such a delicious mollifier
! After having my hands in it for only a few minutes, my fingers felt like eels, and began, as it were, to serpentine and spiralize.
We can obtain the regularized system (25) by using standard mollifier
. Then we can obtain various estimates and local-in-time existence of a solution for (21).
For example, we may choose [mathematical expression not reproducible] where [[eta].sup.([epsilon])] is a standard mollifier
supported in (0, [epsilon]).
where [f.sub.n] (t,x, [lambda]) = f (*, *,[lambda]) * [n.sup.2][omega](nt, nx) is a regularization of the flux f via the standard non-negative mollifier
[omega] [member of] [C.sup.[infinity].sub.c][((-1,1).sup.2]) with total mass one, and ([u.sup.0.sub.n]) is a bounded sequence of functions converging strongly in [L.sup.1.sub.loc] (R) toward [u.sub.0] By multiplying (1.4) by sgn([u.sub.n]t,x) - [lambda]) we get after standard manipulations (see also formula (2.8) of the next section):
Let [[rho].sub.[epsilon]](x,y) = [[epsilon].sup.- (2]) [rho](x/[epsilon],y/[epsilon]) ([epsilon] > 0) be a Friedrichs mollifier
and for any u [member of] [D.sub.0.sup.1]([ohm]) define
where [j.sub.[epsilon]] is a mollifier
. In this case, [phi](u, v) = [phi](r), with r = [square root of ([u.sup.2] + [v.sup.2])].
Karcher, "Riemannian center of mass and mollifier
smoothing," Communications on Pure and Applied Mathematics, vol.
Shaw played the mollifier
at first, but, when Wells grew more and more insistent, Shaw sided against him aning cruthe pragmatists.