DNLW

The D-NLW equation is given by

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

where $u$ is scalar or vector valued, and $F$ is at least quadratic. The 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 the [#dnlw-2 quadratic DNLWs] (which include the Yang-Mills and Maxwell-Klein-Gordon equations) and the [#ddnlw DDNLWs] (which include equations of wave maps type).

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. The precise meaning of this is hard to quantify exactly, but roughly speaking this means that if u is a plane wave then the highest order terms in the non-linearity vanish. In other words, the self-interaction of plane waves is relatively small. The presence of a null structure usually makes the LWP and GWP theory significantly better.

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 quasilinear case].

For small smooth compactly supported data of size $e$ 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 refernces:KlPo1983 KlPo1983, Sa1982, Kl1985b.
• For the $d=2$ case, the results are as follows.
  • One has existence for roughly $e^{-2}$ without a null condition, but at least $exp(C/e^{2})$ with a null condition references:Al1999b 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 $exp(C/e^{2})$); 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 $e^{-4}log(1/e)^{-6}$ De1997b.
    • Without the null form one cannot do better than about $e^{-2}$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 $e^{-2}$. 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 $e^{1-r}log(1/e)^{3-r}$, 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 Gl1981b. 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).

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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 $H^{1}$.

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Quadratic DNLW

By Quadratic DNLW we mean equations with the schematic form

$\Box f=fDf$ .

This equation has the same scaling as cubic NLW, but is more difficult technically because of the derivative in the non-linearity. In practice one can always add a cubic term $f^{3}$ to the non-linearity without disrupting any of the well-posedness theory, as $f^{3}$ is usually much easier to estimate than $fDf$ .

Important examples of this type of equation include the [#mkg Maxwell-Klein-Gordon] and [#ym Yang-Mills] equations (in the Lorentz gauge, at least), as well as the simplified model equations for these equations. The [#ymh Yang-Mills-Higgs] equation is formed by coupling equations of this type to a semi-linear wave equation. The most interesting dimensions are 3 (for physical applications) and 4 (since the energy regularity is then critical). The two-dimensional case appears to be somewhat under-explored. The Yang-Mills and Maxwell-Klein-Gordon equations behave very similarly, but from a technical standpoint the latter is slightly easier to analyze.

In d dimensions, the critical regularity for this equation is $s_{c}=d/2-1$. However, there are no instances of Quadratic DNLW for which any sort of well-posedness is known at this critical regularity (except for those special cases where the equation can be algebraically transformed into a simpler equation such as NLW or the free wave equation).

Energy estimates give local well-posedness for $s>s_{c}+1$. Using Strichartz estimates this can be improved to $s>s_{c}+3/4$ in two dimensions and $s>s_{c}+1/2$ in three and higher dimensions PoSi1993; the point is that these regularity assumptions together with Strichartz allow one to put $f$ into $L_{t}^{2}L_{x}^{\infty }$, hence in $L_{t}^{1}L_{x}^{\infty }$, so that one can then use the energy method.

Using $X^{s,\theta }$ estimates FcKl2000 instead of Strichartz estimates, one can improve this further to $d>s_{c}+1/4$ in four dimensions and to the near-optimal $s>s_{c}$ in five and higher dimensions. In six and higher dimensions one can obtain global well-posedness for small critical Besov space $B_{2,1}^{s_{c}}$ [Stz-p3], and local well-posedness for large Besov data.In four dimensions one has a similar result if one imposes one additional angular derivative of regularity [Stz-p2].

Without any further assumptions on the non-linearity, these results are sharp in 3 dimensions; more precisely, one generically has ill-posedness in $H^{1}$ Lb1993, although one can recover well-posedness in the Besov space B^1_{2,1} Na1999, or when an epsilon of radial regularity is imposed [MacNkrNaOz-p]. It would be interesting to determine what the situation is in the other low dimensions.

If the non-linearity $fDf$ has a null structure then one can improve upon the previous results. For instance, the model equations for the Yang-Mills and Maxwell-Klein-Gordon equations are locally well-posed for $s>s_{c}$ in three KlMa1997 and higher KlTt1999 dimensions. It would be interesting to determine what happens in two dimensions for these equations (probably one can get down to $s>s_{c}+1/4)$. In one dimension the model equation trivially collapses to the free wave equation.

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Two-speed quadratic DNLW

• One can consider two-speed variants of quadratic DNLW ([#two-speed see Overview]), when both F and G have the form U DU.
• The Strichartz and energy estimates carry over without difficulty to this setting. The results obtained by $X^{s,\theta }$ estimates change, however. The null forms are no longer as useful, however the estimates are usually more favourable because of the transversality of the two light cones. Of course, if $F$ contains $uDu$ or $G$ contains $vDv$ then one cannot do any better than the one-speed case.
• For $d=3$ one can obtain LWP for $s\geq 1$ for non-linearities of the form $F=uDv+vDu+vDv$ and $G=uDu+vDu$ OzTyTs1999, Tg2000.
  • When $G$ contains uDv the relevant bilinear estimate generically fails by a logarithmic factor OzTyTs2000
• For $d=2$ one can obtain LWP for $s>1/2$ for non-linearities of the form $F=uDv+vDu+vDv$ and $G=uDu+vDu+uDv$ [Tg-p].
• For $d=1$ one can obtain LWP for $s>1/4$ for non-linearities of the form $F=uDv+vDu$ and $G=uDv+vDu$ [Tg-p].
  • One can improve this to $s>0$ for non-linearities of the form $F=D(uv+v^{2})$ and $G=D(u^{2}+uv)$ [Tg-p]. These expressions are better behaved than the previous ones, for instance their Fourier transform vanishes at the origin.

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