Linear-derivative nonlinear wave equations
From DispersiveWiki
A linear-derivative nonlinear wave equation is a derivative nonlinear wave equation of with the schematic form
An important subclass of such equations are the Yang-Mills-type equations of the form
This equation has the same scaling as cubic NLW, but is more difficult technically because of the derivative term uDu.
Important examples of this type of equation include the Maxwell-Klein-Gordon and Yang-Mills equations (in the Lorenz gauge, at least), as well as the simplified model equations for these equations. The 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 sc = 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 > sc + 1. Using Strichartz estimates this can be improved to s > sc + 3 / 4 in two dimensions and s > sc + 1 / 2 in three and higher dimensions PoSi1993; the point is that these regularity assumptions together with Strichartz allow one to put f into 
, hence in 
, so that one can then use the energy method.
Using Xs,θ estimates FcKl2000 instead of Strichartz estimates, one can improve this further to d > sc + 1 / 4 in four dimensions and to the near-optimal s > sc in five and higher dimensions. In six and higher dimensions one can obtain global well-posedness for small critical Besov space 
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 H1 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 φDφ 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 > sc 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 > sc + 1 / 4). In one dimension the model equation trivially collapses to the free wave equation.
