数学物理学报(英文版)

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Preface

Gui-Qiang G.Chen, Ling Hsiao, Feimin Huang, Athanasios Tzavaras, Zhen Wang

数学物理学报(英文版). 2012 (1): 1-1. DOI: 10.1016/S0252-9602(12)60001-0

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This special issue of Acta Mathematica Scientia is dedicated to the celebration of the 70th birthday of Constantine M. Dafermos. Costas is the Alumni-Alumnae University Professor of Brown University, a Fellow of the American Academy of Arts and Sciences, a Correspondent Member of the Academy of Athens, a Foreign Member of the Academia Nazionale dei Lincei, a SIAM Fellow, and an Honorary Professor of the Chinese Academy of Sciences. He is also an Editor-in-Chief of this journal. Costas has done extensive research at the interface of partial di?erential equations and continuum physics. He is a world leader in nonlinear hyperbolic conservation laws. He has also made fundamental contributions on the mathematical theory of the equa-tions of thermomechanics. Almost uniquely among today’s mathematicians, Costas has seen the greater picture in understanding the fundamental issues of continuum physics and their role in developing new techniques of mathematical analysis. Over the years, Costas has been enthusiastic in advising and helping young mathematicians and is an exemplary figure for many mathematicians around the world. The papers herein are contributed by his former students, postdoctoral fellows, collaborators, and friends from all over the world. They stand as a modest testimony of their appreciation for his scientific eminence and human qualities and of their admi-ration for an exceptional advisor, mentor, collaborator, and colleague. On this special occasion of his 70th birthday, we wish Costas the very best in his scientific endeavours for many decades to come with us.

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REMARKS ON THE CONTRIBUTIONS OF CONSTANTINE M. DAFERMOS TO THE SUBJECT OF ONSERVATION LAWS

Gui-Qiang G. Chen, Athanasios E. Tzavaras

数学物理学报(英文版). 2012 (1): 3-14. DOI: 10.1016/S0252-9602(12)60002-2

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Constantine M. Dafermos has done extensive researchat the interface ofpartial di?erential equations and continuum physics. He is a world leader in nonlinear hyperbolic conservation laws, where he introduced several fundamental methods in the subject including the methods of relative entropy, generalized characteristics, and wave-fronttracking, as well as the entropy rate criterion for the selection of admissible wave fans. He has also made fundamental contributions on the mathematical theory of the equations of thermomechanics as it pertains in modeling and analysis of materials with memory, thermoelasticity, and thermoviscoelasticity. His work is distinctly characterized by an understanding of the fundamental issues of continuum physics and their role in developing new techniques of mathematical analysis.

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A STUDY ON GRADIENT BLOW UP FOR VISCOSITY SOLUTIONS OF FULLY NONLINEAR, UNIFORMLY ELLIPTIC EQUATIONS

Bernd Kawohl, Nikolai Kutev

数学物理学报(英文版). 2012 (1): 15-40. DOI: 10.1016/S0252-9602(12)60003-4

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We investigate sharp conditions for boundary and interior gradient estimates of continuous viscosity solutions to fully nonlinear, uniformly elliptic equations under Dirichlet boundary conditions. When these conditions are violated, there can be blow up of the gradient in the interior or on the boundary of the domain. In particular we de-
rive sharp results on local and global Lipschitz continuity of continuous viscosity solutions under more general growth conditions than before. Lipschitz regularity near the boundary allows us to predict when the Dirichlet condition is satisfied in a classical and not just in a viscosity sense, where detachment can occur. Another consequence is this: if interior gra-dient blow up occurs, Perron-type solutions can in general become discontinuous, so that the Dirichlet problem can become unsolvable in the class of continuous viscosity solutions.

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GLOBAL EXISTENCE, UNIQUENESS, AND STABILITY FOR A NONLINEAR HYPERBOLIC-PARABOLIC PROBLEM IN PULSE COMBUSTION

Olga Terlyga, Hamid Bellout, Frederick Bloom

数学物理学报(英文版). 2012 (1): 41-74. DOI: 10.1016/S0252-9602(12)60004-6

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A global existence theorem is established for an initial-boundary value prob-lem, with time-dependent boundary data, arising in a lumped parameter model of pulse combustion; the model in question gives rise to a nonlinear mixed hyperbolic-parabolic sys-tem. Using results previously established for the associated linear problem, a fixed point argument is employed to prove local existence for a regularized version of the nonlinear problem with artificial viscosity. Appropriate a-priori estimates are then derived which imply that the local existence result can be extended to a global existence theorem for the regularized problem. Finally, a di?erent set of a priori estimates is generated which allows for takingthelimit as theartificial viscosity parameter converges tozero; the corresponding solution of the regularized problem is then proven to converge to the unique solution of the
initial-boundary value problem for the original, nonlinear, hyperbolic-parabolic system.

