Showing papers in "Interfaces and Free Boundaries in 2010"

On the two-phase navier-stokes equations with surface tension

Abstract: The two-phase free boundary problem for the Navier-Stokes system is considered in a situation where the initial interface is close to a halfplane. By means of Lp-maximal regularity of the underlying linear problem we show local well-posedness of the problem, and prove that the solution, in particular the interface, becomes instantaneously real analytic. In this paper we consider a free boundary problem that describes the motion of two viscous incompressible capillary Newtonian fluids. The fluids are separated by an interface that is unknown and has to be determined as part of the problem. Let 1(0) ⊂ R n+1 (n ≥ 1) be a region occupied by a viscous incompressible fluid, fluid1, and let 2(0) be the complement of the closure of 1(0) in R n+1 , corre- sponding to the region occupied by a second incompressible viscous fluid, fluid2. We assume that the two fluids are immiscible. Let 0 be the hypersurface that bounds 1(0) (and hence also 2(0)) and let ( t) denote the position of 0 at time t. Thus, ( t) is a sharp interface which separates the fluids occupying the regions 1(t) and 2(t), respectively, where 2(t) := R n+1 \ 1(t). We denote the normal field on ( t), pointing from 1(t) into 2(t), by ν(t, � ). Moreover, we de- note by V (t, � ) and κ(t, � ) the normal velocity and the mean curvature of ( t) with respect to ν(t, � ), respectively. Here the curvature κ(x, t) is assumed to be negative when 1(t) is convex in a neighborhood of x ∈ ( t). The motion of the fluids is governed by the following system of equations for i = 1,2 :    

Mass conserving Allen-Cahn equation and volume preserving mean curvature flow

Abstract: We dedicate this article to the memory of Michelle Schatzman We consider a mass conserving Allen–Cahn equation ut = ∆u + e−2(f (u) − eλ(t)) in a bounded domain with no flux boundary condition, where eλ(t) is the average of f (u(·, t)) and −f is the derivative of a double equal well potential. Given a smooth hypersurface γ0 contained in the domain, we show that the solution ue with appropriate initial data tends, as e ↘ 0, to a limit which takes only two values, with the jump occurring at the hypersurface obtained from the volume preserving mean curvature flow starting from γ0.

A free boundary problem arising in a simplified tumour growth model of contact inhibition

Abstract: It is observed in vitro and in vivo that when two populations of different types of cells come near to each other, the rate of proliferation of most cells decreases. This phenomenon is often called contact inhibition of growth between two cells. In this paper, we consider a simplified 1-dimensional PDEmodel for normal and abnormal cells, motivated by the paper by Chaplain, Graziano and Preziosi ([5]). We show that if the two populations are initially segregated, then they remain segregated due to the contact inhibition mechanism. In this case the system of PDE’s can be formulated as a free boundary problem.

Parametric Approximation of Surface Clusters driven by Isotropic and Anisotropic Surface Energies

Abstract: We present a variational formulation for the evolution of surface clusters in R3 by mean curvature flow, surface diffusion and their anisotropic variants. We introduce the triple junction line conditions that are induced by the considered gradient flows, and present weak formulations of these flows. In addition, we consider the case where a subset of the boundaries of these clusters are constrained to lie on an external boundary. These formulations lead to unconditionally stable, fully discrete, parametric finite element approximations. The resulting schemes have very good properties with respect to the distribution of mesh points and, if applicable, volume conservation. This is demonstrated by several numerical experiments, including isotropic double, triple and quadruple bubbles, as well as clusters evolving under anisotropic mean curvature flow and anisotropic surface diffusion, including for regularized crystalline surface energy densities.

Stable constant mean curvature hypersurfaces are area minimizing in small l 1 neighborhoods

Abstract: .- We prove that a strictly stable constant-mean-curvature hypersurface ina smooth manifold of dimension less than or equal to 7 is uniquely homologically areaminimizing for fixed volume in a small L 1 neighborhood. 1. IntroductionBy work of White [W] and Grosse-Brauckman [Gr], a strictly stable constant-mean-curvature surface S 0 is minimizing in a small neighborhood U of S 0 amongcompetitor hypersurfaces S ⊂ U enclosing the same volume. Assuming M compact,we extend their results to a small L 1 neighborhood of S 0 , i.e., to hypersurfaces Ssuch that S − S 0 bounds a region with net volume 0 and small total volume.Stable constant-mean-curvature hypersurfaces in M appear in particular as so-lutions of the isoperimetric problem; see for instance [R1]. In the case that theambient space is a flat 3-torus T 3 there is a connection between the isoperimetricproblem and the study of mesoscale phase separation phenomena; see Choksi andSternberg [CS]. One simple model minimizes the Cahn-Hilliard free energyZ

