Lipschitz metric for the Novikov equation

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.