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HÖLDER CONTINUITY AND DIFFERENTIABILITY ON CONVERGING SUBSEQUENCES

Volker Elling

数学物理学报(英文版). 2012 (1): 75-83. DOI: 10.1016/S0252-9602(12)60005-8

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It is shown that an arbitrary function from D ( Rⁿ to R^m will become C^0,α -continuous in almost every x ∈ D after restriction to a certain subset with limit point x. For n ≥ m differentiability can be obtained. Examples show the H¨older exponentn α = min{1, n/m} is optimal.

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ABOUT TWO TYPES OF MICROSTRUCTURES ADAPTED TO HEAT EVACUATION AND ELASTIC STRESS: SNOW FLAKES AND QUASI-CRYSTALS

Luc Tartar

数学物理学报(英文版). 2012 (1): 84-108. DOI: 10.1016/S0252-9602(12)60006-X

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I first met Constantine Dafermos in August 1974, at a meeting at Brown University, where I was invited because my former advisor (Jacques-Louis LIONS) could not come, and he had proposed my name. I was happily surprised that Constantine greeted me as if he knew me well, and since for many years now I have considered him as if he was an older brother, I wonder when this feeling started.
I do not remember when I told him that my paternal grandmother (Cecilia) had a Greek passport, which was not uncommon for Christian families in Syria, since most of them followed the Greek Orthodoxrite, and I now wonder if it played a role in the divorce of my grandparents, after my grandfather (Elias) switched to a more evangelical faith, and became a little excessive about religion: my father (Georges) once mentioned that when he had some argument with his father and he wanted to annoy him, he went to the Sunday offce at the Greek Orthodox¹ church .
Becoming friend with Constantine was natural for human reasons, but also for scientific

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FORMATION OF SINGULARITY FOR COMPRESSIBLE VISCOELASTICITY

Xianpeng Hu, Dehua Wang

数学物理学报(英文版). 2012 (1): 109-128. DOI: 10.1016/S0252-9602(12)60007-1

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The formation of singularity and breakdown of classical solutions to the three-dimensional compressible viscoelasticity and inviscid elasticity are considered. For the compressible inviscid elastic fluids, the finite-time formation of singularity in classical solu-tions is proved for certain initial data. For the compressible viscoelastic fluids, a criterion in term of the temporal integral of the velocity gradient is obtained for the breakdown of smooth solutions.

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HOMOGENIZATION, SYMMETRY, AND PERIODIZATION IN DIFFUSIVE RANDOM MEDIA

Alen Alexanderian, Muruhan Rathinam, Rouben Rostamian

数学物理学报(英文版). 2012 (1): 129-154. DOI: 10.1016/S0252-9602(12)60008-3

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We present a systematic study of homogenization of diffusion in random me-dia with emphasis on tile-based random microstructures. We give detailed examples of several such media starting from their physical descriptions, then construct the associated probability spaces and verify their ergodicity. After a discussion of material symmetries of random media, we derive criteria for the isotropy of the homogenized limits in tile-based
structures. Furthermore, we study the periodization algorithm for the numerical approxi-mation of the homogenized diffusion tensor and study the algorithm’s rate of convergence. For one dimensional tile-based media, we prove a central limit result, giving a concrete rate of convergence for periodization. We also provide numerical evidence for a similar central limit behavior in the case of two dimensional tile-based structures.

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ON A SECOND ORDER DISSIPATIVE ODE IN HILBERT SPACE WITH AN INTEGRABLE SOURCE TERM

Alain Haraux, Mohamed Ali Jendoubi

数学物理学报(英文版). 2012 (1): 155-163. DOI: 10.1016/S0252-9602(12)60009-5

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Asymptotic behaviour of solutions is studied for some second order equations including the model case x¨(t)+γx˙(t)+?Φ(x(t)) = h(t) with γ > 0 and h ∈ L (0,+∞; H), Φ being continuouly di?erentiable with locally Lipschitz continuous gradient and bounded from below. In particular when Φ is convex, all solutions tend to minimize the potential Φ as time tends to infinity and the existence of one bounded trajectory implies the weak convergence of all solutions to equilibrium points.

