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In this paper, we derive the dissipation term in the local energy balance law of weak solutions for a surface growth model arising in the molecular-beam-epitaxy process by using the third-order structure functions. This enables us to address Yang's question posed in [52, J. Differential Equations 283, 2021] , consider generalized Onsager conjecture, and present an upper bound of energy dissipation rate of the form O (ν 3 α − 1 α + 1 ) under the condition that the weak solution h ν belongs to L 3 (0 , T ; B 3 , ∞ α + 1 (T)) with α ∈ (0 , 1) in this model. More importantly, the link between Duchon-Robert's remarkable dissipation term and Lions's classical sufficient condition for energy balance law in the 3D Navier-Stokes equations is illustrated. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the precise relationship between the Hausdorff dimension of possible singular point set S of suitable weak solutions and the parameter α in the nonlinear term in the following parabolic equation h t + h x x x x + ∂ x x | h x | α = f. It is shown that when 5 / 3 ≤ α < 7 / 3 , the 3 α − 5 α − 1 dimensional parabolic Hausdorff measure of S is zero, which generalizes the recent corresponding work of Ozánski and Robinson in [SIAM J. Math. Anal., 51, 228–255 (2019)] for α = 2 and f = 0. The same result is valid for a 3D modified Navier–Stokes system. [ABSTRACT FROM AUTHOR]

By means of Guo and Wang's pure energy method, we present the optimal decay rates of higher-order spatial derivatives of solutions to the surface growth model arising in epitaxial growth of monocrystals, under the smallness condition in scaling-invariant (critical) Sobolev space. We also obtain the optimal decay estimates of higher-order spatial derivatives of the 3D Navier–Stokes system with small initial data in critical Lebesgue space, which is larger than the critical Sobolev space in previous works. The proof relies on our new observation that the solutions of 3D Navier–Stokes equations keep the smallness in critical Lebesgue space as well as the critical Sobolev space. [ABSTRACT FROM AUTHOR]

We show local higher integrability of derivative of a suitable weak solution to the surface growth model, provided a scale-invariant quantity is locally bounded. If additionally our scale-invariant quantity is small, we prove local smoothness of solutions. [ABSTRACT FROM AUTHOR]

We are concerned with one-dimensional scalar surface growth model arising from the physical process of molecular epitaxy. The mathematical theory of the surface growth model is known to share a number of striking similarities with the Navier-Stokes equations, including the results regarding existence and uniqueness of solutions. In this paper, we shall investigate an important subject in mathematical physics: the energy conservation for weak solutions of the surface growth model. As an analogue of the Navier-Stokes equations, we find some sufficient integral conditions that guarantee the validity of energy equality. As far as we know, this is the first result in this aspect. [ABSTRACT FROM AUTHOR]

Abstract The surface growth model, u t + u x x x x + ∂ x x u x 2 = 0 , is a one-dimensional fourth order equation, which shares a number of striking similarities with the three-dimensional incompressible Navier–Stokes equations, including the results regarding existence and uniqueness of solutions and the partial regularity theory. Here we show that a weak solution of this equation is smooth on a space-time cylinder Q if the Serrin condition u x ∈ L q ′ L q (Q) is satisfied, where q , q ′ ∈ [ 1 , ∞ ] are such that either 1 / q + 4 / q ′ < 1 or 1 / q + 4 / q ′ = 1 , q ′ < ∞. [ABSTRACT FROM AUTHOR]

We prove two partial regularity results for the scalar equation ut+uxxxx+@xxu²x = 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 Navie-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(1991), 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. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the higher derivatives estimate of the surface growth equation which has similar features to the 3D Navier–Stokes equations. We extend the well-known apriori estimates for the Navier–Stokes equations due to Foias–Guillope–Temam in C. Foias et al. (1981) to this model. Several new sufficient conditions that guarantee regularity of solutions to the generalized surface growth equation are also presented. As a byproduct, the global well-posedness of the generalized surface growth equation in the sense of Lions's classical work in J.L. Lions (1969) is obtained. [ABSTRACT FROM AUTHOR]

We review results on the existence and uniqueness for a surface growth model with or without space-time white noise. If the surface is a graph, then this model has striking similarities to the three dimensional Navier-Stokes equations in terms of energy estimates and scaling properties, and in both models the question of uniqueness of global weak solutions remains open. In the physically relevant dimension $d=2$ and with the physically relevant space-time white noise driving the equation, the direct fixed-point argument for mild solutions fails, as there is not sufficient regularity for the stochastic forcing. The situation is the simplest case where the method of regularity structures introduced by Martin Hairer can be applied, although we follow here a significantly simpler approach to highlight the key problems. Using spectral Galerkin method or any other type of regularization of the noise, one can give a rigorous meaning to the stochastic PDE and show existence and uniqueness of local solutions in that setting. Moreover, several types of regularization seem to yield all the same solution. We finally comment briefly on possible blow up phenomena and show with a simple argument that many complex-valued solutions actually do blow up in finite time. This shows that energy estimates alone are not enough to verify global uniqueness of solutions. Results in this direction are known already for the 3D-Navier Stokes by Li and Sinai, treating complex valued solutions, and more recently by Tao by constructing an equation of Navier-Stokes type with blow up. [ABSTRACT FROM AUTHOR]

We study local existence and uniqueness for a surface growth model with space-time white noise in 2D. Unfortunately, the direct fixed-point argument for mild solutions fails here, as we do not have sufficient regularity for the stochastic forcing. Nevertheless, one can give a rigorous meaning to the stochastic PDE and show uniqueness of solutions in that setting. Using spectral Galerkin method and any other types of regularization of the noise, we obtain always the same solution. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we establish the local and global existence and uniqueness of solutions for general nonlinear evolution equations with coefficients satisfying some local monotonicity and generalized coercivity conditions. An analogous result is obtained for stochastic evolution equations in Hilbert space with additive noise. As applications, the main results are applied to obtain simpler proofs in known cases as the stochastic 3D Navier–Stokes equation, the tamed 3D Navier–Stokes equation and the Cahn–Hilliard equation, but also to get new results for stochastic surface growth PDE and stochastic power law fluids. [Copyright &y& Elsevier]

We present a new regularity criterion for suitable weak solutions of the one-dimensional surface growth initial-value problem in terms of mixed Lebesgue norms. [ABSTRACT FROM AUTHOR]