Existence of a universal attractor for a fully hyperbolic phase-field system

Here we study a nonlinear hyperbolic integrodifferential system which was proposed by H.G. Rotstein et al. to describe certain peculiar phase transition phenomena. This system governs the evolution of the (relative) temperature and the order parameter (or phase-field) . We first consider an initial and boundary value problem associated with the system and we frame it in a history space setting. This is done by introducing two additional variables accounting for the histories of and . Then we show that the reformulated problem generates a dissipative dynamical system in a suitable infinite-dimensional phase space. Finally, we prove the existence of a universal attractor.     

The ignition problem for a scalar nonconvex combustion model

The ignition problem for the scalar Chapman-Jouguet combustion model without convexity is considered. Under the pointwise and global entropy conditions, we constructively obtain the existence and uniqueness of the solution and show that the unburnt state is stable (unstable) when the binding energy is small (large), which is the desired property for a combustion model. The transitions between deflagration and detonation are shown, which do not appear in the convex case.     

Robust exponential attractors for a phase-field system with memory

H.G. Rotstein et al. proposed a nonconserved phase-field system characterized by the presence of memory terms both in the heat conduction and in the order parameter dynamics. These hereditary effects are represented by time convolution integrals whose relaxation kernels k and h are nonnegative, smooth and decreasing. Rescaling k and h properly, we obtain a system of coupled partial integrodifferential equations depending on two relaxation times ɛ and σ. When ɛ and σ tend to 0, the formal limiting system is the well-known nonconserved phase-field model proposed by G. Caginalp. Assuming the exponential decay of the relaxation kernels, the rescaled system, endowed with homogeneous Neumann boundary conditions, generates a dissipative strongly continuous semigroup S_{ɛ, σ}(t) on a suitable phase space, which accounts for the past histories of the temperature as well as of the order parameter. Our main result consists in proving the existence of a family of exponential attractors for S_{ɛ, σ}(t), with ɛ, σ ∈ [0, 1], whose symmetric Hausdorff distance from tends to 0 in an explicitly controlled way.     

Weak solutions for the fully hyperbolic phase-field system of conserved type

We consider a conserved phase-field system coupling two nonlinear hyperbolic integro-differential equations. The model results from the assumption that the material undergoing phase transition exhibits some thermal memory effects (cf. [15]) and that the response of the order parameter to the variation of the free-energy functional is delayed (cf. [10, 23]). We prove the existence of the solution to the corresponding initial-boundary value problem associated with the resulting PDE system and a (conditioned) continuous dependence estimate of the solution with respect to the data of the problem. This work is partially supported by the Italian Ministero dell’Istruzione, dell’Università e della Ricerca, PRIN no. 2004011204, Project Analisi Matematica nei Problemi Inversi     

Reconstruction of two time independent coefficients in an inverse problem for a phase field system

In this paper we present stability results concerning the inverse problem of determining two time independent coefficients for a phase field system in a bounded domain Ω⊂Rⁿ for the dimension n≤3 with a single observation on a subdomain ω?Ω and the Sobolev norm of certain partial derivatives of the solutions at a fixed positive time θ∈(0,T) over the whole spatial domain. The proof of these results relies on an appropriate Carleman estimate for the phase field system.     

Existence and continuous dependence of large solutions for the magnetohydrodynamic equations

Global solutions of the nonlinear magnetohydrodynamic (MHD) equations with general large initial data are investigated. First the existence and uniqueness of global solutions are established with large initial data in H ¹. It is shown that neither shock waves nor vacuum and concentration are developed in a finite time, although there is a complex interaction between the hydrodynamic and magnetodynamic effects. Then the continuous dependence of solutions upon the initial data is proved. The equivalence between the well-posedness problems of the system in Euler and Lagrangian coordinates is also showed.     

Existence and uniqueness for a nonlinear discrete hyperbolic system

The existence, uniqueness and asymptotic behaviour of the solutions to a nonlinear discrete hyperbolic system subject to some extreme conditions and initial data are investigated in a real Hilbert space.     

Transonic shock wave in an infinite nozzle asymptotically converging to a cylinder

We construct a single transonic shock wave pattern in an infinite nozzle asymptotically converging to a cylinder, which is close to a uniform transonic shock wave. In other words, suppose there is a uniform transonic shock wave in an infinite cylinder nozzle which can be constructed easily, if we perturbed the supersonic incoming flow and the infinite nozzle a little bit, we can obtain a transonic wave near the uniform one. As a consequence, we can show that the uniform transonic wave is stable with respect to the perturbation of the incoming flow and nozzle wall. Based on the theory of [G.Q. Chen, M. Feldman, Existence and stability of multi-dimensional transonic flows through an infinite nozzle of arbitrary cross-sections, Arch. Ration. Mech. Anal. 184 (2007) 185-242], the crucial parts of this paper are to derive the uniform Schauder estimates of the linear elliptic equation for the infinite nozzle asymptotically converging to a cylinder.     

Uniqueness of transonic shock solutions in a duct for steady potential flow

We study the uniqueness of solutions with a transonic shock in a duct in a class of transonic shock solutions, which are not necessarily small perturbations of the background solution, for steady potential flow. We prove that, for given uniform supersonic upstream flow in a straight duct, there exists a unique uniform pressure at the exit of the duct such that a transonic shock solution exists in the duct, which is unique modulo translation. For any other given uniform pressure at the exit, there exists no transonic shock solution in the duct. This is equivalent to establishing a uniqueness theorem for a free boundary problem of a partial differential equation of second order in a bounded or unbounded duct. The proof is based on the maximum/comparison principle and a judicious choice of special transonic shock solutions as a comparison solution.     

