Stefan Problem with Phase Relaxation

Taking account of a microscopical model for dynamical supercoolingand superheating effects, the usual equilibrium condition prescribinga fixed temperature at the interface between two phases is replacedby relaxation dynamics for the phase variable x, representingthe concentration of one of the two phases. At first the approach of "non-equilibrium thermodynamics" isfollowed and a parabolic system is formulated; existence, uniquenessand regularity properties of the L²-solution are obtained bymeans of the theory of non-linear semigroups of contractions.These developments are also generalized to phase transitionsin heterogeneous systems. Then the heat diffusion equation is coupled with the relaxationdynamics for x and the well-posedness of an initial- and boundary-valueproblem is proved. The standard Stefan problem is obtained asa limit case. Also, a model for glass formation by very fastcooling is proposed. Finally, Fourier's conduction law is replaced by relaxationdynamics for the heat flux; this corresponds to assuming wavepropagation for the heat. An existence result is proved forthe corresponding problem and the limit behaviour as the relaxationtime vanishes is studied.     

A dual-dual mixed formulation for nonlinear exterior transmission problems

We combine a dual-mixed finite element method with a Dirichlet-to-Neumann mapping (derived by the boundary integral equation method) to study the solvability and Galerkin approximations of a class of exterior nonlinear transmission problems in the plane. As a model problem, we consider a nonlinear elliptic equation in divergence form coupled with the Laplace equation in an unbounded region of the plane. Our combined approach leads to what we call a dual-dual mixed variational formulation since the main operator involved has itself a dual-type structure. We establish existence and uniqueness of solution for the continuous and discrete formulations, and provide the corresponding error analysis by using Raviart-Thomas elements. The main tool of our analysis is given by a generalization of the usual Babuska-Brezzi theory to a class of nonlinear variational problems with constraints.

     

Modeling,shape analysis and computation of the equilibrium pore shape near a PEM–PEM intersection

In this paper we study the equilibrium shape of an interface that represents the lateral boundary of a pore channel embedded in an elastomer. The model consists of a system of PDEs, comprising a linear elasticity equation for displacements within the elastomer and a nonlinear Poisson equation for the electric potential within the channel (filled with protons and water). To determine the equilibrium interface, a variational approach is employed. We analyze: (i) the existence and uniqueness of the electrical potential, (ii) the shape derivatives of state variables and (iii) the shape differentiability of the corresponding energy and the corresponding Euler–Lagrange equation. The latter leads to a modified Young–Laplace equation on the interface. This modified equation is compared with the classical Young–Laplace equation by computing several equilibrium shapes, using a fixed point algorithm.     

Existence Theorems for the Linear, Space-inhomogeneous Transport Equation

This paper studies existence problems in L¹ for the linear,space-inhomogeneous Boltzmann equation with periodic or (perfectly)absorbing boundary conditions under realistic assumptions onthe cross-sections. By an iteration technique, solutions arefirst constructed to an integral equation variant of the transportequation in the case of bounded impact parameters and an L¹type of cross-sections. They are then used to study the existenceof solutions of a measure form of the transport equation inthe case of unbounded impact parameters. These solutions conservemass. Estimates of their higher moments are also given. In particularthe results hold for inverse kth-power forces with 3 < k 5.     

Viscosity solutions to a Cauchy problem of a phase-field model for solid-solid phase transitions

We investigate a partial differential equation which models solid-solid phase transitions. This model is for martensitic phase transitions driven by configurational force and its counterpart is for interface motion by mean curvature. Mathematically, this equation is a second-order nonlinear degenerate parabolic equation. And in multidimensional case, its principal part cannot be written into divergence form . We prove the existence and uniqueness of viscosity solution to a Cauchy problem for this model.     

Superconvergence of a Collocation-type Method for Hummerstein Equations

This paper considers the numerical solution of Hammerstein equationsof the form by a collocation method applied not to this equation, but ratherto an equivalent equation for z(t) :=g(t, y(t)). The desiredapproximation to y is then obtained by use of the (exact) equation In an earlier paper, questions of existence and optimal convergenceof the respective approximations to z and y were examined. Inthis sequel, collocation approximations to z are sought in certainpiecewise polynomial function spaces, and analogous of knownsuperconvergence results for the iterated collocation solutionof (linear) second-kind Fredhoim integral equations are statedand proved for the approximation to y.     

