Stability of non-isothermal phase transitions in a steady van der Waals flow

This paper is concerned with the multidimensional stability of non-isothermal subsonic phase transitions in a steady supersonic flow of van der Waals type. For the sake of seeking physical admissible planar waves, the viscosity–capillarity criterion (Slemrod in Arch Ration Mech Anal 81(4):301–315, 1983) is chosen to be the admissible criterion. By showing the Lopatinski determinant being non-zero, we prove that subsonic phase transitions are uniformly stable in the sense of Majda (Mem Am Math Soc 41(275):1–95, 1983) under both one-dimensional and multidimensional perturbations.     

Stability and Existence of Multidimensional Subsonic Phase Transitions

The purpose of this paper is to prove the uniform stability of multidimensional subsonic phase transitions satisfying the viscosity-capillarity criterion in a van der Waals fluid, and further to establish the local existence of phase transition solutions.     

Existence of multidimensional phase transitions in a non-isothermal van der Waals fluid

In this paper, we prove the existence of multidimensional subsonic phase transitions in a non-isothermal van der Waals fluid. The argument is based on the result of the existence of travelling waves given in [S.-Y. Zhang, Existence of travelling waves in non-isothermal phase dynamics, J. Hyperbolic Differ. Equ. 4 (3) (2007) 391–400] and the result of multidimensional stability given in [S.-Y. Zhang, Stability of multidimensional subsonic phase transition in a non-isothermal van der Waals fluid, Preprint]. An iteration technique [A. Majda, The existence of multi-dimensional shock fronts, Mem. Amer. Math. Soc. 281 (1983) 1–93] is applied to achieve the result.     

Existence of multidimensional shock waves and phase boundaries

In this paper, a kind of Riemann problem for the Euler equations in a van der Waals fluid is considered. We constructed the weak solution in multidimensional space which contains one shock front and one subsonic phase boundary. We mainly follow the arguments of Majda's [A. Majda, The stability of multi-dimensional shock fronts, Mem. Amer. Math. Soc. 275 (1983) 1-95; A. Majda, The existence of multi-dimensional shock fronts, Mem. Amer. Math. Soc. 281 (1983) 1-93] and Métivier's [G. Métivier, Interaction de deux chocs pour un système de deux lois de conservation, en dimension deux d'espace, Trans. Amer. Math. Soc. 296 (1986) 431-479] work. The linear stability results are based on Majda's [A. Majda, The stability of multi-dimensional shock fronts, Mem. Amer. Math. Soc. 275 (1983) 1-95] work for the single shock front and Wang and Xin's [Y.-G. Wang, Z. Xin, Stability and existence of multidimensional subsonic phase transitions, Acta Math. Appl. Sin. 19 (2003) 529-558] work for the single phase boundary. The initial boundary value problem concerned in this paper is different from the boundary value problem for double shock fronts concerned in [G. Métivier, Interaction de deux chocs pour un système de deux lois de conservation, en dimension deux d'espace, Trans. Amer. Math. Soc. 296 (1986) 431-479], we slightly modified Métivier's frame work to establish the existence for the solution to the nonlinear problem.     

Reflection of Shock Fronts in a van der Waals Fluid

In this paper, the reflection phenomenon of a vapor shock front （both sides of the front are in the vapor phase） in a van der Waals fluid is considered. Both the 1-dimensional case and the multidimensional case are investigated. The authors find that under certain conditions, the reflected wave can be a single shock, or a single subsonic phase boundary, or one weak shock together with one subsonic phase boundary, which depends on the strength of the incident shock. This is different from the known result for the reflection of shock fronts in a gas dynamical system due to Chen in 1989.     

Admissible Riemann Solvers for Genuinely Nonlinear p-Systems of Mixed Type

We consider in this paper the Riemann problem for p-systems of mixed type that define two hyperbolic phases with a stress function satisfying general genuinely nonlinear hypotheses. We describe here all the global Riemann solvers that are continuous for the L¹ distance with respect to initial data while conserving the natural symmetry properties of the p-system and coinciding with the Lax solution when defined: these Riemann solvers can be described entirely by a kinetic function, used to select a manifold of subsonic phase transitions and a corresponding set of supersonic phase transitions.     

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.     

A variational inequality approach to generalized two-phase Stefan problem in several space variables

Summary The paper offers a study of a broad class of multidimensional two- phase problems of Stefan type by means of variational inequality techniques. The problems for quasilinear equations of alternatively parabolic or mixed parabolicelliptic type, mixed type nonlinear conditions at the fixed lateral boundary, involving free boundary conditions corresponding to phase transitions of both first (latent heat positive) and second kind (latent heat equal to zero) are taken into consideration. Results concerning existence of weak solutions, their uniqueness and stability are established.The preparation of the paper was partially carried out while the author's visiting Istituto di Analisi Numerica del C.N.R., Pavia, due to support of the C.N.R.     

