Mathematical analysis of a simplified Hookean dumbbells model arising from viscoelastic flows

A stochastic model corresponding to a simplified Hookean dumbbells viscoelastic fluid is considered, the convective terms being disregarded. Existence on a fixed time interval is proved provided the data are small enough, using the implicit function theorem and a maximum regularity property for a three fields Stokes problem.     

Numerical analysis of a second order algorithm for simplified magnetohydrodynamic flows

In this paper, we construct a second order algorithm based on the spectral deferred correction method for the time-dependent magnetohydrodynamics flows at a low magnetic Reynolds number. We present a complete theoretical analysis to prove that this algorithm is unconditionally stable, consistent and second order accuracy. Finally, two numerical examples are given to illustrate the convergence and effectiveness of our algorithm.     

A simplified model for unsteady pressure driven flows in circular microchannels of variable cross-section

In this paper, we present a fast and accurate model for unsteady pressure-driven flows in circular microchannels of variable cross-section. The model is developed for channels of small diameter to length ratio, but allows for large variations in the channel’s diameter along the axis. A key feature of the model is that it puts no restriction on the time dependence of the forcing, in terms of shape and frequency. The only condition on the forcing is such that the advective component of the inertia term is small. This is a major departure from many previous expositions which assume harmonic forcing. The model is based on an extended and unsteady lubrication approximation in the aspect ratio of the channel. The resulting equations for each order are solved analytically using a finite Hankel transform, except for the implicit pressure profile, which is solved numerically with a recursive time scheme. Compared to classical CFD simulations, the reduced order semi-analytic method is two orders of magnitude faster, owing in part to the fact that the number of modes required for the convergence of these expressions is not too large. The numerical simulations reveal that the model is accurate for a large class of channels and a fairly wide range of Reynolds numbers. This, combined with the fact that it imposes no conditions on the shape and frequency of the unsteady forcing, renders the model a valuable tool for rapidly simulating large fluidic circuits (as in lab-on-a-chip, μTAS and the human body), thereby allowing significant reduction in the design parameters space.     

Global well-posedness and relaxation limits of a model for radiating gas

We study the initial value problem for a hyperbolic-elliptic coupled system with arbitrary large discontinuous initial data. We prove existence and uniqueness for that model by means of L¹-contraction and comparison properties. Moreover, after suitable scalings, we study both the hyperbolic-parabolic and the hyperbolic-hyperbolic relaxation limits for that system.     

Decay of a model system of radiating gas

This paper concerns the Cauchy problem of a model system to the radiating gas in . Large time behaviors of classical solutions to the Cauchy problem are studied without needing the smallness assumption of initial perturbation in L¹‐norm. We obtain the optimal H^N‐norm time‐decay rates of the solutions in with 1 ≤ n ≤ 4 by applying the Fourier splitting method introduced by Schonbek (1980) with a slight modification and an energy method. Furthermore, when initial perturbation is bounded in L^p‐norm (p ∈ (1,2]), optimal L^p‐ L² decay estimates of the derivatives of solutions are shown. Copyright © 2013 John Wiley & Sons, Ltd.     

On the Cauchy problem for macroscopic model of pedestrian flows

In this paper, we discuss the limit behavior of hyperbolic systems of conservation laws with stiff relaxation terms to the local systems as the relaxation time tends to zero. The prototype is crowd models derived from crowd dynamics according to macroscopic scaling when the flow of crowds is supposed to satisfy the paradigms of continuum mechanics. Under an appropriate structural stability condition, the asymptotic expansion is obtained when one assumes the existence of a smooth solution to the equilibrium system. In this case, the local existence of a classical solution is also shown.     

A newton-penalty method for a simplified liquid crystal model

In this paper we are concerned with the computation of a liquid crystal model defined by a simplified Oseen-Frank energy functional and a (sphere) nonlinear constraint. A particular case of this model defines the well known harmonic maps. We design a new iterative method for solving such a minimization problem with the nonlinear constraint. The main ideas are to linearize the nonlinear constraint by Newton’s method and to define a suitable penalty functional associated with the original minimization problem. It is shown that the solution sequence of the new minimization problems with the linear constraints converges to the desired solutions provided that the penalty parameters are chosen by a suitable rule. Numerical results confirm the efficiency of the new method.     

Convergence to the rarefaction wave for a model of radiating gas in one-dimension

In this paper, a sequence of solutions to the one-dimensional motion of a radiating gas are constructed. Furthermore, when the absorption coefficient α tends to ∞, the above solutions converge to the rarefaction wave, which is an elementary wave pattern of gas dynamics, with a convergence rate \(\alpha ^{ - \tfrac{1}{3}} \left| {\ln \alpha } \right|^2\).     

