Local minimizers and planar interfaces in a phase-transition model with interfacial energy

Interfacial energy is often incorporated into variational solid-solid phase transition models via a perturbation of the elastic energy functional involving second gradients of the deformation. We study consequences of such higher-gradient terms for local minimizers and for interfaces. First it is shown that at slightly sub-critical temperatures, a phase which globally minimizes the elastic energy density at super-critical temperatures is an L ¹-local minimizer of the functional including interfacial energy, whereas it is typically only a W ^1,??-local minimizer of the purely elastic functional. The second part deals with the existence and uniqueness of smooth interfaces between different wells of the multi-well elastic energy density. Attention is focussed on so-called planar interfaces, for which the deformation depends on a single direction x · N and the deformation gradient then satisfies a rank-one ansatz of the form ${Dy(x) = A + u(x \cdot N) \otimes N}$ , where A and ${B=A+a \otimes N}$ are the gradients connected by the interface.     

Global solutions of the Navier-Stokes equations for compressible flow with density-dependent viscosity and discontinuous initial data

In this paper, we study the evolutions of the interfaces between the gas and the vacuum for viscous one-dimensional isentropic gas motions. We prove the global existence and uniqueness for discontinuous solutions of the Navier-Stokes equations for compressible flow with density-dependent viscosity coefficient. Precisely, the viscosity coefficient μ is proportional to ρ^θ with 0<θ<1. Specifically, we require that the initial density be piecewise smooth with arbitrarily large jump discontinuities, bounded above and below away from zero, in the interior of gas. We show that the discontinuities in the density persist for all time, and give a decay result for the density as t→+∞.     

Stability of discontinuity structures described by a generalized KdV–Burgers equation

The stability of discontinuities representing solutions of a model generalized KdV–Burgers equation with a nonmonotone potential of the form φ(u) = u⁴–u² is analyzed. Among these solutions, there are ones corresponding to special discontinuities. A discontinuity is called special if its structure represents a heteroclinic phase curve joining two saddle-type special points (of which one is the state ahead of the discontinuity and the other is the state behind the discontinuity).The spectral (linear) stability of the structure of special discontinuities was previously studied. It was shown that only a special discontinuity with a monotone structure is stable, whereas special discontinuities with a nonmonotone structure are unstable. In this paper, the spectral stability of nonspecial discontinuities is investigated. The structure of a nonspecial discontinuity represents a phase curve joining two special points: a saddle (the state ahead of the discontinuity) and a focus or node (the state behind the discontinuity). The set of nonspecial discontinuities is examined depending on the dispersion and dissipation parameters. A set of stable nonspecial discontinuities is found.     

Broken solutions of homogeneous variational problems

A real-valued function L on the tangent bundle of $\text{R}$ⁿ gives rise to variational problems as follows: for two points x₀, x₁ in $\text{R}$ⁿ and a time interval [0, T] to determine a curve γ: [O,T] → $\text{R}$ⁿ, connecting x₀ with x₁ which minimizes ∫₀^TL(γ(t), gg(t)) dt. We consider the associated Hamiltonian vectorfield on the cotangent bundle. If L is not convex on each fibre then the corresponding Hamiltonian vectorfield is not continuous. For homogeneous L and n = 2 restriction to an energy level gives an essentially three-dimensional vectorfield. In this case we list the possible discontinuities for generic L. Then we observe that there exits an open class of such variational problems, which admit no minimizing solution.     

Analysis of a two-phase cell division model

In this paper we mainly study an equal mitosis two-phase cell division model. By using the C₀-semigroup theory, we prove that this model is the well-posed in L¹[0, 1] × L¹[0, 1], and exhibits asynchronous exponential growth phenomenon as time goes to infinity. We also give a comparison between this two-phase model with the classical one-phase model. Finally, we briefly study the asymmetric two-phase cell division model, and show that similar results hold for it.     

Isogeometric Analysis of a Phase Field Model for Darcy Flows with Discontinuous Data

The authors consider a phase field model for Darcy flows with discontinuous data in porous media; specifically,they adopt the Hele-Shaw-Cahn-Hillard equations of[Lee,Lowengrub,Goodman,Physics of Fluids,2002] to model flows in the Hele-Shaw cell through a phase field formulation which incorporates discontinuities of physical data,namely density and viscosity,across interfaces. For the spatial approximation of the problem,the authors use NURBS—based isogeometric analysis in the framework of the Galerkin method,a computational framework which is particularly advantageous for the solution of high order partial differential equations and phase field problems which exhibit sharp but smooth interfaces. In this paper,the authors verify through numerical tests the sharp interface limit of the phase field model which in fact leads to an internal discontinuity interface problem; finally,they show the efficiency of isogeometric analysis for the numerical approximation of the model by solving a benchmark problem,the so-called"rising bubble" problem.     

