# QNLW

In a local co-ordinate chart, quasilinear wave equations (QNLW) take the form

${\displaystyle \partial _{\alpha }g^{\alpha \beta }(u)\partial _{\beta }u=F(u,Du)}$.

One could also consider equations where the metric depends on derivatives of ${\displaystyle u}$, but one can reduce to this case (giving up a derivative) by differentiating the equation. One can also reduce to the case ${\displaystyle g^{00}=1}$, ${\displaystyle g^{0i}=g^{i0}=0}$ by a suitable change of variables. ${\displaystyle F}$ is usually quadratic in the derivatives ${\displaystyle Du}$, as this formulation is then robust under many types of changes of variables.

## Specific equations

Quasilinear NLWs appear frequently in general relativity. Examples include

The most interesting dimension is of course the physical dimension d=3.

## Wellposedness theory

Classically one has LWP for ${\displaystyle H^{s}}$ when ${\displaystyle s>d/2+1}$ (HuKaMar1977), but the semilinear theory suggests that we should be able to improve this to ${\displaystyle s>s_{c}=d/2}$ with a null condition, and to ${\displaystyle s>d/2+max(1/2,(d-5)/4)}$ without one (these results would be sharp even in the semilinear case). In principle Strichartz estimates should be able to push down to ${\displaystyle s>d/2+1/2}$, but only partial results of this type are known. Specifically:

• When ${\displaystyle d=2}$ one has LWP in the expected range ${\displaystyle s>d/2+3/4}$ without a null condition (SmhTt-p).
• For ${\displaystyle s>d/2+3/4+1/12}$ this is in Tt-p5 (using the FBI transform).
• For ${\displaystyle s>d/2+3/4+1/8}$ this is in BaCh1999 (using FIOs) and Tt2000 (using the FBI transform).
• When ${\displaystyle d=3,4,5}$ one has LWP for ${\displaystyle s>d/2+1/2}$ SmhTt-p (using parametrices and the equation for the metric); in the specific case of the Einstein equations see KlRo-p3, KlRo-p4, KlRo-p5 (using vector fields and the equation for the metric)
• For ${\displaystyle s>d/2+1/2+1/7}$ (approx) and ${\displaystyle d=3}$ this is in KlRo-p2 (vector fields and the equation for the metric)
• For ${\displaystyle s>d/2+1/2+1/6}$ and ${\displaystyle d=3}$ this is in Tt-p5 (using the FBI transform).
• For ${\displaystyle s>d/2+1/2+1/5}$ (approx) and ${\displaystyle d=3}$ this is in Kl-p2 (vector fields methods).
• For ${\displaystyle s>d/2+1/2+1/4}$ and ${\displaystyle d\geq 3}$ this is in BaCh1999 (using FIOs) and Tt2000 (using the FBI transform). See also BaCh1999b.

A special type of QNLW is the cubic equations, where the metric g itself obeys an elliptic equation of the form Delta ${\displaystyle g=|Du|^{2}}$, and the non-linearity is of the form ${\displaystyle DgDu}$. For such equations, we have LWP for ${\displaystyle s>d/2+1/6}$ when ${\displaystyle d\geq 4}$ BaCh 2003, BaCh2002. This equation has some similarity with the differentiated wave maps equation in the Coulomb gauge.

For small smooth compactly supported data of size ${\displaystyle \epsilon }$ and smooth non-linearities, the GWP theory for QNLW is as follows.

• If the non-linearity is a null form, then one has GWP for ${\displaystyle d\geq 3}$; in fact one can take the data in a weighted Sobolev space ${\displaystyle H^{4,3}xH^{3,4}}$ Cd1986.
• Without the null structure, one has almost GWP in d=3 Kl1985b, and this is sharp Jo1981, Si1983
• In the semi-linear case and when the nonlinearity is quadratic in the derivatives, this is also true outside of a compact non-trapping obstacle KeSmhSo-p2. This has been generalized to the quasi-linear case in KeSmhSo-p3 (and non-linear Dirichlet wave equations are also treated there, as are multiple speeds).
• With a null structure and outside a star-shaped obstacle with Dirichlet conditions and ${\displaystyle d=3}$, one has GWP for small data in ${\displaystyle H^{9,8}\times H^{8,9}}$ which are compatible with the boundary [KeSmhSo-p]. Earlier work in this direction is in Dt1990.
• For ${\displaystyle d>3}$ or for cubic nonlinearities one has GWP regardless of the null structure KlPo1983, Sa1982, Kl1985b.
• In the exterior of an nontrapping obstacle with Dirichlet conditions, with multiple speeds, one has GWP for sufficiently smooth and decaying data obeying the usual compatibility conditions at the boundary MetSo-p2, if the quasilinear terms obey some symmetry conditions and the semilinear terms are quadratic in ${\displaystyle Du}$
2. For ${\displaystyle d\geq 6}$ outside of a starshaped obstacle this is in ShbTs1984, ShbTs1986.