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LOW MACH NUMBER LIMIT ON EXTERIOR DOMAINS

Donatella Donatelli, Pierangelo Marcati

数学物理学报(英文版). 2012 (1): 164-176. DOI: 10.1016/S0252-9602(12)60010-1

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This paper is concerned with the low Mach number limit for the compressible Navier-Stokes equations in an exterior domain. We present here an approach based on Strichartz estimate defined on a non trapping exterior domain and we will be able to show the compactness and strong convergence of the velocity vector field.

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NONLOCAL CROWD DYNAMICS MODELS FOR SEVERAL POPULATIONS

Rinaldo M. Colombo, Magali L′ecureux-Mercier

数学物理学报(英文版). 2012 (1): 177-196. DOI: 10.1016/S0252-9602(12)60011-3

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This paper develops the basic analytical theory related to some recently intro-duced crowd dynamics models. Where well posedness was known only locally in time, it is here extended to all of R⁺ . The results on the stability with respect to the equations are improved. Moreover, here the case of several populations is considered, obtaining the well posedness of systems of multi-D non-local conservation laws. The basic analytical tools are provided by the classical Kruˇzkov theory of scalar conservation laws in several space dimensions.

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BOUND STATES FOR A CLASS OF QUASILINEAR SCALAR FIELD EQUATIONS WITH POTENTIALS VANISHING AT INFINITY

Athanasios N. Lyberopoulos

数学物理学报(英文版). 2012 (1): 197-208. DOI: 10.1016/S0252-9602(12)60012-5

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We study the existence and non-existence of bound states (i.e., solutions in W^{1, p} (R^N )) for a class of quasilinear scalar field equations of the form

-?_pu+V(x)|u| ^p-2u = a(x)|u| ^q-2u, x ∈ R^N , 1 < p < N,

when the potentials V(·) 0 and a(·) decay to zero at infinity.

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ESTIMATES ON THE FINITE ELEMENT APPROXIMATIONS OF A QUADRATIC FUNCTIONAL

Xiaqi Ding

数学物理学报(英文版). 2012 (1): 209-218. DOI: 10.1016/S0252-9602(12)60013-7

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This paper gives some estimates of a quadratic functional related with Rie-mann Hypothesis.

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LARGE TIME BEHAVIOR OF SOLUTIONS TO NONLINEAR VISCOELASTIC MODEL WITH FADING MEMORY

Yanni Zeng

数学物理学报(英文版). 2012 (1): 219-236. DOI: 10.1016/S0252-9602(12)60014-9

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We study the Cauchy problem of a one-dimensional nonlinear viscoelastic model with fading memory. By introducing appropriate new variables we convert the integro-partial differential equations into a hyperbolic system of balance laws. When it is a perturbation of a constant state, the solution is shown time asymptotically approach-ing to predetermined di?usion waves. Pointwise estimates on the convergence details are obtained.

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THE ONE-DIMENSIONAL HUGHES MODEL FOR EDESTRIAN FLOW: RIEMANN-TYPE SOLUTIONS

Debora Amadori, M. Di Francesco

数学物理学报(英文版). 2012 (1): 259-280. DOI: 10.1016/S0252-9602(12)60016-2

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This paper deals with a coupled system consisting of a scalar conservation law
and an eikonal equation, called the Hughes model. Introduced in [24], this model attempts
to describe the motion of pedestrians in a densely crowded region, in which they are seen
as a ‘thinking’ (continuum) fluid. The main mathematical diffculty is the discontinuous
gradient of the solution to the eikonal equation appearing in the flux of the conservation
law. On a one dimensional interval with zero Dirichlet conditions (the two edges of the
interval are interpreted as ‘targets’), the model can be decoupled in a way to consider
two classical conservation laws on two sub-domains separated by a turning point at which
the pedestrians change their direction. We shall consider solutions with a possible jump
discontinuity around the turning point. For simplicity, we shall assume they are locally
constant on both sides of the discontinuity. We provide a detailed description of the local-
in-time behavior of the solution in terms of a ‘global’ qualitative property of the pedestrian
density (that we call ‘relative evacuation rate’), which can be interpreted as the attitude
of the pedestrians to direct towards the left or the right target. We complement our result
with explicitly computable examples.

