Global well-posedness of solutions for the epitaxy thin film growth model

In this paper, we consider the global well-posedness of solutions for the initial-boundary value problems of the epitaxy growth model. We first construct the local smooth solution, then by combining some a priori estimates, continuity argument, the local smooth solutions are extended step by step to all t > 0, provided that the initial datums sufficiently small and the smooth nonlinear functions satisfy certain local growth conditions

Well-posedness and stability for a class of fourth-order nonlinear parabolic equations

In this paper we examine well-posedness for a class of fourth-order nonlinear parabolic equation $\partial_t u + (-\Delta)^2 u = \nabla \cdot F(\nabla u)$, where $F$ satisfies a cubic growth conditions. We establish existence and uniqueness of the solution for small initial data in local BMO spaces. In the cubic case $F(\xi) = \pm \lvert \xi \rvert^2 \xi$ we also examine the large time behaivour and stability of global solutions for arbitrary and small initial data in VMO, respectively

Rigorous Numerical Verification of Uniqueness and Smoothness in a Surface Growth Model

Based on numerical data and a-posteriori analysis we verify rigorously the uniqueness and smoothness of global solutions to a scalar surface growth model with striking similarities to the 3D Navier--Stokes equations, for certain initial data for which analytical approaches fail. The key point is the derivation of a scalar ODE controlling the norm of the solution, whose coefficients depend on the numerical data. Instead of solving this ODE explicitly, we explore three different numerical methods that provide rigorous upper bounds for its solutio

A sufficient integral condition for local regularity of solutions to the surface growth model

The surface growth model, $u_t + u_{xxxx} + \partial_{xx} u_x^2 =0$, 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 $u_x\in L^{q'}L^q (Q)$ is satisfied, where $q,q'\in [1,\infty ]$ are such that either $1/q+4/q'<1$ or $1/q+4/q'=1$, $q'<\infty$.Comment: 18 page

Mathematical Theory of Water Waves

Water waves, that is waves on the surface of a fluid (or the interface between different fluids) are omnipresent phenomena. However, as Feynman wrote in his lecture, water waves that are easily seen by everyone, and which are usually used as an example of waves in elementary courses, are the worst possible example; they have all the complications that waves can have. These complications make mathematical investigations particularly challenging and the physics particularly rich. Indeed, expertise gained in modelling, mathematical analysis and numerical simulation of water waves can be expected to lead to progress in issues of high societal impact (renewable energies in marine environments, vorticity generation and wave breaking, macro-vortices and coastal erosion, ocean shipping and near-shore navigation, tsunamis and hurricane-generated waves, floating airports, ice-sea interactions, ferrofluids in high-technology applications, ...). The workshop was mostly devoted to rigorous mathematical theory for the exact hydrodynamic equations; numerical simulations, modelling and experimental issues were included insofar as they had an evident synergy effect

Global dynamics of a fourth-order parabolic equation describing crystal surface growth

In this paper, we study the global dynamics for the solution semiflow of a fourth-order parabolic equation describing crystal surface growth. We show that the equation has a global attractor in H4per(Ω) when the initial value belongs to H1per(Ω)

A sufficient integral condition for local regularity of solutions to the surface growth model

The surface growth model, , 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 is satisfied, where are such that either or ,