The Interface Between Fresh and Salt Groundwater: A Numerical Study

Present address: Laboratoire National d'Hydraulique, 6 quai Watier 78401 Chatou, France. In this paper the two-dimensional flow of fresh and salt waterthrough a homogeneous aquifer is considered. The two fluidsare assumed to be separated by a sharp interface. They differonly in their specific weight. This difference induces a flowin the aquifer which in turn causes a motion of the interface. We present a mathematical formulation of this problem whichconsists of a Poisson equation for the stream function coupledto a time evolution equation for the moving interface. The equationfor the stream function is solved by means of a finite-elementmethod while a predictor-corrector method (the S^ßscheme) is used for the discretization of the equation for theinterface.     

Optimal consumption models in economic growth

We study the optimal consumption problem in the one-sector model of economic growth under uncertainty. We show the existence of a classical solution of the Hamilton-Jacobi-Bellman equation associated with the stochastic optimization problem, and then give an optimal consumption policy in terms of its solution.     

Existence and Uniqueness of Fixed Points for Markov Operators and Markov Processes

This paper concerns a Markov operator T on a space L₁, and aMarkov process P which defines a Markov operator on a spaceM of finite signed measures. For T, the paper presents necessaryand sufficient conditions for:

a the existence of invariant probabilitydensities (IPDs)

b the existence of strictly positive IPDs,and

c the existence and uniqueness of IPDs.

Similar resultson invariant probability measures for P are presented. The basicapproach is to pose a fixed-point problem as the problem ofsolving a certain linear equation in a suitable Banach space,and then obtain necessary and sufficient conditions for thisequation to have a solution. 1991 Mathematics Subject Classification:60J05, 47B65, 47N30.     

Discontinuous travelling wave solutions for certain hyperbolic systems

In an earlier paper on a malignant cell invasion model (Marchantet al., SIAM J. Appl. Math, 60, 2000) we introduced a novelform of discontinuous travelling wave solution. These solutionscould be studied easily by combining behaviour within a phaseplane with the Rankine–Hugoniot shock conditions, whichdescribe properties (such as the ratio of the jump discontinuitiesto the speed of propagation) that solutions may possess. Theseresults were new for several reasons. The shock conditions relateto hyperbolic equations (which the model is) but were appliedin a travelling wave ordinary differential equation phase planeusing techniques that usually apply to parabolic reaction–diffusionsystems. In addition the solutions possess singular behaviournear several points in the phase plane but in spite of thisthere exists a robust and stable family of physically interestingsolutions. In this paper we discuss two previously studied models, oneof detonation theory and one of angiogenesis. We show that eachof these models also possesses a family of discontinuous travellingwave solutions which was not previously discovered. Of particularinterest is the solution which has a blunt interface at thefront of the invading profile. In all three models it is thissolution that is seen to stably evolve from physically relevantinitial data, and for physically relevant parameter values. This work confirms the robustness of these novel travellingwave solutions and their applicability to a wider range of mathematicalmodelling situations.     

Optimal Consumption in a Growth Model with the CES Production Function

Abstract

This article studies the optimization problem of maximizing the expected discounted present value of lifetime utility of consumption in the framework of one-sector neoclassical growth model with the Constant Elasticity of Substitution (CES) production function. We establish the existence of a classical solution of the Hamilton–Jacobi–Bellman (HJB) equation associated with this problem by the technique of viscosity solutions under the strict concavity of the utility function, and hence derive an optimal consumption from the optimality conditions in the HJB equation.     

The Helmholtz equation in a locally perturbed half-plane with passive boundary

^** Email: mduran{at}ing.puc.cl^*** Email: ignacio.muga{at}ucv.cl^**** Email: nedelec{at}cmapx.polytechnique.fr In this article, we study the existence and uniqueness of outgoingsolutions for the Helmholtz equation in locally perturbed half-planeswith passive boundary. We establish an explicit outgoing radiationcondition which is somewhat different from the usual Sommerfeld'sone due to the appearance of surface waves. We work with thehelp of Fourier analysis and a half-plane Green's function framework.This is an extended and detailed version of the previous articleDurán et al. (2005, The Helmholtz equation with impedancein a half-plane. C. R. Acad. Sci. Paris, Ser. I, 340, 483–488).     

Sliding mode control for a diffuse interface tumor growth model coupling a Cahn–Hilliard equation with a reaction–diffusion equation

We consider the sliding mode control (SMC) problem for a diffuse interface tumor growth model coupling a Cahn–Hilliard equation with a reaction–diffusion equation perturbed by a maximal monotone nonlinearity. We prove existence and regularity of strong solutions and, under further assumptions, a uniqueness result. Then, we show that the chosen SMC law forces the system to reach within finite time a sliding manifold that we chose in order that the tumor phase remains constant in time.     

