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.

$\Box f=G(f)DfDf$ .

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

$\Box f=f''(f)Q_{0}(f,f)$

where $Q_{0}$ is the null form

$Q_{0}(\phi ,\psi ):=\partial _{\alpha }\phi \partial ^{\alpha }\psi =\nabla \phi \cdot \nabla \psi -\phi _{t}\psi _{t}.$

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 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 ( $f$ , $Df$ ). The reader should compare the results below with the 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
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

$\Box f=-fQ_{0}(f,f)$

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

$\Box f=fQ_{0}(f,f)$

is ill-posed in $H^{d/2}$; this is basically because the non-linear operator $f\rightarrow exp(if)$ is continuous on (real-valued) $H^{d/2}$, while $f\rightarrow exp(f)$ 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 $L_{t}^{2}L_{x}^{\infty }$ (or $L_{t}^{4}L_{x}^{\infty }$ 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 $H^{2}$ (Lb1993), although one can recover well-posedness in the Besov space $B_{2,1}^{2}$ (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 general theory of quasilinear NLW.

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