  We consider the Lipschitz continuous dependence of solutions for the Novikov
equation with respect to the initial data. In particular, we construct a
Finsler type optimal transport metric which renders the solution map Lipschitz
continuous on bounded set of $H^1(R)\cap W^{1,4}(R)$, although it is not
Lipschitz continuous under the natural Sobolev metric from energy law due to
the finite time gradient blowup. By an application of Thom's transversality
Theorem, we also prove that when the initial data are in an open dense set of
$H^1(R)\cap W^{1,4}(R)$, the solution is piecewise smooth. This generic
regularity result helps us extend the Lipschitz continuous metric to the
general weak solutions.
