Divergence‐Measure Fields and Hyperbolic Conservation Laws

. We analyze a class of vector fields, called divergence‐measure fields. We establish the Gauss‐Green formula, the normal traces over subsets of Lipschitz boundaries, and the product rule for this class of fields. Then we apply this theory to analyze entropy solutions of initial‐boundary‐value problems for hyperbolic conservation laws and to study the ways in which the solutions assume their initial and boundary data. The examples of conservation laws include multidimensional scalar equations, the system of nonlinear elasticity, and a class of systems with affine characteristic hypersurfaces. The analysis in also extends to . (Accepted July 16, 1998)     

On the Initial‐Boundary‐Value Problem for the Linearized Equations of Magnetohydrodynamics

We study the initial‐boundary‐value problem for the linearized equations of ideal magnetohydrodynamics defined on a bounded domain whose boundary is a regular magnetic surface. We show that it is well‐posed in and that the loss of regularity of solutions always occurs for some initial data in (Accepted October 23, 1997)     

Constitutive Relations for Monatomic Gases¶Based on a¶Generalized Rational Approximation¶to the Sum of the Chapman-Enskog Expansion

. We study the asymptotic behavior as time goes to infinity of solutions to the initial‐boundary‐value problem on the half space for a one‐dimensional model system for the isentropic flow of a compressible viscous gas, the so‐called p‐system with viscosity. As boundary conditions, we prescribe the constant state at infinity and require that the velocity be zero at the boundary . When the velocity at infinity is negative and satisfies a condition on the magnitude, we prove that if the initial data are suitably close to those for the corresponding outgoing viscous shock profile, which is suitably far from the boundary, then a unique solution exists globally in time and tends toward the properly shifted viscous shock profile as the time goes to infinity. The proof is given by an elementary energy method. (Accepted March 2, 1998)     

Materials with Internal Variables and Relaxation to Conservation Laws

.The theory of materials with internal state variables of Coleman & Gurtin [CG] provides a natural framework to investigate the structure of relaxation approximations of conservation laws from the viewpoint of continuum thermomechanics. After reviewing the requirements imposed on constitutive theories by the principle of consistency with the Clausius‐Duhem inequality, we pursue two specific theories pertaining to stress relaxation and relaxation of internal energy. They each lead to a relaxation framework for the theory of thermoelastic non‐conductors of heat, equipped with globally defined “entropy” functions for the associated relaxation process. Next, we consider a semilinear model problem of stress relaxation. We discuss uniform stability and compactness for solutions of the relaxation system in the zero‐relaxation limit, and establish convergence to the system of isothermal elastodynamics by using compensated compactness. Finally, we prove a strong dissipation estimate for the relaxation approximations proposed in Jin & Xin [JX] when the limit system is equipped with a strictly convex entropy. (Accepted June 17, 1998)     

Quasi‐One‐Dimensional Riemann Problems and Their Role in Self‐Similar Two‐Dimensional Problems

We study two‐dimensional Riemann problems with piecewise constant data. We identify a class of two‐dimensional systems, including many standard equations of compressible flow, which are simplified by a transformation to similarity variables. For equations in this class, a two‐dimensional Riemann problem with sectorially constant data becomes a boundary‐value problem in the finite plane. For data leading to shock interactions, this problem separates into two parts: a quasi‐one‐dimensional problem in supersonic regions, and an equation of mixed type in subsonic regions. We prove a theorem on local existence of solutions of quasi‐one‐dimensional Riemann problems. For 2 × 2 systems, we generalize a theorem of Courant & Friedrichs, that any hyperbolic state adjacent to a constant state must be a simple wave. In the subsonic regions, where the governing equation is of mixed hyperbolic‐elliptic type, we show that the elliptic part is degenerate at the boundary, with a nonlinear variant of a degeneracy first described by Keldysh. (Accepted December 4, 1997)     

Initial Layers and Uniqueness of¶Weak Entropy Solutions to¶Hyperbolic Conservation Laws

