Existence of martingale solutions and large-time behavior for a stochastic mean curvature flow of graphs
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Existence of martingale solutions and large-time behavior for a stochastic mean curvature flow of graphs Nils Dabrock1 · Martina Hofmanová2 · Matthias Röger1 Received: 19 September 2019 / Revised: 15 September 2020 / Accepted: 12 October 2020 © The Author(s) 2020
Abstract We are concerned with a stochastic mean curvature flow of graphs over a periodic domain of any space dimension. For the first time, we are able to construct martingale solutions which satisfy the equation pointwise and not only in a generalized (distributional or viscosity) sense. Moreover, we study their large-time behavior. Our analysis is based on a viscous approximation and new global bounds, namely, an L ∞ ω,x,t estimate for the gradient and an L 2ω,x,t bound for the Hessian. The proof makes essential use of the delicate interplay between the deterministic mean curvature part and the stochastic perturbation, which permits to show that certain gradient-dependent energies are supermartingales. Our energy bounds in particular imply that solutions become asymptotically spatially homogeneous and approach a Brownian motion perturbed by a random constant. Keywords Stochastic mean curvature flow · Variational SPDE · Martingale solutions · Energy estimates · Large-time behavior Mathematics Subject Classification 60H15 · 60H30 · 53C44
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Martina Hofmanová [email protected] Nils Dabrock [email protected] Matthias Röger [email protected]
1
Fakultät für Mathematik, Technische Universität Dortmund, Vogelpothsweg 87, 44227 Dortmund, Germany
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Faculty of Mathematics, Bielefeld University, Universitätsstrasse 25, 33615 Bielefeld, Germany
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N. Dabrock et al.
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Existence of viscous approximation . . . . . . . . . . . . . . 5 A priori estimates . . . . . . . . . . . . . . . . . . . . . . . 6 Vanishing viscosity limit . . . . . . . . . . . . . . . . . . . . 7 Large-time behavior . . . . . . . . . . . . . . . . . . . . . . Appendix A. Variational SPDE under a compactness assumption Data availability statement . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1 Introduction The mean curvature flow (MCF) of hypersurfaces is one key example of a geometric evolution law and is of major importance both for applications and for the mathematical theory of surface evolution equations, see for example [17,42,55] or [
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