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A MATHEMATICAL THEORY FOR LES CONVERGENCE

H. Lim, T. Kaman, Y. Yu, V. Mahadeo, Y. Xu, H. Zhang, J. Glimm, S. Dutta, D. H. Sharp, B. Plohr

数学物理学报(英文版). 2012 (1): 237-258. DOI: 10.1016/S0252-9602(12)60015-0

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Practical simulations of turbulent processes are generally cutoff, with a grid
resolution that stops within the inertial range, meaning that multiple active regions and
length scales occur below the grid level and are not resolved. This is the regime of large
eddy simulations (LES), in which the larger but not the smaller of the turbulent length
scales are resolved. Solutions of the fluid Navier-Stokes equations, when considered in the
inertial regime, are conventionally regarded as solutions of the Euler equations. In other
words, the viscous and di?usive transport terms in the Navier-Stokes equations can be
neglected in the inertial regime and in LES simulations, while the Euler equation becomes
fundamental.
For such simulations, significant new solution details emerge as the grid is refined. It
follows that conventional notions of grid convergence are at risk of failure, and that a new,
and weaker notion of convergence may be appropriate. It is generally understood that the
LES or inertial regime is inherently fluctuating and its description must be statistical in
nature. Here we develop such a point of view systematically, based on Young measures,
which are measures depending on or indexed by space time points. In the Young measure
d (ξ)ν ξ x,t, the random variable ξ of the measure is a solution state variable, i.e., a solution
dependent variable, representing momentum, density, energy and species concentrations,
while the space time coordinates,x,t , serve to index the measure.

Theoretical evidence suggests that convergence via Young measures is suffciently weak to
encompass the LES/inertial regime; numerical and theoretical evidence suggests that this
notion may be required for passive scalar concentration and thermal degrees of freedom.
Our objective in this research is twofold: turbulent simulations without recourse to ad-
justable parameters (calibration) and extension to more complex physics, without use of
additional models or parameters, in both cases with validation through comparison to
experimental data.

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LESSER KNOWN MIRACLES OF BURGERS EQUATION

Govind Menon

数学物理学报(英文版). 2012 (1): 281-294. DOI: 10.1016/S0252-9602(12)60017-4

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This article is a short introduction to the surprising appearance of Burgers equation in some basic probabilistic models.

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ANALYSIS OF A COMPRESSIBLE GAS-LIQUID MODEL MOTIVATED BY OIL WELL CONTROL OPERATIONS

Steinar Evje, K.H. Karlsen

数学物理学报(英文版). 2012 (1): 295-314. DOI: 10.1016/S0252-9602(12)60018-6

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We are interested in a viscous two-phase gas-liquid mixture model relevant for modeling of well control operations within the petroleum industry. We focus on a simplified mixture model and provide an existence result within an appropriate class of weak solutions. We demonstrate that upper and lower limits can be obtained for the gas and liquid masses which ensure that transition to single-phase regions do not occur. This is used together with appropriate a prior estimates to obtain convergence to a weak solution for a sequence of approximate solutions corresponding to mollified initial data. Moreover, by imposing an additional regularity condition on the initial masses, a uniqueness result is obtained. The framework herein seems useful for further investigations of more realistic versions of the gas-liquid model that take into account di?erent flow regimes.

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CAUCHY PROBLEM FOR THE ONE-DIMENSIONAL COMPRESSIBLE NAVIER-STOKES EQUATIONS

连汝续, 刘健, 李海梁, 肖玲

数学物理学报(英文版). 2012 (1): 315-324. DOI: 10.1016/S0252-9602(12)60019-8

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We consider the Cauchy problem for one-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosity coe?cient. For regular initial data, we show that the unique strong solution exits globally in time and converges to the equilibrium state time asymptotically. When initial density is piecewise regular with jump discontinuity, we show that there exists a unique global piecewise regular solution. In
particular, the jump discontinuity of the density decays exponentially and the piecewise regular solution tends to the equilibrium state as t → +∞.

