View source for DDNLW

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>

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>

where Q_0 is the [[null form]]

<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]].

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]].

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

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

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

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

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 <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]]).

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.

If the quadratic non-linearity 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  G ( f ) is constant and d 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]].
* Without an obstacle, one can use the [[QNLW|general theory of quasilinear NLW]].

The [[two-speed DDNLW|two-speed analogue]] of these equations has also been studied.

[[Category:wave]] 
[[Category:Equations]]