Nonexistence of Slow Heteroclinic Travelling Waves for a Bistable Hamiltonian Lattice Model

The nonexistence of heteroclinic travelling waves in an atomistic model for martensitic phase transitions is the focus of this study. The elastic energy is assumed to be piecewise quadratic, with two wells representing two stable phases. We demonstrate that there is no travelling wave joining bounded strains in the different wells of this potential for a range of wave speeds significantly lower than the speed of sound. We achieve this using a profile-corrector method previously used to show existence of travelling waves for the same model at higher subsonic velocities.     

Subsonic irrotational flows in a two-dimensional finitely long curved nozzle

This is a continuation of our previous paper Du et al. (http://www.ims.cuhk.edu.hk/publications/reports/2012-06.pdf), where we have characterized a set of physical boundary conditions that ensures the existence and uniqueness of subsonic irrotational flow in a flat nozzle. In this paper, we will investigate the influence of the incoming flow angle and the geometry structure of the nozzle walls on subsonic flows in a finitely long curved nozzle. It turns out to be interesting that the incoming flow angle and the angle of inclination of nozzle walls play the same role as the end pressure for the stabilization of subsonic flows. In other words, the L ² and L ^∞ bounds of the derivative of these two quantities cannot be too large, similar as we have indicated in Du et al. (http://www.ims.cuhk.edu.hk/publications/reports/2012-06.pdf) for the end pressure. The curvatures of the nozzle walls will also play an important role in the stability of the subsonic flow.     

Persistence of undercompressive phase boundaries for isothermal Euler equations including configurational forces and surface tension

The persistence of subsonic phase boundaries in a multidimensional Van der Waals fluid is analyzed. The phase boundary is considered as a sharp free boundary that connects liquid and vapor bulk phase dynamics given by the isothermal Euler equations. The evolution of the boundary is driven by effects of configurational forces as well as surface tension. To analyze this problem, the equations and trace conditions are linearized such that one obtains a general hyperbolic initial boundary value problem with higher‐order boundary conditions. A global existence theorem for the linearized system with constant coefficients is shown. The proof relies on the normal mode analysis and a linear form in suitable spaces that is defined using an associated adjoint problem. Especially, the associated adjoint problem satisfies the uniform backward in time Kreiss–Lopatinski? condition. A new energy‐like estimate that also includes surface energy terms leads finally to the uniqueness and regularity for the found solutions of the problem in weighted spaces. Copyright © 2016 John Wiley & Sons, Ltd.     

Macroscopic Limits and Phase Transition in a System of Self-propelled Particles

We investigate systems of self-propelled particles with alignment interaction. Compared to previous work (Degond and Motsch, Math. Models Methods Appl. Sci. 18:1193–1215, 2008a; Frouvelle, Math. Models Methods Appl. Sci., 2012), the force acting on the particles is not normalized, and this modification gives rise to phase transitions from disordered states at low density to aligned states at high densities. This model is the space-inhomogeneous extension of (Frouvelle and Liu, Dynamics in a kinetic model of oriented particles with phase transition, 2012), in which the existence and stability of the equilibrium states were investigated. When the density is lower than a threshold value, the dynamics is described by a nonlinear diffusion equation. By contrast, when the density is larger than this threshold value, the dynamics is described by a similar hydrodynamic model for self-alignment interactions as derived in (Degond and Motsch, Math. Models Methods Appl. Sci. 18:1193–1215, 2008a; Frouvelle, Math. Models Methods Appl. Sci., 2012). However, the modified normalization of the force gives rise to different convection speeds, and the resulting model may lose its hyperbolicity in some regions of the state space.     

Subsonic solutions for compressible transonic potential flows

We establish an existence theorem for transonic isentropic potential flows where the subsonic region is bounded by the sonic line and thus the governing equation may become degenerate on the boundary partly or entirely. It has been conjectured by experiments and numerical studies that the self-similar multidimensional flow changes its type, namely, hyperbolic far from the origin (supersonic region) and elliptic near the origin (subsonic region). Furthermore, the potential equation has a different nonlinearity compared to other transonic problems such as the unsteady transonic small disturbance equation, the nonlinear wave equation, and the pressure gradient equation. Namely, the coefficients of the potential equation depend on the gradients while others are independent of the gradients. We provide techniques to handle the gradients, establish interior and boundary gradient estimates for the potential flow in a convex region, and answer the conjecture, that is, the flow is strictly elliptic and the region is subsonic.     