Existence results for some Kirchhoff-Carrier problems

Analyzing the viscoelastic problem for small vibrations of elastic strings, Kirchhoff and Carrier proposed two different models of nonlinear partial differential equations. By combining these two models, we deal here with some nonlocal hyperbolic problems that cover a large class of Kirchhoff and Carrier type problems. The existence of local solutions of degenerate problems as well as local and nonlocal solutions of nondegenerate problems is established. The proofs are based on the combination of the Schauder fixed point theorem with some asymptotic method.     

Universal Attractor for a Penrose-Fife System with Special Heat Flux Law

A Penrose-Fife system for non isothermal phase transitions with non conserved order parameter is introduced. A linear growth of the latent heat density with respect to the phase field is allowed. Continous dependence on data and the existence of the universal attractor for the associated nonlinear semigroup are shown. These properties hold with respect to a strong metric accounting for the nonlinear and even singular terms characterizing the system. The present analysis extends a former result by the same authors, holding in the case of a constant latent heat.     

Global solutions of nonconcave hyperbolic conservation laws with relaxation arising from traffic flow

We establish global solutions of nonconcave hyperbolic equations with relaxation arising from traffic flow. One of the characteristic fields of the system is neither linearly degenerate nor genuinely nonlinear. Furthermore, there is no dissipative mechanism in the relaxation system. Characteristics travel no faster than traffic. The global existence and uniqueness of the solution to the Cauchy problem are established by means of a finite difference approximation. To deal with the nonconcavity, we use a modified argument of Oleinik (Amer. Math. Soc. Translations 26 (1963) 95). It is also shown that the zero relaxation limit of the solutions exists and is the unique entropy solution of the equilibrium equation.     

Dispersion estimates for the two-dimensional Vlasov-Yukawa system with small data

We present dispersion estimates for the two-dimensional Vlasov-Yukawa system with small data. When the initial data are sufficiently regular and small, we show that the local mass density and the Yukawa force field decay to zero algebraically fast in time. These dispersion estimates are not known for the two-dimensional Vlasov-Poisson system. For the dispersion estimates, we effectively use the short-range character of the Yukawa potential and the optimal gradient estimates introduced by Hwang, Rendall and Velázquez for the three-dimensional Vlasov-Poisson system.     

Nanofluid flow and heat transfer in a Brinkman porous channel with variable porosity

The problem of forced convection in a channel filled with a nanofluidsaturated porous medium is investigated, numerically. A finite difference Computational Fluid Dynamics (CFD) model with structured uniform grid system is employed to solve the momentum and energy equations. In modeling flow in the channel, the effects of flow inertia, variable porosity and Brinkman friction are taken into account. Studies are carried out for different nanoparticles with different volume fractions in the range 0%-4% and different nanoparticle diameters. Comparison made between our numerical and semi analytical Differential Transform Method (DTM) results with those in previous published research is found to be appropriate. Results show that increasing either nanoparticls volume fraction or pressure gradient parameter improves heat transfer. Further, for large quantities of nanoparticle concentration and pressure gradient, the channeling phenomenon is intensified.     

Sharp decay estimates for hyperbolic balance laws

We consider strictly hyperbolic and genuinely nonlinear systems of hyperbolic balance laws in one-space dimension. Sharp decay estimates are derived for the positive waves in an entropy weak solution. The result is obtained by introducing a partial ordering within the family of positive Radon measures, using symmetric rearrangements and a comparison with a solution of Burgers's equation with impulsive sources as well as lower semicontinuity properties of continuous Glimm-type functionals.     

Global solution to a phase transition model based on a microforce balance

Our aim in this article is to prove the existence and the uniqueness of a global solution to a nonisothermal Ginzburg-Landau (Allen-Cahn) system. This system is obtained by considering, in addition to the fundamental laws of thermodynamics, a balance law for internal microforces proposed by M. Gurtin.     

Long-time behavior of solutions to a nonlinear hyperbolic relaxation system

We study the large time behavior of solutions of a one-dimensional hyperbolic relaxation system that may be written as a nonlinear damped wave equation. First, we prove the global existence of a unique solution and their decay properties for sufficiently small initial data. We also show that for some large initial data, solutions blow-up in finite time. For quadratic nonlinearities, we prove that the large time behavior of solutions is given by the fundamental solution of the viscous Burgers equation. In some other cases, the convection term is too weak and the large time behavior is given by the linear heat kernel.     

Global solution for a one-dimensional model problem in thermally radiative magnetohydrodynamics

The dynamics of gaseous stars is often described by magnetic fields coupled to self-gravitation and radiation effects. In this paper we consider an initial-boundary value problem for nonlinear planar magnetohydrodynamics (MHD) in the case that the effect of self-gravitation as well as the influence of radiation on the dynamics at high temperature regimes are taken into account. Based on the fundamental local existence results and global-in-time a priori estimates, we establish the global existence of a unique classical solution with large initial data to the initial-boundary value problem under quite general assumptions on the heat conductivity.     

On the incompressible limits for the full magnetohydrodynamics flows

In this article the incompressible limits of weak solutions to the governing equations for magnetohydrodynamics flows on both bounded and unbounded domains are established. The governing equations for magnetohydrodynamic flows are expressed by the full Navier-Stokes system for compressible fluids enhanced by forces due to the presence of the magnetic field as well as the gravity and with an additional equation which describes the evolution of the magnetic field. The scaled analogues of the governing equations for magnetohydrodynamic flows involve the Mach number, Froude number and Alfven number. In the case of bounded domains the establishment of the singular limit relies on a detail analysis of the eigenvalues of the acoustic operator, whereas the case of unbounded domains is being treated by their suitable approximation by a family of bounded domains and the derivation of uniform bounds.