Convergence rates toward the travelling waves for a model system of the radiating gas

The present paper is concerned with an asymptotics of a solution to the model system of radiating gas. The previous researches have shown that the solution converges to a travelling wave with a rate (1 + t)^?1/4 as time t tends to infinity provided that an initial data is given by a small perturbation from the travelling wave in the suitable Sobolev space and the perturbation is integrable. In this paper, we make more elaborate analysis under suitable assumptions on initial data in order to obtain shaper convergence rates than previous researches. The first result is that if the initial data decays at the spatial asymptotic point with a certain algebraic rate, then this rate reflects the time asymptotic convergence rate. Precisely, this convergence rate is completely same as the spatial convergence rate of the initial perturbation. The second result is that if the initial data is given by the Riemann data, an admissible weak solution, which has a discontinuity, converges to the travelling wave exponentially fast. Both of two results are proved by obtaining decay estimates in time through energy methods with suitably chosen weight functions. Copyright © 2006 John Wiley & Sons, Ltd.     

Justification of a simplified model for shells in nonlinear elasticity

We introduce a simplified model for the minimization of the elastic energy in thin shells. The thickness of the shell remains a parameter in this new model.     

On a two-fluid model of two-phase compressible flows and its numerical approximation

We consider a two-fluid model of two-phase compressible flows. First, we derive several forms of the model and of the equations of state. The governing equations in all the forms contain source terms representing the exchanges of momentum and energy between the two phases. These source terms cause unstability for standard numerical schemes. Using the above forms of equations of state, we construct a stable numerical approximation for this two-fluid model. That only the source terms cause the oscillations suggests us to minimize the effects of source terms by reducing their amount. By an algebraic operator, we transform the system to a new one which contains only one source term. Then, we discretize the source term by making use of stationary solutions. We also present many numerical tests to show that while standard numerical schemes give oscillations, our scheme is stable and numerically convergent.     

Numerical approximation for a Baer-Nunziato model of two-phase flows

We present a well-balanced numerical scheme for approximating the solution of the Baer-Nunziato model of two-phase flows by balancing the source terms and discretizing the compaction dynamics equation. First, the system is transformed into a new one of three subsystems: the first subsystem consists of the balance laws in the gas phase, the second subsystem consists of the conservation law of the mass in the solid phase and the conservation law of the momentum of the mixture, and the compaction dynamic equation is considered as the third subsystem. In the first subsystem, stationary waves are used to build up a well-balanced scheme which can capture equilibrium states. The second subsystem is of conservative form and thus can be numerically treated in a standard way. For the third subsystem, the fact that the solid velocity is constant across the solid contact suggests us to compose the technique of the Engquist-Osher scheme. We show that our scheme is capable of capturing exactly equilibrium states. Moreover, numerical tests show the convergence of approximate solutions to the exact solution.     

On the multiplicity of solutions of a simplified tumor growth model with a free boundary

We consider a mathematical model of tumor growth taking into account the spatial structure and include a different phases because of the lack of the nutriments. This model is expressed as a PDE on a spherical domain describing the tumor region. The nonlinearity of this PDE is discontinuous, and our problem can be regarded as a free boundary problem. In fact, the boundary of the tumor and another part in the interior are unknown. We obtain a multiplicity result of solutions together with some properties of the associated free boundaries.     

A simplified quantum energy-transport model for semiconductors

The existence of global-in-time weak solutions to a quantum energy-transport model for semiconductors is proved. The equations are formally derived from the quantum hydrodynamic model in the large-time and small-velocity regime. They consist of a nonlinear parabolic fourth-order equation for the electron density, including temperature gradients; an elliptic nonlinear heat equation for the electron temperature; and the Poisson equation for the electric potential. The equations are solved in a bounded domain with periodic boundary conditions. The existence proof is based on an entropy-type estimate, exponential variable transformations, and a fixed-point argument. Furthermore, we discretize the equations by central finite differences and present some numerical simulations of a one-dimensional ballistic diode.     

Analysis of a simplified model of drop combustion

Summary A simplified model of drop combustion is studied; its main mathematical difficulty is a nonlinearity which does not satisfy a Lipschitz condition. The existence of a solution is proved and a uniqueness result is given. Moreover, an algorithm is stated and its convergence is proved.Supported by the Fonds National Suisse de la Recherche Scientifique     

Existence analysis for a simplified transient energy‐transport model for semiconductors

A simplified transient energy‐transport system for semiconductors subject to mixed Dirichlet–Neumann boundary conditions is analyzed. The model is formally derived from the non‐isothermal hydrodynamic equations in a particular vanishing momentum relaxation limit. It consists of a drift‐diffusion‐type equation for the electron density, involving temperature gradients, a nonlinear heat equation for the electron temperature, and the Poisson equation for the electric potential. The global‐in‐time existence of bounded weak solutions is proved. The proof is based on the Stampacchia truncation method and a careful use of the temperature equation. Under some regularity assumptions on the gradients of the variables, the uniqueness of solutions is shown. Finally, numerical simulations for a ballistic diode in one space dimension illustrate the behavior of the solutions. Copyright © 2012 John Wiley & Sons, Ltd.     

On a class of special flows

Summary A special flow, in which the basic automorphism is a Bernoulli automorphism and the ceiling function depends only on the present position of the Bernoulli sequence and is not lattice distributed, is a K-flow.This paper represents the result, in a revised form, which was presented at the Symposium on Ergodic Theory held at Mathematisches Forschungsinstitut Oberwolfach from August 4 to 10, 1968.