On the solvability for the mixed-type Lyapunov equation

^** Email: xsf{at}math.pku.edu.cn^*** Email: mscheng{at}math.pku.edu.cn In this paper, the linear matrix equation X = AXB^* + BXA^* +Q is considered, which is called the mixed-type Lyapunov equation.Some necessary and sufficient conditions for the existence ofa unique solution are presented. Since a Hermitian positivesemidefinite solution is important from the application pointof view, some sufficient conditions for the existence of a Hermitianpositive semidefinite solution are derived.     

On a nonlocal Cahn-Hilliard/Navier-Stokes system with degenerate mobility and singular potential for incompressible fluids with different densities

We consider a diffuse interface model describing flow and phase separation of a binary isothermal mixture of (partially) immiscible viscous incompressible Newtonian fluids having different densities. The model is the nonlocal version of the one derived by Abels, Garcke and Grün and consists in a inhomogeneous Navier-Stokes type system coupled with a convective nonlocal Cahn-Hilliard equation. This model was already analyzed in a paper by the same author, for the case of singular potential and non-degenerate mobility. Here, we address the physically more relevant situation of degenerate mobility and we prove existence of global weak solutions satisfying an energy inequality. The proof relies on a regularization technique based on a careful approximation of the singular potential. Existence and regularity of the pressure field is also discussed. Moreover, in two dimensions and for slightly more regular solutions, we establish the validity of the energy identity. We point out that in none of the existing contributions dealing with the original (local) Abels, Garcke Grün model, an energy identity in two dimensions is derived (only existence of weak solutions has been proven so far).     

A Phase-Change Model with a Zone of Coexistence of Phases

A mathematical model for change of phase is presented, accountingfor the presence of regions in which liquid and solid coexist.The basic variables are temperature and solid fraction v. Westart from a relationship of the type =(v), supposed valid inthermodynamical equilibrium. Then for dynamical processes weintroduce a perturbation causing v to be less than its equilibriumvalue in any solidification process. This solid fraction deficiencyis governed by an ordinary differential equation containing_t, in the forcing term. The heat-balance equation is in turncoupled to the ordinary differential equation through the termv_t, ( is latent heat). Some existence and uniqueness resultsare proved and some monotonicity properties are described forpure melting or pure solidification processes.     

A nonlocal equation arising in the theory of wet combustion

A steady-state equation describing the self-heating of a materialdue to oxidation, hydrolysis, and the evaporation and condensationof water was introduced recently by Sisson et al. (1993, IMAJ. Appl. Math. 50, 285), and some preliminary existence anduniqueness results were derived. In this paper a numerical studyof the equation is undertaken. Critical behaviour and the disappearanceof criticality is observed.     

A hyperbolic model of two-phase flow: global solutions for large initial data

The paper deals with a simple nonlinear hyperbolic system of conservation laws modeling the flow of an inviscid fluid. The model is given by a standard isothermal p-system of the gasdynamics, for which phase transitions of the fluid are taken into consideration via a third homogeneous equation. We focus on the case of initial data consisting of two different phases separated by an interface. By means of an adapted version of the front tracking algorithm, we prove the global-in time existence of weak entropic solutions under suitable assumptions on the (possibly large) initial data.     

Well‐posedness and exact controllability of the mass balance equations for an extrusion process

In this paper, we study the well‐posedness and exact controllability of a physical model for an extrusion process in the isothermal case. The model expresses the mass balance in the extruder chamber and consists of a hyperbolic partial differential equation (PDE) and a nonlinear ordinary differential equation (ODE) whose dynamics describes the evolution of a moving interface. By suitable change of coordinates and fixed point arguments, we prove the existence, uniqueness, and regularity of the solution and finally, the exact controllability of the coupled system. Copyright © 2015 John Wiley & Sons, Ltd.     

Trapped acoustic modes

A constructive proof is given for the existence of trapped acousticmodes in the vicinity of a strip of length 2a parallel to, andmidway between, the bounding lines of a two-dimensional waveguide.The modes, which are shown to exist for sufficiently large a,satisfy Neumann conditions on the strip and the bounding linesand a Dirichlet condition on the midline outside the strip,and may be either symmetric or antisymmetric about a line throughthe centre of, and perpendicular to, the strip. When a is large,the equation determining the wavenumber of the modes reducesto that proposed by Evans and Linton (1991) using nonrigorousarguments