We consider initial layers and uniqueness of weak entropy solutions to hyperbolic conservation laws through the scalar case. The entropy solutions we address assume their initial data only in the sense of weak-star in L ^∞ as t→0₊ and satisfy the entropy inequality in the sense of distributions for t>0. We prove that, if the flux function has weakly genuine nonlinearity, then the entropy solutions are always unique and the initial layers do not appear. We also discuss applications to the zero relaxation limit for hyperbolic systems of conservation laws with relaxation. Accepted: October 26, 1999     

On an extension of the d’Alembert solution to initial–boundary value problems in multi-layered,multi-material domains

This work introduces a method for the exact solution of initial–boundary value problems for linear, one-dimensional conservation laws in multi-layered, multi-material domains. The method is based on the geometry of the solutions of such conservation laws and represents an extension of the d’Alembert solution to initial–boundary value problems in multi-layered, multi-material domains.     

Young‐Measure Solutions for a Viscoelastically Damped Wave Equation with Nonmonotone Stress‐Strain Relation

We study the viscoelastically damped wave equation with a nonmonotone stress‐strain relation σ. This system describes the dynamics of phase transitions, which is closely related to the creation of microstructures. In order to analyze the dynamic behavior of microstructures we consider highly oscillatory initial states. Two questions are addressed in this work: How do oscillations propagate in space and time? What can be said about the long‐time behavior? An appropriate tool to deal with oscillations are Young measures. They describe the local distribution or one‐point statistics of a sequence of fast fluctuating functions. We demonstrate that highly oscillatory initial states generate in a unique fashion an evolution in the space of Young measures and we derive the determining equations. Further on we prove a generalized dissipation identity for Young‐measure solutions. As a consequence, it is shown that every low‐energy solution converges to a Young‐measure equilibrium as t→∞. This is a generalization of G. Friesecke's & J. B. McLeod's [FM96] convergence result for classical solutions to the case of Young‐measure solutions. (Accepted November 12, 1997)     

Propagating phase boundaries: Formulation of the problem and existence via the Glimm method

This paper treats the hyperbolic-elliptic system of two conservation laws which describes the dynamics of an elastic material having a non-monotone strain-stress function. FollowingAbeyaratne &Knowles, we propose a notion of admissible weak solution for this system in the class of functions of bounded variation. The formulation includes an entropy inequality, a kinetic relation (imposed along any subsonic phase boundary) and an initiation criterion (for the appearance of new phase boundaries). We prove theL ¹-continuous dependence of the solution to the Riemann problem. Our main result yields the existence and the stability of propagating phase boundaries. The proofs are based onGlimm's scheme and in particular on the techniques ofGlimm andLax. In order to deal with the kinetic relation, we prove a result of pointwise convergence of the phase boundary.     

Derivation of the LSW‐Theory for Ostwald Ripening by Homogenization Methods

. The coarsening of a solid phase in an undercooled liquid is described by a Stefan problem with surface tension. The solid phase is characterized by a large number of balls with small volume fraction and small capacity. We solve the quasi‐static equation as well as the parabolic problem and construct approximate solutions by means of comparison principles. The framework of Young measures is used to pass to the limit of infinitely many particles. We obtain that the capacity is the crucial quantity to get the conservation law for the particle size distribution which is studied in the classical theory for Ostwald ripening by Lifshitz, Slyozov & Wagner (LSW). (Accepted June 25, 1998)     

An asymptotic method for a singular hyperbolic equation

Ho & Meyer [1] have sketched an argument for arriving at asymptotic properties of solutions of a pair of non-linear conservation laws with floating, shocktype boundary condition of importance in gas dynamics and oceanography. To clarify the mathematical basis of their method of inequalities, a proof of their main asymptotic results is given here. It depends on substantially weakened hypotheses.     

