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- W2963280278 abstract "We prove two partial regularity results for the scalar equation $u_t+u_{xxxx}+partial_{xx}u_x^2=0$, a model of surface growth arising from the physical process of molecular epitaxy. We show that the set of space-time singularities has (upper) box-counting dimension no larger than 7/6 and 1-dimensional (parabolic) Hausdorff measure zero. These parallel the results available for the three-dimensional Navier--Stokes equations. In fact the mathematical theory of the surface growth model is known to share a number of striking similarities with the Navier--Stokes equations, and the partial regularity results are the next step towards understanding this remarkable similarity. As far as we know the surface growth model is the only lower-dimensional “mini-model” of the Navier--Stokes equations for which such an analogue of the partial regularity theory has been proved. In the course of our proof, which is inspired by the rescaling analysis of Lin [Comm. Pure Appl. Math., 51(1998), pp. 241--257] and Ladyzhenskaya and Seregin [J. Math. Fluid Mech., 1(1999), pp. 356--387], we develop certain nonlinear parabolic Poincaré inequality, which is a concept of independent interest. We believe that similar inequalities could be applicable in other parabolic equations." @default.
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- W2963280278 date "2019-01-01" @default.
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- W2963280278 title "Partial Regularity for a Surface Growth Model" @default.
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- W2963280278 doi "https://doi.org/10.1137/18m1166821" @default.
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