Solutions to a Scalar Discontinuous Conservation Law in a Limit Case of Phase Transitions

For a particular discontinuous flux function that can be associated to the limit case of a phase transition, we introduce an appropriate notion of entropy weak solution to the Cauchy problem for the corresponding conservation law. Then, for a class of initial data, that includes the Riemann data, we prove, by the vanishing viscosity method and with a suitable regularisation of the flux function, the existence of an entropy weak solution. This result can be easily extended to more general flux functions.     

A Quantitative Modulus of Continuity for the Two-Phase Stefan Problem

We derive the quantitative modulus of continuity $$\omega(r)=\left[ p+\ln \left( \frac{r_0}{r}\right)\right]^{-\alpha (n, p)},$$ which we conjecture to be optimal for solutions of the p-degenerate two-phase Stefan problem. Even in the classical case p = 2, this represents a twofold improvement with respect to the early 1980’s state-of-the-art results by Caffarelli– Evans (Arch Rational Mech Anal 81(3):199–220, 1983) and DiBenedetto (Ann Mat Pura Appl 103(4):131–176, 1982), in the sense that we discard one logarithm iteration and obtain an explicit value for the exponent α(n, p).     

Orthogonality Conditions and Asymptotic Stability in the Stefan Problem with Surface Tension

We prove nonlinear asymptotic stability of steady spheres in the two-phase Stefan problem with surface tension. Our method relies on the introduction of appropriate orthogonality conditions in conjunction with a high-order energy method.     

Phase Transitions in Gas- and Liquid-Saturated Porous Solids

Phase transitions in porous media consisting of a porous solid filled with liquid and gas constituents can occur, for example, due to freezing and drying processes. Although these phenomena are of certain relevance in soil mechanics and material sciences, a general thermo-dynamical theory is still awaited. Based on recent findings in the porous media theory, this paper is concerned with the development of thermodynamic restrictions for the constitutive relations of an elastic, incompressible porous solid, filled with an incompressible liquid and a compressible gas. The investigations show that mass conversions are related to the differences of the chemical potentials and energy transitions to the differences of temperatures. Thus, they confirm well-known results in classical thermodynamics of gases.     

Chirality Transitions in Frustrated Ferromagnetic Spin Chains: A Link with the Gradient Theory of Phase Transitions

We study chirality transitions in frustrated ferromagnetic spin chains, in view of a possible connection with the theory of Liquid Crystals. A variational approach to the study of these systems has been recently proposed by Cicalese and Solombrino, focusing close to the helimagnet/ferromagnet transition point corresponding to the critical value of the frustration parameter \(\alpha=4\). We reformulate this problem for any \(\alpha\geq0\) in the framework of surface energies in nonconvex discrete systems with nearest neighbours ferromagnetic and next-to-nearest neighbours antiferromagnetic interactions and we link it to the gradient theory of phase transitions, by showing a uniform equivalence by \(\varGamma \)-convergence on \([0,4]\) with Modica-Mortola type functionals.     

Collapsing and Explosion Waves in Phase Transitions with Metastability,Existence, Stability and Related Riemann Problems

Collapsing waves were observed numerically before and were used to explain the ring formations in dynamic flows involving phase transitions with metastability. In this paper, necessary and sufficient conditions for collapsing type of waves to exist are given. The conditions are that the wave speed of the collapsing wave is not less than a number and is supersonic on both sides of the wave. Existence and non-existence conditions for the explosion waves are also found. The stability of these waves are studied numerically. Although there are infinitely many collapsing (or explosion) waves for a fixed downstream state, the collapsing (or explosion) wave appeared in the solution of Riemann problem is numerically verified to be the one with the slowest speed. Although a Riemann problem in the zero viscosity limit may have two solutions, one with, the other without, a collapsing (or explosion) wave, from the vanishing viscosity point of view, the one with a collapsing (or explosion) wave is numerically verified to be admissible.     

Stefan Number Effect on the Transition from Stationary to Oscillatory Convection in a Solidifying Mushy Layer Subjected to Rotation

The current study investigates the Stefan number effect on the transition from stationary to oscillatory convection in a rotating mushy layer where the near eutectic approximation is applied. It is found that for rotating solidifying systems exhibiting a Stefan number of unit order (i.e., St=1), stationary convection is only possible up to Ta=3. Beyond Ta=3, for St=1, it is found that the oscillatory mode is the most dangerous mode of convection. A map showing the region of occurrence of the oscillatory mode is also presented for a range of Stefan numbers. The map reveals that the oscillatory mode is the most dangerous mode for intermediate values of Stefan number whilst the stationary mode is the most dangerous mode for very small and very large values of Stefan number. It is also demonstrated that increasing the rotation rate serves to render the oscillatory mode as the becoming the most dangerous mode of convection.     

Spherical and Cylindrical Low-Amplitude Waves in Polydisperse Fogs with Phase Transitions

A linear theory of propagation of spherical and cylindrical disturbances in polydisperse gas-vapor-drop mixtures is developed. Unsteady and non-equilibrium effects in the interphase mass, momentum, and energy exchange are taken into account. A general dispersion relation determining the propagation of plane, spherical, and cylindrical harmonic disturbances in polydisperse gas-vapor-drop systems is obtained. Using the fast Fourier transform, the propagation of pulse disturbances of different shapes in mixtures of air with water vapor and water drops is calculated. The effect of the geometry and interphase heat and mass transfer on the evolution of weak pulses in polydisperse air fogs is investigated.     

