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 [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.