Abstract
We prove the existence of non-negative martingale solutions to a class of stochastic degenerate-parabolic fourth-order PDEs arising in surface-tension driven thin-film flow influenced by thermal noise. The construction applies to a range of mobilites including the cubic one which occurs under the assumption of a no-slip condition at the liquid-solid interface. Since their introduction more than 15 years ago, by Davidovitch, Moro, and Stone and by Grün, Mecke, and Rauscher, the existence of solutions to stochastic thin-film equations for cubic mobilities has been an open problem, even in the case of sufficiently regular noise. Our proof of global-in-time solutions relies on a careful combination of entropy and energy estimates in conjunction with a tailor-made approximation procedure to control the formation of shocks caused by the nonlinear stochastic scalar conservation law structure of the noise.
Highlights
In this work, we consider the stochastic thin-film equation du = −∂x M(u) ∂x3u dt + ∂x M(u) ◦ dW in QT, (1.1)where u = u(t, x) denotes the height of a thin viscous film depending on the independent variables time t ∈ [0, T ], where T ∈ (0, ∞) is fixed, and lateral position x ∈ T, where T is the one-dimensional torus of length L:= |T|, and QT :=[0, T ] × T
Where u = u(t, x) denotes the height of a thin viscous film depending on the independent variables time t ∈ [0, T ], where T ∈ (0, ∞) is fixed, and lateral position x ∈ T, where T is the one-dimensional torus of length L:= |T|, and QT :=[0, T ] × T
Equation (1.1) describes the spreading of viscous thin films driven by capillary forces and thermal noise and decelerated by friction
Summary
We give a brief account on the literature for the deterministic thin-film equation: A theory of existence of weak solutions for the deterministic thin-film equation has been developed in [1,4,6] and [5,43,45] for zero and nonzero contact angles at the intersection of the liquid-gas and liquid-solid interfaces, respectively, while the higher-dimensional version of (1.1) with W = 0 in [0, T ] × T and zero contact angles has been the subject of [11,32] For these solutions, a number of quantitative results has been obtained – including optimal estimates on spreading rates of free boundaries, that is, the triple lines separating liquid, gas, and solid, see [2,18,30,34], optimal conditions on the occurrence of waiting time phenomena [12], as well as scaling laws for the size of waiting times [19,20].
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