A Regularized Sharp Interface Model for Phase Transformation Accounting for Prescribed Sharp Interface Kinetics

The macroscopic mechanical behavior of multi-phasic materials depends on the formation and evolution of their microstructure by means of phase transformation. In case of martensitic transformations, the resulting phase boundaries are sharp interfaces. We carry out a geometrically motivated discussion of the regularization of such sharp interfaces by use of an order parameter/phase-field and exploit the results for a regularized sharp interface model for two-phase elastic materials with evolving phase boundaries. To account for the dissipative effects during phase transition, we model the material as a generalized standard medium with energy storage and a dissipation function that determines the evolution of the regularized interface. Making use of the level-set equation, we are thereby able to directly translate prescribed sharp interface kinetic relations to the constitutive model in the regularized setting. We develop a suitable incremental variational three-field framework for the dissipative phase transformation problem. Finally, the modeling capability and the associated numerical solution techniques are demonstrated by means of a representative numerical example. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Viscous limits for piecewise smooth solutions of the p-system

We study the zero-dissipation problem for a one-dimensional model system for the isentropic flow of a compressible viscous gas, the so-called p-system with viscosity. When the solution of the inviscid problem is piecewise smooth and having finitely many noninteracting shocks satisfying the entropy condition, there exists unique solution to the viscous problem which converges to the given inviscid solution away from shock discontinuities at a rate of order ε as the viscosity coefficient ε goes to zero. The proof is given by a matched asymptotic analysis and an elementary energy method. And we do not need the smallness condition on the shock strength.     

Synchronized oscillations on a Kuramoto ring and their entrainment under periodic driving

We consider a finite number of coupled oscillators on a ring as an adaptation of the Kuramoto model of populations of oscillators. The synchronized solutions are characterized by an integer m, the winding number, and a second integer l, with solutions of type (m, l = 0) being all stable. Following a number of recent works (see below) we indicate how the various solutions emerge as the coupling strength K is varied, presenting a perturbative expression for these for large K. The low K scenario is also briefly outlined, where the onset of synchronization by a tangent bifurcation is explained. The simplest situation involving three oscillators is described, where more than one tangent bifurcations are involved. Immediately before the tangent bifurcation leading to synchronization, the system exhibits the phenomenon of frequency- (or phase) splitting where more than one (usually two) phase clusters are involved. All the synchronized solutions are seen to be entrained by an external periodic driving, provided that the driving frequency is sufficiently close to the frequency of the synchronized population. A perturbative approach is outlined for the construction of the entrained solutions. Under a periodic driving with an appropriately limited detuning, there occurs entrainment of the phase-split solutions as well.     

Existence of weak solutions for a non-classical sharp interface model for a two-phase flow of viscous,incompressible fluids

We introduce a new sharp interface model for the flow of two immiscible, viscous, incompressible fluids. In contrast to classical models for two-phase flows we prescribe an evolution law for the interfaces that takes diffusional effects into account. This leads to a coupled system of Navier–Stokes and Mullins–Sekerka type parts that coincides with the asymptotic limit of a diffuse interface model. We prove the long-time existence of weak solutions, which is an open problem for the classical two-phase model. We show that the phase interfaces have in almost all points a generalized mean curvature.     

Polynomial Models of Degree 5 and Algebras of Their Automorphisms

In classifying and studying holomorphic automorphisms of surfaces, it is often convenient to pass to tangent model surfaces. This method is well developed for surfaces of type (n,K), where K≤ ² ; for such surfaces, tangent quadrics (i.e., surfaces determined by equations of degree 2) with a number of useful properties have been constructed. In recent years, for surfaces of higher codimensions, tangent model surfaces of degrees 3 and 4 with similar properties were constructed. However, this construction imposes new constraints on the codimension. In this paper, the same method is applied to surfaces of even higher codimension. Model surfaces of the fifth degree are constructed. It is shown that all the basic useful properties of model surfaces are preserved, in spite of a number of technical difficulties.     