Global Existence for Reaction‐Diffusion Systems Modelling Ignition

The pair of parabolic equations , , with a>0 and b>0 models the temperature and concentration for an exothermic chemical reaction for which just one species controls the reaction rate f. Of particular interest is the case where , which appears in the Frank‐Kamenetskii approximation to Arrhenius‐type reactions. We show here that for a large choice of the nonlinearity f(u,v) in (1), (2)(including the model case (3)), the corresponding initial‐value problem for (1), (2) in the whole space with bounded initial data has a solution which exists for all times. Finite‐time blow‐up might occur, though, for other choices of function f(u,v), and we discuss here a linear example which strongly hints at such behaviour. (Accepted September 16, 1996)     

Decay Rates for the Three‐Dimensional Linear System of Thermoelasticity

. We consider the two and three‐dimensional system of linear thermoelasticity in a bounded smooth domain with Dirichlet boundary conditions. We analyze whether the energy of solutions decays exponentially uniformly to zero as . First of all, by a decoupling method, we reduce the problem to an observability inequality for the Lamé system in linear elasticity and more precisely to whether the total energy of the solutions can be estimated in terms of the energy concentrated on its longitudinal component. We show that when the domain is convex, the decay rate is never uniform. In fact, the lack of uniform decay holds in a more general class of domains in which there exist rays of geometric optics of arbitrarily large length that are always reflected perpendicularly or almost tangentially on the boundary. We also show that, in three space dimensions, the lack of uniform decay may also be due to a critical polarization of the energy on the transversal component of the displacement. In two space dimensions we prove a sufficient (and almost necessary) condition for the uniform decay to hold in terms of the propagation of the transversal characteristic rays, under the further assumption that the boundary of the domain does not have contacts of infinite order with its tangents. We also give an example, due to D. Hulin, in which these geometric properties hold. In three space dimensions we indicate (without proof) how a careful analysis of the polarization of singularities may lead to sharp sufficient conditions for the uniform decay to hold. In two space dimensions we prove that smooth solutions decay polynomially in the energy space to a finite‐dimensional subspace of solutions except when the domain is a ball or an annulus. Finally we discuss some closely related controllability and spectral issues. (Accepted May 14, 1998)     

A Multi-Fluid Compressible System as the Limit of Weak Solutions of the Isentropic Compressible Navier–Stokes Equations

This paper mainly concerns the mathematical justification of a viscous compressible multi-fluid model linked to the Baer-Nunziato model used by engineers, see for instance Ishii (Thermo-fluid dynamic theory of two-phase flow, Eyrolles, Paris, 1975), under a “stratification” assumption. More precisely, we show that some approximate finite-energy weak solutions of the isentropic compressible Navier–Stokes equations converge, on a short time interval, to the strong solution of this viscous compressible multi-fluid model, provided the initial density sequence is uniformly bounded with corresponding Young measures which are linear convex combinations of m Dirac measures. To the authors’ knowledge, this provides, in the multidimensional in space case, a first positive answer to an open question, see Hillairet (J Math Fluid Mech 9:343–376, 2007), with a stratification assumption. The proof is based on the weak solutions constructed by Desjardins (Commun Partial Differ Equ 22(5–6):977–1008, 1997) and on the existence and uniqueness of a local strong solution for the multi-fluid model established by Hillairet assuming initial density to be far from vacuum. In a first step, adapting the ideas from Hoff and Santos (Arch Ration Mech Anal 188:509–543, 2008), we prove that the sequence of weak solutions built by Desjardins has extra regularity linked to the divergence of the velocity without any relation assumption between λ and μ. Coupled with the uniform bound of the density property, this allows us to use appropriate defect measures and their nice properties introduced and proved by Hillairet (Aspects interactifs de la m’ecanique des fluides, PhD Thesis, ENS Lyon, 2005) in order to prove that the Young measure associated to the weak limit is the convex combination of m Dirac measures. Finally, under a non-degeneracy assumption of this combination (“stratification” assumption), this provides a multi-fluid system. Using a weak–strong uniqueness argument, we prove that this convex combination is the one corresponding to the strong solution of the multi-fluid model built by Hillairet, if initial data are equal. We will briefly discuss this assumption. To complete the paper, we also present a blow-up criterion for this multi-fluid system following (Huang et al. in Serrin type criterion for the three-dimensional viscous compressible flows, arXiv, 2010).     

