DNLW

Derivative NLW (D-NLW) equations take the form

$\Box u=F(u,Du)$

where $u$ is scalar or vector valued, and $F$ is at least quadratic. The derivative NLKG (D-NLKG) equation is similar:

$\Box u=u+F(u,Du)$.

From a LWP perspective the two equations are virtually equivalent, but the NLKG is slightly better behaved for the GWP (it decays like the NLW at one higher dimension).

Among the more intensively studied derivative NLWs are

• linear-derivative nonlinear wave equations (which include the Yang-Mills equations and the Maxwell-Klein-Gordon equations as spcial cases)
• quadratic-derivative nonlinear wave equations such as wave maps.

From energy estimates one can always obtain LWP in $H^{s}$ for $s>n/2+1$. (or $s>n/2$ if the non-linearity is at most linear in $Du$). However, this is rarely best possible.

In many cases the non-linearity $F$ has a null structure, which often causes the LWP and GWP theory to improve significantly.

The GWP theory for small data is usually accomplished by vector fields methods (or similar methods which try to capture the decay, and proximity to the light cone, of the global solution), or via conformal compactification. The method of normal forms is also often useful, as it can eliminate the worst terms in a non-linearity.

As a general principle, the small data GWP theory becomes better whenever the order of the non-linearity increases (because this makes the non-linearity even smaller) or when the dimension increases (because there is more decay). There is rarely any need to distinguish between u and Du in the small data GWP theory. In many cases the theory is robust enough to carry over to the quasilinear case.

For small smooth compactly supported data of size $\epsilon$ and smooth non-linearities, the GWP theory for D-NLW 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 $H^{2,1}\times H^{1,2}$ Cd1986
  • This is also true in the multi-speed (i.e. nonrelativistic) case KlSi1996 (see also Yk2000). Earlier related work appears in Ko1987, Ko1989
    • In fact, one does not need the compact support condition, and can take data small in H^9 SiTu-p.
  • Without the null structure, one has almost GWP in d=3 Kl1985b, and this is sharp Jo1981, Si1983
  • With the null structure and outside a convex obstacle one has GWP in $H^{2,1}\times H^{1,2}$ KeSmhSo2000, assuming the standard compatibility conditions on the data. Earlier work in this direction is in Dt1990.
    • For radial data and obstacle this was obtained in Go1995; see also Ha1995.
    • GWP for small smooth data outside a star-shaped obstacle was shown for $d\geq 6$ and non-linearities quadratic in $Du$ in ShbTs1984, ShbTs1986.
  • For $d>3$ or for cubic nonlinearities one has GWP regardless of the null structure KlPo1983, Sa1982, Kl1985b.
• For the $d=2$ case, the results are as follows.
  • One has existence for roughly $\epsilon ^{-2}$ without a null condition, but at least $exp(C/\epsilon ^{2})$ with a null condition Al1999, Al1999b, Al2001, Al2001b
    • For semilinear equations these are in Gd1993
    • For cubic nonlinearities these are in Hg1995; furthermore one has global existence assuming a "second null condition"
    • For spherically symmetric data this is in Lad1999; furthermore one has global existence assuming a "second null condition"
  • Earlier results are in Ky1993. Non-relativistic variants are in Hg1998, HgKu2000

For small smooth compactly supported data of size $e$ and smooth non-linearities, the GWP theory for D-NLKG is as follows.

• One has GWP for $d\geq 2$ SnTl1993, OzTyTs1996.
  • For $d\geq 3$ this was proven in Kl1985 (by vector fields) and Sa1985 (by normal forms).
  • For $d=3$ this result is extended to Klein-Gordon-Zakharov systems in OzTyTs1995
• For $d=1$ one has GWP for quartic and higher non-linearities LbSo1996.
  • For cubic non-linearities one has almost global existence (for time $exp(C/\epsilon ^{2})$); see Ho1997. This is sharp KeTa1999, [Yo-p]. Explicit lower bounds on the constant $C$ are in De1999.
  • Using normal forms one can push the almost global existence to quadratic non-linearities MrTwSr1997.
  • For septic and higher nonlinearities one has global existence Mr1997, Yag1994
  • For quadratic non-linearities of null form type one has existence for time roughly $e^{-4}$ De1997
    • The compact support condition can be weakened substantially De1997. One can even just assume small $H^{3}$ data, but then the time of existence shrinks slightly, to $\epsilon ^{-4}log(1/\epsilon )^{-6}$ De1997b.
    • Without the null form one cannot do better than about $\epsilon ^{-2}$KeTa1999.
    • A necessary and sufficient condition on the nonlinearity has been obtained as to whether one can obtain global existence for small data De2001

For small smooth data of size $e<math>onthecircle<math>T$, and smooth non-linearities, the GWP theory for D-NLKG is as follows.

• For quadratic non-linearities one has existence for time $\epsilon ^{-2}$. De1998.This was extended to higher dimensional tori and to quasilinear NLKG in DeSze-p
• For higher non-linearities of order r, one has existence for time $\epsilon ^{1-r}log(1/\epsilon )^{3-r}$, and in some cases this bound is sharp. De1998
• One has a similar result (with more technical time of existence) when the domain is a sphere, although now one must exclude a set of masses of measure zero. DeSze-p

For non-smooth non-linearities of order $p$, one has blow-up examples for D-NLKG from small smooth compactly supported data whenever $p\leq 1+2/d$ KeTa1999, although in certain cases (esp. coercive Hamiltonian systems) one still has GWP Ca1985. One also has failure of scattering for this range of powers Gs1981b. It would be interesting to see if one could obtain GWP for D-NLKG the $p>1+2/d$ case (though in the non-smooth non-linearity case one probably is restricted to $d=1,2,3$ in order to keep the non-linearity sufficiently smooth).

Damping DNLW

The equation

$\Box u=u^{2}u_{t}$

in three dimensions is known to be locally well-posed in any sub-critical regularity $s>1$, and has scattering in $H^{3}$ Smh-p. It would be interesting to see whether one has local well-posedness in the critical energy regularity $H^{1}$.