Abstract
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 sets of \({H^1(\mathbb{R})\cap W^{1,4}(\mathbb{R})}\), although it is not Lipschitz continuous under the natural Sobolev metric from an 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 is in an open dense subset of \({H^1(\mathbb{R})\cap W^{1,4}(\mathbb{R})}\), the solution is piecewise smooth. This generic regularity result helps us extend the Lipschitz continuous metric to the general weak solutions. Our method of constructing the metric can be used to treat other kinds of quasi-linear equations, provided a good knowledge about the energy concentration.
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Cai, H., Chen, G., Chen, R.M. et al. Lipschitz Metric for the Novikov Equation. Arch Rational Mech Anal 229, 1091–1137 (2018). https://doi.org/10.1007/s00205-018-1234-4
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DOI: https://doi.org/10.1007/s00205-018-1234-4