On the existence of mean curvature flow with transport term

Abstract: We prove the global-in-time existence of weak solution for a hypersurface evolution problem where the velocity is the sum of the mean curvature and arbitrarily given non-smooth vector field in a suitable Sobolev space. The approximate solution is obtained by the Allen–Cahn equation with transport term. By establishing the density ratio upper bound on the phase boundary measure it is shown that the limiting surface moves with the desired velocity in the sense of Brakke.

Wavelet analogue of the Ginzburg–Landau energy and its Γ-convergence

Abstract: Fourier analysis provides many elegant approaches to differential operators and related tools in PDE-based image processing. Our work develops the idea of using a more localized basis than the Fourier one in the context of variational methods based on diffuse interfaces ([17], [5], [6]). Our philosophy involves looking for new types of pseudo-differential energy functionals, which inherit important properties of classical functionals, but leave out the computational drawbacks associated with the discrete differentiation. Wavelets appeared in the variational context in a number of works (e.g. [8] and [13]). We use a well-known characterization of function regularity in terms of the wavelet coefficients, we also take a widely used PDE-based functional to become a prototype of the new energy we design. However, our approach is conceptually different from the wavelet-PDE techniques that use wavelets to solve PDEs numerically [4], [11] as well as those involving differentiation in the wavelet domain [9]. The energy functional we study is entirely “derivative-free” as it is defined; nevertheless, it exhibits behavior analogous to the ones of energies used in material science and fluid dynamics. In classical fluid models, an interface between two fluids is treated as infinitely thin and sharp, and is endowed with properties such as surface tension. Diffuse-interface theories replace this sharp interface with continuous variations of an order parameter, such as density, in a way consistent with microscopic theories of the interface. In the inhomogeneous systems which involve domains of well-defined phases separated by a distinct interface, the diffuse-interface description assumes the smoothness of the transition between phases and approaches the sharp interface model asymptotically. At the same time, if used in signal processing applications, diffuse-interface models

Convergence of a large time-step scheme for mean curvature motion

Abstract: We propose a new scheme for the level set approximation of motion by mean curvature (MCM). The scheme originates from a representation formula recently given by Soner and Touzi, which allows us to construct large time-step, Godunov-type schemes. One such scheme is presented and its consistency is analyzed. We also provide and discuss some numerical tests.

A posteriori error controlled local resolution of evolving interfaces for generalized Cahn–Hilliard equations

Abstract: For equations of generalized Cahn–Hilliard type we present an a posteriori error analysis that is robust with respect to a small interface length scale γ . We propose the solution of a fourth order elliptic eigenvalue problem in each time step to gain a fully computable error bound, which only depends polynomially (of low order) on the inverse of γ . A posteriori and a priori error bounds for the eigenvalue problem are also derived. In numerical examples we demonstrate that this approach extends the applicability of robust a posteriori error estimation as it removes restrictive conditions on the initial data. Moreover we show that the computation of the principal eigenvalue allows the detection of critical points during the time evolution that limit the validity of the estimate.

Mixed finite element method for electrowetting on dielectric with contact line pinning

Abstract: We present a mixed finite element method for a model of the flow in a Hele-Shaw cell of 2-D fluid droplets surrounded by air driven by surface tension and actuated by an electric field. The application of interest regards a micro-fluidic device called ElectroWetting on Dielectric (EWOD). Our analysis first focuses on the time discrete (continuous in space) problem and is presented in a mixed variational framework, which incorporates curvature as a natural boundary condition. The model includes a viscous damping term for interface motion, as well as contact line pinning (sticking of the interface) and is captured in our formulation by a variational inequality. The semi-discrete problem uses a semiimplicit time discretization of curvature. We prove the well-posedness of the semi-discrete problem and fully discrete problem when discretized with iso-parametric finite elements. We derive a priori error estimates for the space discretization. We also prove the convergence of an Uzawa algorithm for solving the semi-discrete EWOD system with inequality constraint. We conclude with a discussion about experimental orders of convergence.