Integrability Criterion of Geodesical Equivalence. Hierarchies

We prove that the Riemannian metrics g and (given in `general position") are geodesically equivalent if and only if some canonically given functions are pairwise commuting integrals of the geodesic flow of the metric g. This theorem is a multidimensional generalization of the well-known Dini theorem proved in the two-dimensional case. A hierarchy of completely integrable Riemannian metrics is assigned to any pair of geodesically equivalent Riemannian metrics. We show that the metrics of the standard ellipsoid and the Poisson sphere lie in such an hierarchy.     

Consistency of iterative operator‐splitting methods: Theory and applications

In this article, we describe a different operator‐splitting method for decoupling complex equations with multidimensional and multiphysical processes for applications for porous media and phase‐transitions. We introduce different operator‐splitting methods with respect to their usability and applicability in computer codes. The error‐analysis for the iterative operator‐splitting methods is discussed. Numerical examples are presented. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

The zero-electron-mass limit for a stationary nonisentropic hydrodynamic semiconductor model

In this paper, we study a general multidimensional nonisentropic hydrodynamical model for semiconductors. The model is self-consistent in the sense that the electric field, which forms a forcing term in the momentum equation, is determined by the coupled Poisson equation. For steady state, subsonic and potential flows, we discuss the zero-electron-mass limit of system by using the method of asymptotic expansions. We show the existence and uniqueness of profiles, and justify the asymptotic expansions up to any order.     

Degenerate random environments

We consider connectivity properties of certain i.i.d. random environments on , where at each location some steps may not be available. Site percolation and oriented percolation are examples of such environments. In these models, one of the quantities most often studied is the (random) set of vertices that can be reached from the origin by following a connected path. More generally, for the models we consider, multiple different types of connectivity are of interest, including: the set of vertices that can be reached from the origin; the set of vertices from which the origin can be reached; the intersection of the two. As with percolation models, many of the models we consider admit, or are expected to admit phase transitions. Among the main results of the paper is a proof of the existence of phase transitions for some two‐dimensional models that are non‐monotone in their underlying parameter, and an improved bound on the critical value for oriented site percolation on the triangular lattice. The connectivity of the random directed graphs provides a foundation for understanding the asymptotic properties of random walks in these random environments, which we study in a second paper. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 45, 111–137, 2014     

Singularly Perturbed Parabolic Problems with Multidimensional Boundary Layers

The first boundary value problem for a multidimensional parabolic differential equation with a small parameter ε multiplying all derivatives is studied. A complete (i.e., of any order with respect to the parameter) regularized asymptotics of the solution is constructed, which contains a multidimensional boundary layer function that is bounded for x = (x¹, x₂) = 0 and tends to zero as ε → +0 for x ≠ 0. In addition, it contains corner boundary layer functions described by the product of a boundary layer function of the exponential type by a multidimensional parabolic boundary layer function.     

Shock-Induced Phase Transitions in Systems of Hard Spheres with Attractive Interactions

Shock-induced phase transitions are studied by adopting the recently-developed theoretical framework, which is applicable for shock waves in three phases (gas, liquid, and solid), based on the system of hard spheres with mutual attractive interactions. The Rankine-Hugoniot conditions derived from the system of Euler equations with caloric and thermal equations of state are studied, and the admissibility (stability) of a shock wave is analyzed. Two typical scenarios of the shock-induced phase transitions from gas phase to solid phase are found. A scenario of shock-induced phase transitions involving three phases simultaneously near the triple point is also found.     

A highly accurate explicit symplectic ERKN method for multi-frequency and multidimensional oscillatory Hamiltonian systems

The numerical integration of Hamiltonian systems with multi-frequency and multidimensional oscillatory solutions is encountered frequently in many fields of the applied sciences. In this paper, we firstly summarize the extended Runge–Kutta–Nyström (ERKN) methods proposed by Wu et al. (Comput. Phys. Comm. 181:1873–1887, (2010)) for multi-frequency and multidimensional oscillatory systems and restate the order conditions and symplecticity conditions for the explicit ERKN methods. Secondly, we devote to exploring the explicit symplectic multi-frequency and multidimensional ERKN methods of order five based on the symplecticity conditions and order conditions. A five-stage explicit symplectic multi-frequency and multidimensional ERKN method of order five with some small residuals is proposed and its stability and phase properties are analyzed. It is shown that the new method is dispersive of order six. Numerical experiments are carried out and the numerical results demonstrate that the new method is much more efficient than the methods appeared in the scientific literature.