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.

Quasilinear NLWs appear frequently in general relativity. The most famous example is the Einstein equations, but there are others (coming from relativistic elasticity, hydrodynamics, minimal surfaces, etc. [Ed: anyone willing to contribute information on these other equations (even just their name and form) would be greatly appreciated.]). The most interesting dimension is of course the physical dimension d=3.

Classically one has LWP for H^s when s > d/2+1 (HuKaMar1977), but the [DNLW|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 (SmTt-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 SmTt-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 Dg Du. 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\geq3; in fact one can take the data in a weighted Sobolev space H^{4,3} x H^{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} x 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.