# Linear-derivative nonlinear wave equations

A linear-derivative nonlinear wave equation is a derivative nonlinear wave equation of with the schematic form

${\displaystyle \Box u=F(u)Du+G(u)}$

An important subclass of such equations are the Yang-Mills-type equations of the form

${\displaystyle \Box u=uDu+u^{3}}$

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

Using ${\displaystyle X^{s,\theta }}$ estimates FcKl2000 instead of Strichartz estimates, one can improve this further to ${\displaystyle d>s_{c}+1/4}$ in four dimensions and to the near-optimal ${\displaystyle s>s_{c}}$ in five and higher dimensions. In six and higher dimensions one can obtain global well-posedness for small critical Besov space ${\displaystyle B_{2,1}^{s_{c}}}$ 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 ${\displaystyle H^{1}}$ 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 ${\displaystyle \phi D\phi }$ 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 ${\displaystyle s>s_{c}}$ 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 ${\displaystyle s>s_{c}+1/4)}$. In one dimension the model equation trivially collapses to the free wave equation.