Fractal First-Order Partial Differential Equations

The present paper is concerned with semi-linear partial differential equations involving a particular pseudo-differential operator. It investigates both fractal conservation laws and non-local Hamilton–Jacobi equations. The idea is to combine an integral representation of the operator and Duhamel's formula to prove, on the one hand, the key a priori estimates for the scalar conservation law and the Hamilton–Jacobi equation and, on the other hand, the smoothing effect of the operator. As far as Hamilton–Jacobi equations are concerned, a non-local vanishing viscosity method is used to construct a (viscosity) solution when existence of regular solutions fails, and a rate of convergence is provided. Turning to conservation laws, global-in-time existence and uniqueness are established. We also show that our formula allows us to obtain entropy inequalities for the non-local conservation law, and thus to prove the convergence of the solution, as the non-local term vanishes, toward the entropy solution of the pure conservation law.     

Symmetry of Ground States of Quasilinear Elliptic Equations

. We consider the problem of radial symmetry for non‐negative continuously differentiable weak solutions of elliptic equations of the form under the ground state condition Using the well‐known moving plane method of Alexandrov and Serrin, we show, under suitable conditions on A and f, that all ground states of (1) are radially symmetric about some origin O in . In particular, we obtain radial symmetry for compactly supported ground states and give sufficient conditions for the positivity of ground states in terms of the given operator A and the nonlinearity f. (Accepted September 21, 1998)     

Non-Classical Shocks and Kinetic Relations: Scalar Conservation Laws

This paper analyzes the non-classical shock waves which arise as limits of certain diffusive-dispersive approximations to hyperbolic conservation laws. Such shocks occur for non-convex fluxes and connect regions of different convexity. They have negative entropy dissipation for a single convex entropy function, but not all convex entropies, and do not obey the classical Oleinik entropy criterion. We derive necessary conditions for the existence of non-classical shock waves, and construct them as limits of traveling-wave solutions for several diffusive-dispersive approximations. We introduce a “kinetic relation” to act as a selection principle for choosing a unique non-classical solution to the Riemann problem. The convergence to non-classical weak solutions for the Cauchy problem is investigated. Using numerical experiments, we demonstrate that, for the cubic flux-function, the Beam-Warming scheme produces non-classical shocks while no such shocks are observed with the Lax-Wendroff scheme. All of these results depend crucially on the sign of the dispersion coefficient. (Accepted February 8, 1996)     

A non-oscillatory balanced scheme for an idealized tropical climate model

We propose a non-oscillatory balanced numerical scheme for a simplified tropical climate model with a crude vertical resolution, reduced to the barotropic and the first baroclinic modes. The two modes exchange energy through highly nonlinear interaction terms. We consider a periodic channel domain, oriented zonally and centered around the equator and adopt a fractional stepping–splitting strategy, for the governing system of equations, dividing it into three natural pieces which independently preserve energy. We obtain a scheme which preserves geostrophic steady states with minimal ad hoc dissipation by using state of the art numerical methods for each piece: The f-wave algorithm for conservation laws with varying flux functions and source terms of Bale et al. (2002) for the advected baroclinic waves and the Riemann solver-free non-oscillatory central scheme of Levy and Tadmor (1997) for the barotropic-dispersive waves. Unlike the traditional use of a time splitting procedure for conservation laws with source terms (here, the Coriolis forces), the class of balanced schemes to which the f-wave algorithm belongs are able to preserve exactly, to the machine precision, hydrostatic (geostrophic) numerical-steady states. The interaction terms are gathered into a single second order accurate predictor-corrector scheme to minimize energy leakage. Validation tests utilizing known exact solutions consisting of baroclinic Kelvin, Yanai, and equatorial Rossby waves and barotropic Rossby wave packets are given.     

求解双曲守恒律方程的WENO型熵相容格式

通过在单元交界面处进行高阶WENO重构,得到了一种求解双曲型守恒律方程的WENO型熵相容格式。用该格式对一维Burgers方程和Euler方程进行数值模拟,结果表明,该格式具有高精度、基本无振荡性等特点。