DDNLW
DDNLW
We use DDNLW to denote a semi-linear wave equation whose non-linear term is quadratic in the derivatives, i.e.
A fairly trivial example of such equations arise by considering fields of the form f = f(u), where f is a given smooth function (e.g. f(x) = exp(ix)) and u is a scalar solution to the free wave equation. In this case f solves the equation
where Q_0 is the null form
The above equation can be viewed as the wave equation on the one-dimensional manifold f(R), with the induced metric from R. The higher-dimensional version of this equation is known as the [#wm wave map equation].
DDNLW behaves like DNLW but with all fields requiring one more derivative of regularity. One explicit way to make this connection is to differentiate
DDNLW and view the resulting as an instance of DNLW for the system of fields ( f , D f ). The reader should compare the results below with the [#dnlw-2 corresponding results for quadratic DNLW].
The critical regularity is s_c = d/2. For subcritical regularities s > s_c, f has some Holder continuity, and so one heuristically expects the G ( f ) terms
to be negligible. However, this term must play a crucial role in the critical case s=s_c. For instance, Nirenberg [ref?] observed that the real scalar equation
is globally well-posed in H^{d/2}, but the equation
is ill-posed in H^{d/2}; this is basically because the non-linear operator f -> exp(if) is continuous on (real-valued) H^{d/2}, while f -> exp(f) is not.
Energy estimates show that the general DDNLW equation is locally well-posed in H^s 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; the point is that these regularity assumptions together with Strichartz allow one to put f into L2_t L^ ¥ _x (or L^4_t L^ ¥ _x in two dimensions), so that one can then use the energy method.
Using X^{s,b} estimates FcKl2000 instead of Strichartz estimates, one can improve this further to s > s_c + 1/4 in four dimensions and to the near-optimal s > s_c in five and higher dimensions Tt1999.
Without any further assumptions on the non-linearity, these results are sharp in 3 dimensions; more precisely, one generically has ill-posedness in H2 Lb1993, although one can recover well-posedness in the Besov space B2_{2,1} Na1999, or with an epsilon of radial regularity [MacNkrNaOz-p]. It would be interesting to determine what the situation is in the other low dimensions.
If the quadratic non-linearity ( Ñf )2 is of the form Q_0( f , f ) (or of a null form of similar strength) then the LWP theory can be pushed to s > s_c in all dimensions (see [KlMa1997], [KlMa1997b] for d >= 4, KlSb1997 for d \geq 2, and KeTa1998b for d=1).
If G ( f ) is constant and d is at least 4, then one has GWP outside of convex obstacles [Met-p2]
- For d \geq 6 this is in ShbTs1986; for d \geq 4 and the case of a ball this is in Ha1995.
- Without an obstacle, one can use the [#gwp_qnlw general theory of quasilinear NLW].