The thin film equation with backwards second order diffusion

Abstract: where D 1; n > 0; M > m; and A > 0: Global existence of weak nonnegative solutions is proven whenm n > 2 andA > 0 or D 1; and when 2 3=2 if D 1: A local energy estimate is obtained when 2 6 n n=2; for the case of “strong slippage”, 0 < n < 2; when D 1 based on local entropy estimates, and for the case of “weak slippage”, 2 6n < 3; when D 1 based on local entropy and energy estimates.

Cauchy problems for noncoercive Hamilton–Jacobi–Isaacs equations with discontinuous coefficients

Abstract: We study the Cauchy problem for a homogeneous and not necessarily coercive Hamilton‐Jacobi‐ Isaacs equation with anx-dependent, piecewise continuous coefficient. We prove that under suitable assumptions there exists a unique and continuous viscosity solution. The result applies in particular to the Carnot‐Caratheodory eikonal equation with discontinuous refraction index of a family of vector fields satisfying the Hormander condition. Our results are also of interest in connection with geometric flows with discontinuous velocity in anisotropic media with a non-euclidian ambient space.

Convexity breaking of the free boundary for porous medium equations

Abstract: Is then Pφ(t) convex for every t > 0? (See Section 2 for the definition of α-concavity.) The porous medium equation provides a simple model in many physical situations, in particular, the flow of an isentropic gas through a porous medium; in such a case, u and um−1 represent the density and the pressure of the gas, respectively. Due to its practical interest, regularity and geometric properties of the free boundary ∂Pφ(t) have been extensively studied by many

Well-posedness of a parabolic moving-boundary problem in the setting of Wasserstein gradient flows

Abstract: We develop a gradient-flow framework based on the Wasserstein metric for a parabolic moving-boundary problem that models crystal dissolution and precipitation. In doing so we derive a new weak formulation for this moving-boundary problem and we show that this formulation is well-posed. In addition, we develop a new uniqueness technique based on the framework of gradient flows with respect to the Wasserstein metric. With this uniqueness technique, the Wasserstein framework becomes a complete well-posedness setting for this parabolic moving-boundary problem.

Bifurcation and secondary bifurcation of heavy periodic hydroelastic travelling waves

Abstract: The existence question for two-dimensional symmetric steady waves travelling on the surface of a deep ocean beneath a heavy elastic membrane is analyzed as a problem in bifurcation theory. The behaviour of the two-dimensional cross-section of the membrane is modelled as a thin (unshearable), heavy, hyperelastic extensible rod, and the fluid beneath is supposed to be in steady two-dimensional irrotational motion under gravity. When the wavelength has been normalized to be 2 , and when gravity and the density of the undeformed membrane are prescribed, there are two free parameters in the problem: the speed of the wave and the drift velocity of the membrane. It is observed that the problem, when linearized about uniform horizontal flow, has at most two independent solutions for any values of the parameters. When the linearized problem has only one normalized solution, it is shown that the full nonlinear problem has a sheet of solutions consisting of a family of curves bifurcating from simple eigenvalues. Here one of the problem’s parameters is used to index a family of bifurcation problems in which the other is the bifurcation parameter. When the linearized problem has two solutions, with wave numbers k and l such that maxfk;lg=minfk;lg = 2 Z, it is shown that there are three two-dimensional sheets of bifurcating solutions. One consists of “special” solutions with minimal period 2=k ; another consists of “special” solutions with minimal period 2=l ; and the third, apart from those on the curves where it intersects the “special” sheets, consists of “general” solutions with minimal period 2 . The two sheets of “special” solutions are rather similar to those that occur when the linearized problem has only one solution. However, points where the first sheet or the second sheet intersects the third sheet are period-multiplying (or symmetry-breaking) secondary bifurcation points on primary branches of “special” solutions. This phenomenon is analogous to that of Wilton ripples, which arises in the classical water-wave problem when the surface tension has special values. In the case of Wilton ripples, the coefficient of surface tension and the wave speed are the problem’s two parameters. In the present context, there are two speed parameters, meaning that the membrane elasticity does not need to be highly specified for this symmetry-breaking phenomenon to occur. 2010 Mathematics Subject Classification: 35R35, 74B20, 74F10, 76B07, 37G40.