A compactness result for Landau state in thin-film micromagnetics

We deal with a nonconvex and nonlocal variational problem coming from thin-film micromagnetics. It consists in a free-energy functional depending on two small parameters ε and η and defined over vector fields m:Ω⊂R²→S² that are tangent at the boundary ∂Ω. We are interested in the behavior of minimizers as ε,η→0. They tend to be in-plane away from a region of length scale ε (generically, an interior vortex ball or two boundary vortex balls) and of vanishing divergence, so that S¹-transition layers of length scale η (Néel walls) are enforced by the boundary condition. We first prove an upper bound for the minimal energy that corresponds to the cost of a vortex and the configuration of Néel walls associated to the viscosity solution, so-called Landau state. Our main result concerns the compactness of vector fields {mε,η}_ε,η↓0 of energies close to the Landau state in the regime where a vortex is energetically more expensive than a Néel wall. Our method uses techniques developed for the Ginzburg-Landau type problems for the concentration of energy on vortex balls, together with an approximation argument of S²-vector fields by S¹-vector fields away from the vortex balls.     

Cycle analysis of a two-phase queueing model with threshold

We consider a single-server, two-phase queueing system with N-policy. Customers arrive at the system according to a Poisson process and receive batch service in the first phase followed by individual services in the second phase. If the system becomes empty at the moment of the completion of the second-phase services, it is turned off. After an idle period, when the queue length reaches N (threshold), the server is turned on and begins to serve customers. We obtain the system size distribution and show that the system size decomposes into three random variables. The system sojourn time is provided. Analysis for the gated batch service model is also provided. Finally we derive a condition under which the optimal operating policy is achieved.     

Wild double tangent ball embeddings of spheres in En

It has been known for years that a 2-sphere Σ in E³ must be flatly embedded if it has double tangent balls on opposite sides of Σ at each of its points. However, when the double tangent balls are not required to be on opposite sides of Σ, pathological embeddings exist. This paper details the allowable wild embeddings of spheres having these indiscrete double tangent balls and discusses the higher dimensional analogues.     

An example of a <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript>-smooth function whose gradient range is an arc with no tangent at any point

We construct an example of a C ¹-smooth real function of two variables whose gradient range is an arc with no tangent at any point.     

A mathematical model for induction hardening including mechanical effects

We investigate a mathematical model for induction hardening of steel. It accounts for electromagnetic effects that lead to the heating of the workpiece as well as thermomechanical effects that cause the hardening of the workpiece. The new contribution of this paper is that we put a special emphasis on the thermomechanical effects caused by the phase transitions. We take care of effects like transformation strain and transformation plasticity induced by the phase transitions and allow for physical parameters depending on the respective phase volume fractions.The coupling between the electromagnetic and the thermomechanical part of the model is given through the temperature-dependent electric conductivity on the one hand and through the Joule heating term on the other hand, which appears in the energy balance and leads to the rise in temperature. Owing to the quadratic Joule heat term and a quadratic mechanical dissipation term in the energy balance, we obtain a parabolic equation with L¹ data. We prove existence of a weak solution to the complete system using a truncation argument.     

Cell membranes,Lipid Bilayers,and the Elastica Functional

We study an energy functional that arises in a simplified two-dimensional model for lipid bilayer membranes. We demonstrate that this functional, defined on a class of spatial mass densities, favours concentrations on ‘thin structures’. Stretching, fracture and bending of such structures all carry an energy penalty. In this sense we show that the models captures essential features of lipid bilayers, namely partial localisation and a solid-like behaviour. Our findings are made precise in a Gamma-convergence result. We prove that a rescaled version of the energy functional converges in the ‘zero thickness limit’ to a functional that is defined on a class of planar curves. Finiteness of the limit value enforces both optimal thickness and non-fracture; if these conditions are met, then the value of this functional is given by the classical Elastica (bending) energy. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Approximating tensor product Bézier surfaces with tangent plane continuity

We present a simple method for degree reduction of tensor product Bézier surfaces with tangent plane continuity in L₂-norm. Continuity constraints at the four corners of surfaces are considered, so that the boundary curves preserve endpoints continuity of any order α. We obtain matrix representations for the control points of the degree reduced surfaces by the least-squares method. A simple optimization scheme that minimizes the perturbations of some related control points is proposed, and the surface patches after adjustment are C^∞ continuous in the interior and G¹ continuous at the common boundaries. We show that this scheme is applicable to surface patches defined on chessboard-like domains.