Graphical mean curvature flow with zero boundary data is locally well-posed for approximable Lipschitz graphs, globally well-posed for small initial slopes, and decays exponentially on bounded domains.
Geometric flows with rough initial data
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abstract
We show the existence of a global unique and analytic solution for the mean curvature flow, the surface diffusion flow and the Willmore flow of entire graphs for Lipschitz initial data with small Lipschitz norm. We also show the existence of a global unique and analytic solution to the Ricci-DeTurck flow on euclidean space for bounded initial metrics which are close to the euclidean metric in $L^\infty$ and to the harmonic map flow for initial maps whose image is contained in a small geodesic ball.
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Well-posedness for the mean curvature flow on the half-space and on bounded domains
Graphical mean curvature flow with zero boundary data is locally well-posed for approximable Lipschitz graphs, globally well-posed for small initial slopes, and decays exponentially on bounded domains.