
The critical regularity is <math>s_c = d/2</math>. For subcritical regularities <math>s > s_c</math>,  f  has some Holder continuity, and so one heuristically expects the  <math>G ( f )</math> terms <br /> to be negligible. However, this term must play a crucial role in the critical case <math>s=s_c</math>. For instance, Nirenberg [ref?] observed that the real scalar equation


<center>\Box  f  = - f  Q_0( f ,  f )</center>
<center><math>\Box  f  = - f  Q_0( f ,  f )</math></center>


is globally well-posed in H^{d/2}, but the equation
is globally well-posed in <math>H^{d/2}</math>, but the equation


<center>\Box  f  =  f  Q_0( f ,  f )</center>
<center><math>\Box  f  =  f  Q_0( f ,  f )</math></center>


is ill-posed in H^{d/2}; this is basically because the non-linear operator f -> exp(if) is continuous on (real-valued) H^{d/2}, while f -> exp(f) is not.
is ill-posed in <math>H^{d/2}</math>; this is basically because the non-linear operator <math>f \rightarrow exp(if)</math> is continuous on (real-valued) <math>H^{d/2}</math>, while <math>f \rightarrow exp(f)</math> is not.


Energy estimates show that the general DDNLW equation is locally well-posed in H^s for s > s_c + 1. Using Strichartz estimates this can be improved to s > s_c + 3/4 in two dimensions and s > s_c + 1/2 in three and higher dimensions; the point is that these regularity assumptions together with Strichartz allow one to put  f  into L<sup>2</sup>_t L^ ¥ _x (or L^4_t L^ ¥ _x in two dimensions), so that one can then use the energy method.
Energy estimates show that the general DDNLW equation is locally well-posed in <math>H^s</math> for <math>s > s_c + 1</math>. Using Strichartz estimates this can be improved to <math>s > s_c + 3/4</math> in two dimensions and <math>s > s_c + 1/2</math> in three and higher dimensions; the point is that these regularity assumptions together with Strichartz allow one to put  <math>f</math> into <math>L^2_t L^\infty_x</math> (or <math>L^4_t L^\infty_x</math> in two dimensions), so that one can then use the energy method.


Using X^{s,b} estimates ([[FcKl2000]]) instead of Strichartz estimates, one can improve this further to s > s_c + 1/4 in four dimensions and to the near-optimal s > s_c in five and higher dimensions ([[Tt1999]]).
Using <math>X^{s,b}</math> estimates ([[FcKl2000]]) instead of Strichartz estimates, one can improve this further to <math>s > s_c + 1/4</math> in four dimensions and to the near-optimal <math>s > s_c</math> in five and higher dimensions ([[Tt1999]]).


Without any further assumptions on the non-linearity, these results are sharp in 3 dimensions; more precisely, one generically has ill-posedness in H<sup>2</sup> ([[Lb1993]]), although one can recover well-posedness in the Besov space B<sup>2</sup>_{2,1} ([[Na1999]]), or with an epsilon of radial regularity ([[MacNkrNaOz-p]]). It would be interesting to determine what the situation is in the other low dimensions.
Without any further assumptions on the non-linearity, these results are sharp in 3 dimensions; more precisely, one generically has ill-posedness in <math>H^2</math> ([[Lb1993]]), although one can recover well-posedness in the Besov space <math>B^2_{2,1}</math> ([[Na1999]]), or with an epsilon of radial regularity ([[MacNkrNaOz-p]]). It would be interesting to determine what the situation is in the other low dimensions.


If the quadratic non-linearity ( Ñf )<sup>2</sup> is of the form Q_0( f , f ) (or of a null form of similar strength) then the LWP theory can be pushed to s > s_c in all dimensions (see [[KlMa1997], [[KlMa1997b]] for d >= 4, [[KlSb1997]] for d \geq 2, and [[KeTa1998b]] for d=1).
If the quadratic non-linearity is of the form <math>Q_0( f , f )</math> (or of a null form of similar strength) then the LWP theory can be pushed to <math>s > s_c</math> in all dimensions (see [[KlMa1997], [[KlMa1997b]] for <math>d >= 4</math>, [[KlSb1997]] for <math>d \geq 2</math>, and [[KeTa1998b]] for <math>d=1</math>).


If  G ( f ) is constant and d is at least 4, then one has GWP outside of convex obstacles [[Met-p2]]
If  <math>G ( f )</math> is constant and <math>d</math> is at least 4, then one has GWP outside of convex obstacles [[Met-p2]]


* For d \geq 6 this is in [[ShbTs1986]]; for d \geq 4 and the case of a ball this is in [[Ha1995]].
* For <math>d \geq 6</math> this is in [[ShbTs1986]]; for <math>d \geq 4</math> and the case of a ball this is in [[Ha1995]].
* Without an obstacle, one can use the [[QNLW|general theory of quasilinear NLW]].
* Without an obstacle, one can use the [[QNLW|general theory of quasilinear NLW]].



Latest revision as of 20:10, 3 August 2006

By quadratic-derivative nonlinear wave equation (DDNLW), we mean a semi-linear wave equation whose non-linear term is quadratic in the derivatives, i.e.

.

A fairly trivial example of such equations arise by considering fields of the form f = f(u), where f is a given smooth function (e.g. f(x) = exp(ix)) and u is a scalar solution to the free wave equation. In this case f solves the equation

where is the null form

The above equation can be viewed as the wave equation on the one-dimensional manifold , with the induced metric from . The higher-dimensional version of this equation is known as the wave map equation.

Quadratic-derivative nonlinear wave equations behave like their linear-derivative counterparts, with all fields requiring one more derivative of regularity. One explicit way to make this connection is to differentiate the DDNLW equation and view the resulting as an instance of a linear-derivative NLW for the system of fields ( , ). The reader should compare the results below with the linear-derivative counterparts.

The critical regularity is . For subcritical regularities , f has some Holder continuity, and so one heuristically expects the terms
to be negligible. However, this term must play a crucial role in the critical case . For instance, Nirenberg [ref?] observed that the real scalar equation

is globally well-posed in , but the equation

is ill-posed in ; this is basically because the non-linear operator is continuous on (real-valued) , while is not.

Energy estimates show that the general DDNLW equation is locally well-posed in for . Using Strichartz estimates this can be improved to in two dimensions and in three and higher dimensions; the point is that these regularity assumptions together with Strichartz allow one to put into (or in two dimensions), so that one can then use the energy method.

Using estimates (FcKl2000) instead of Strichartz estimates, one can improve this further to in four dimensions and to the near-optimal in five and higher dimensions (Tt1999).

Without any further assumptions on the non-linearity, these results are sharp in 3 dimensions; more precisely, one generically has ill-posedness in (Lb1993), although one can recover well-posedness in the Besov space (Na1999), or with an epsilon of radial regularity (MacNkrNaOz-p). It would be interesting to determine what the situation is in the other low dimensions.

If the quadratic non-linearity is of the form (or of a null form of similar strength) then the LWP theory can be pushed to in all dimensions (see [[KlMa1997], KlMa1997b for , KlSb1997 for , and KeTa1998b for ).

If is constant and is at least 4, then one has GWP outside of convex obstacles Met-p2

The two-speed analogue of these equations has also been studied.