QNLW

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

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

One could also consider equations where the metric depends on derivatives of $u$, but one can reduce to this case (giving up a derivative) by differentiating the equation. One can also reduce to the case $g^{00}=1$, $g^{0i}=g^{i0}=0$ by a suitable change of variables. $F$ is usually quadratic in the derivatives $Du$, 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 Einstein equations
• Equations of relativistic elasticity
• Equations of relativistic hydrodynamics
• The minimal surface equation

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

Wellposedness theory

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

• When $d=2$ one has LWP in the expected range $s>d/2+3/4$ without a null condition (SmhTt-p).
  • For $s>d/2+3/4+1/12$ this is in Tt-p5 (using the FBI transform).
  • For $s>d/2+3/4+1/8$ this is in BaCh1999 (using FIOs) and Tt2000 (using the FBI transform).
• When $d=3,4,5$ one has LWP for $s>d/2+1/2$ 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 $s>d/2+1/2+1/7$ (approx) and $d=3$ this is in KlRo-p2 (vector fields and the equation for the metric)
  • For $s>d/2+1/2+1/6$ and $d=3$ this is in Tt-p5 (using the FBI transform).
  • For $s>d/2+1/2+1/5$ (approx) and $d=3$ this is in Kl-p2 (vector fields methods).
  • For $s>d/2+1/2+1/4$ and $d\geq 3$ 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 $g=|Du|^{2}$, and the non-linearity is of the form $DgDu$. For such equations, we have LWP for $s>d/2+1/6$ when $d\geq 4$ BaCh-p, BaCh2002. This equation has some similarity with the differentiated wave maps equation in the Coulomb gauge.

For small smooth compactly supported data of size $\epsilon$ 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 $d\geq 3$; in fact one can take the data in a weighted Sobolev space $H^{4,3}xH^{3,4}$ 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 $d=3$, one has GWP for small data in $H^{9,8}\times H^{8,9}$ which are compatible with the boundary [KeSmhSo-p]. Earlier work in this direction is in Dt1990.
    • For radial data and obstacle this was obtained in Go1995; see also Ha1995, Ha2000.
    • In the semilinear case, the non-trapping condition was removed in MetSo-p, even in the multiple speed case, provided one has an exponential decay result near the obstacle (this is true, for instance, if the obstacle is a union of a finite number of sufficiently separated strictly convex bodies).
  • For $d>3$ or for cubic nonlinearities one has GWP regardless of the null structure KlPo1983, Sa1982, Kl1985b.
    • In three dimensions with a null structure, for systems with multiple wave speeds, one has GWP So2001
    • 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 $Du$
      1. When the obstacle is a ball this is in Ha1995.
      2. For $d\geq 6$ outside of a starshaped obstacle this is in ShbTs1984, ShbTs1986.