Determining Modes, Nodes and Volume Elements for Stationary Solutions of the Navier-Stokes Problem Past a Three-Dimensional Body

In this paper we show that every solution of the three-dimensional exterior Navier-Stokes boundary-value problem, corresponding to a given non-zero, constant velocity at infinity (flow past a body) and belonging to a very general functional class, , can be determined by a finite number of parameters. Our results extend the analogous classical results by Foiaş & Temam [6, 7], and by Jones & Titi [14] for the interior problem. This extension is by no means trivial, in that all fundamental tools used in the case of the interior problem – such as compactness of the Sobolev embeddings, Poincaré's inequality, and the special basis constituted by eigenfunctions of the Stokes operator – are no longer available for the exterior problem. An important consequence of our results is that any solution in is uniquely determined by the knowledge of the associated velocity field only ``near' the boundary. Just how ``near' it has to be depends only on the Reynolds number and on the body. Dedicated to John Heywood on the occasion of his 65th birthday     

Homogenization of the Process of Phase Transitions in Multidimensional Heterogeneous Periodic Media

The homogenization of the Stefan multidimensional problem is carried out in the case where the medium is a composite consisting of two different substances with an –periodic structure. The averaged problem is deduced by asymptotic methods. It is shown that its solution is the limit of solutions of –problems.     

频域滤波及相移技术在ESPI中的应用

本文针对电子散斑条纹颗粒性强这一特点,采用频域同态滤波技术,对ESPI 条纹进行了滤波处理,实验表明,该方法对改善 ESPI 条纹质量非常有效.另外,本文还推出了一种自行设计的新型相移器,它结构简单,操作简便,并且稳定性好,可适用于各种光力学干涉光路.文章最后使用双光束电子散斑法,应用以上技术,对一个带方孔板的平面应力模型的面内位移进行了定量分析.     

一维六方准晶中星形静态裂纹和运动裂纹的解析解

利用复变函数方法研究了一维六方准晶中星形静态裂纹和运动裂纹的反平面剪切问题,得到了星形裂纹尖端处应力强度因子和动应力强度因子的解析解.当裂纹条数给定时,由此可得到直线裂纹,Griffith裂纹,共点均匀分布三裂纹,对称十字形裂纹,米字型裂纹(对称八裂纹)静力学和动力学问题的解析解.当k=4时,用数值算例讨论了声子场-相位子场耦合系数和裂纹运动速度对动应力强度因子的影响.当速度趋于0时,运动裂纹的解可以退化为静态裂纹的解.     

Low‐Dimensional Lattice Groups for the Continuum Mechanics of Phase Transitions in Crystals

First I review and adapt some classical and recent formulations of problems relating to the classification of two‐ and three‐dimensional lattices. Then I use these methods to calculate lattice groups that are of interest in the continuum mechanics of phase transitions in crystals, and to construct invariant neighbourhoods that are useful in the same context. The paper is designed to provide a perspective of similar calculations which employ methods of traditional crystallography. (Accepted February 9, 1998)     

Some Problems of Linear Thermoelasticity in the Ginzburg–Landau Theory of Phase Transitions

The well-posedness of the problem of linear thermoelasticity in the Ginzburg–Landau theory of phase transitions is proved.     

Sharp Existence Results¶for the Stationary Navier-Stokes Problem¶in Three-Dimensional Exterior Domains

The stationary Navier-Stokes problem in a three-dimensional connected exterior domain is formulated in a new functional setting. For the case of a constant but nonzero velocity at infinity and of vanishing boundary fluxes, the problem involves a proper Fredholm operator of index 0. Topological degree arguments provide the existence of solutions and the Sard-Smale theorem yields their finiteness in number for generic external forces and boundary velocities. Accepted March 20, 2000?Published online September 18, 2000     

On the Stability of Damped Timoshenko Systems: Cattaneo Versus Fourier Law

We consider hyperbolic Timoshenko-type vibrating systems that are coupled to a heat equation modeling an expectedly dissipative effect through heat conduction. While exponential stability under the Fourier law of heat conduction holds, it turns out that the coupling via the Cattaneo law does not yield an exponentially stable system. This seems to be the first example that a removal of the paradox of infinite propagation speed inherent in Fourier’s law by changing to the Cattaneo law causes a loss of the exponential stability property. Actually, for systems with history, the Fourier law keeps the exponential stability known for the pure Timoshenko system without heat conduction, but introducing the Cattaneo coupling even destroys this property. This work was supported by the DFG-project “Hyperbolic Thermoelasticity” (RA 504/3-1).     

On a Stationary Problem of the Stokes Equation in an Infinite Layer in Sobolev and Besov Spaces

This paper is concerned with the stationary problem of the Stokes equation in an infinite layer and provides a condition on the external force sufficient for the existence of the solution. Since the Poiseuille flow is a solution to the homogeneous equation, the solution is not unique when p = ∞. It is also proved that, under some suitable conditions, solutions to the homogeneous equation are limited only to the Poiseuille flow.     

Dynamics of Phase Transitions and Hysteresis in a Viscoelastic Ericksen's Bar on an Elastic Foundation

This work is a follow-up on the study [32] of interface dynamics and hysteresis in materials undergoing solid-solid phase transitions. We consider the dynamics of a viscoelastic bar with a nonmonotone stress-strain relation and viscous stress linearly proportional to the strain rate. The bar is placed on an elastic foundation with stiffness β mimicking the interaction of phases in higher dimensions. Time-dependent displacement-controlled loading of the bar results in a tilted and serrated hysteresis loop, in qualitative agreement with some experimental observations in shape-memory alloys. The model exhibits three phase transition processes: phase nucleation, interface slip and phase annihilation. Between these dynamic processes the system gets stuck in local minimizers of the potential energy. As β increases from zero, a slip-dominated solution behavior transforms to the one where slip and annihilation events are preceded by a step-by-step nucleation process. We show that this transition is caused by an interplay between the slip-favoring inertia term and the nucleation-favoring elastic foundation terms. This revised version was published online in August 2006 with corrections to the Cover Date.