Error analysis for the approximation of axisymmetric Willmore flow by C1-finite elements

Abstract: We consider the Willmore flow of axially symmetric surfaces subject to Dirichlet boundary conditions. The corresponding evolution is described by a nonlinear parabolic PDE of fourth order for the radius function. A suitable weak form of the equation, which is based on the first variation of the Willmore energy, leads to a semidiscrete scheme, in which we employ piecewise cubic C1-finite elements for the one-dimensional approximation in space. We prove optimal error bounds in Sobolev norms for the solution and its time derivative and present numerical test examples.

Transport of interfaces with surface tension by 2D viscous flows

Abstract: We consider the problem of finding a global weak solution for two-dimensional, incompressible viscous flow on a torus, containing a surface-tension bearing curve transported by the flow. This is the simplest case of a class of two-phase flows considered by Plotnikov in [16] and Abels in [1]. Our work complements Abels’ analysis by examining this special case in detail. We construct a family of approximations and show that the limit of these approximations satisfies, globally in time, an incomplete set of equations in the weak sense. In addition, we examine criteria for closure of the limit system, we find conditions which imply nontrivial dependence of the limiting solution on the surface tension parameter, and we obtain a new system of evolution equations which models our flowinterface problem, in a form that may be useful for further analysis and for numerical simulations.

The Neumann problem in an irregular domain

Abstract: We ask the question of patterns’ stability for the reaction-diffusion equation with Neumann boundary conditions in an irregular domain in R , N ≥ 2, the model example being two convex regions connected by a small ’hole’ in their boundaries. By patterns we mean solutions having an interface, i.e. a transition layer between two constants. It is well known that in 1D domains and in many 2D domains ’patterns’ are unstable for this equation. We show that, unlike the 1D case, but as in 2D dumbbell domains, stable patterns exist. In a more general way, we prove invariance of stability properties for steady states when a sequence of domains Ωn converges to our limit domain Ω in the sense of Mosco. We illustrate the theoretical results by numerical simulations of evolving and persisting interfaces. ∗To whom correspondence should be addressed

Boundary regularity for a parabolic obstacle type problem

Abstract: We study the regularity of the free boundary, near contact points with the fixed boundary, for a parabolic free boundary problem Lambda u - partial derivative u/partial derivative t = chi({u not equal o}) in Q(r)(+) = {(x, t) is an element of B-r x (-r(2), 0); x(1) > 0}, u = f(x, t) on {x(1) = 0} boolean AND Q(r). We will show that under certain regularity assumptions on the boundary data f the free boundary is a C-1 manifold up to the fixed boundary. We also show that the C-1 modulus of continuity is uniform for a certain, and specified, subclass of solutions.

Long-time behaviour of two-phase solutions to a class of forward-backward parabolic equations

Abstract: We consider two-phase solutions to the Neumann initial-boundary value problem for the parabolic equation ut = [φ(u)]xx , where φ is a nonmonotone cubic-like function. First, we prove global existence for a restricted class of initial data u0, showing that two-phase solutions can be obtained as limiting points of the family of solutions to the Neumann initial-boundary value problem for the regularized equation ut = [φ(u )]xx + eutxx (e > 0). Then, assuming global existence, we study the long-time behaviour of two-phase solutions for any initial datum u0.

Nonlinear stability analysis of a two-dimensional diffusive free boundary problem

Abstract: We explore global existence and stability of planar solutions to a multi-dimensional Case II polymer diffusion model which takes the form of a one-phase free boundary problem with phase onset. Due to a particular boundary condition, convergence can not be expected on the whole domain. A boundary integral formulation derived in [16] is shown to remain valid in the present context and allows us to circumvent this difficulty by restricting the analysis to the free boundary. The integral operators arising in the boundary integral formulation are analyzed by methods of pseudodifferential calculus. This is possible as explicit symbols are available for the relevant kernels. Spectral analysis of the linearization can then be combined with a known principle of linearized stability [13] to obtain local exponential stability of planar solutions with respect to two-dimensional perturbations.

On a free boundary problem describing the phase transition in an incompressible viscous fluid

Abstract: Ice melts at 0 C under a pressure of 1 atm, and increasing the pressure decreases the melting temperature. In the present paper, a new problem is posed that describes the process of phase transition in an incompressible viscous fluid, taking into account the above-described pressure effect. This problem is described as a free boundary problem in terms of the Navier‐Stokes equations coupled with the heat equation, where the equilibrium temperature is assumed to be related to the pressure by the Clapeyron‐Clausius equation. We prove the existence of a global-in-time solution.