View source for QNLW

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

<center><math>\partial_\alpha  g^{\alpha\beta}(u) \partial_\beta  u = F(u, Du)</math>.</center>

One could also consider equations where the metric depends on derivatives of <math>u</math>, but one can reduce to this case (giving up a derivative) by differentiating the equation. One can also reduce to the case <math>g^{00} = 1</math>, <math>g^{0i} = g^{i0} = 0</math> by a suitable change of variables. <math>F</math> is usually quadratic in the derivatives <math>Du</math>, 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 <math>H^s</math> when <math>s > d/2+1</math> ([[HuKaMar1977]]), but the [DNLW|semilinear theory] suggests that we should be able to improve this to <math>s > s_c = d/2</math> with a null condition, and to <math>s > d/2 + max(1/2, (d-5)/4)</math> without one (these results would be sharp even in the semilinear case). In principle Strichartz estimates should be able to push down to <math>s > d/2 + 1/2</math>, but only partial results of this type are known. Specifically:

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

For small smooth compactly supported data of size <math>\epsilon</math> 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 <math>d\geq3</math>; in fact one can take the data in a weighted Sobolev space <math>H^{4,3} x H^{3,4}</math> [[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 <math>d=3</math>, one has GWP for small data in <math>H^{9,8} \times H^{8,9}</math> 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 <math>d>3</math> 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 <math>Du</math>
***# When the obstacle is a ball this is in [[Ha1995]].
***# For <math>d \geq 6</math> outside of a starshaped obstacle this is in [[ShbTs1984]], [[ShbTs1986]].

[[Category:Wave]]
[[Category:Equations]]