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VARIATIONAL PRINCIPLES FOR NONLOCAL CONTINUUM MODEL OF ORTHOTROPIC GRAPHENE SHEETS EMBEDDED IN AN ELASTIC MEDIUM

Sarp Adali

数学物理学报(英文版). 2012 (1): 325-338. DOI: 10.1016/S0252-9602(12)60020-4

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Equations governing the vibrations and buckling of multilayered orthotropic graphene sheets can be expressed as a system of n partial differential equations where n refers to the number of sheets. This description is based on the continuum model of the graphene sheets which can also take the small scale effects into account by employing a nonlocal theory. In the present article a variational principle is derived for the nonlocal elastic theory of rectangular graphene sheets embedded in an elastic medium and undergo-ingtransverse vibrations. Moreover the graphene sheets are subject to biaxial compression. Rayleigh quotients are obtained for the frequencies of freely vibrating graphene sheets and for the buckling load. The influence of small scale effects on the frequencies and the buckling load can be observed qualiatively from the expressions of the Rayleigh quotients. Elastic medium is modeled as a combination of Winkler and Pasternak foundations acting on the top and bottom layers of the mutilayered nano-structure. Natural boundary con-ditions of the problem are derived using the variational principle formulated in the study. It is observed that free boundaries lead to coupled boundary conditions due to nonlocal theory used in the continuum formulation while the local (classical) elasticity theory leads to uncoupled boundary conditions. The mathematical methods used in the study involve
calculus of variations and the semi-inverse method for deriving the variational integrals.

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THE VACUUM IN NONISENTROPIC GAS DYNAMICS

Geng Chen, Robin Young

数学物理学报(英文版). 2012 (1): 339-351. DOI: 10.1016/S0252-9602(12)60021-6

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We investigate the vacuum in nonisentropic gas dynamics in one space vari-able, with the most general equation of states allowed by thermodynamics. We recall physical constraints on the equations of state and give explicit and easily checkable condi-tions under which vacuums occur in the solution of the Riemann problem. We then present a class of models for which the Riemann problem admits unique global solutions without
vacuums.

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DECAY OF POSITIVE WAVES OF HYPERBOLIC BALANCE LAWS

Cleopatra Christoforou, Konstantina Trivisa

数学物理学报(英文版). 2012 (1): 352-366. DOI: 10.1016/S0252-9602(12)60022-8

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Historically, decay rates have been used to provide quantitative and quali-tative information on the solutions to hyperbolic conservation laws. Quantitative results include the establishment of convergence rates for approximating procedures and numer-ical schemes. Qualitative results include the establishment of results on uniqueness and regularity as well as the ability to visualize the waves and their evolution in time. This work presents two decay estimates on the positive waves for systems of hyperbolic and gen-uinely nonlinear balance laws satisfying a dissipative mechanism. The result is obtained by employing the continuity of Glimm-type functionals and the method of generalized characteristics [7, 17, 24].

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NONUNIQUENESS AND SINGULAR RADIAL SOLUTIONS OF SYSTEMS OF CONSERVATION LAWS

Michael G. Hilgers

数学物理学报(英文版). 2012 (1): 367-379. DOI: 10.1016/S0252-9602(12)60023-X

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The case of a radial initial state for a family of hyperbolic systems of con-servation laws with several spatial dimensions is considered. It will be shown that the singularity at the origin introduces multiple solutions outside of the traditional admissible classes.

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SBV REGULARITY OF GENUINELY NONLINEAR HYPERBOLIC SYSTEMS OF CONSERVATION LAWS IN ONE SPACE DIMENSION

Stefano Bianchini

数学物理学报(英文版). 2012 (1): 380-388. DOI: 10.1016/S0252-9602(12)60024-1

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The problem of the presence of Cantor part in the derivative of a solution to a hyperbolic system of conservation laws is considered. An overview of the techniques involved in the proof is given, and a collection of related problems concludes the paper. Key words hyperbolic systems; conservation laws; SBV; regularity

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ASYMPTOTIC BEHAVIOR OF SOLUTIONS TOWARD THE SUPERPOSITION OF CONTACT DISCONTINUITY AND SHOCK WAVE FOR COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH FREE BOUNDARY

Hakho Hong, Feimin Huang

数学物理学报(英文版). 2012 (1): 389-412. DOI: 10.1016/S0252-9602(12)60025-3

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A free boundary problem for the one-dimensional compressible Navier-Stokes equations is investigated. The asymptotic behavior of solutions toward the superposition of contact discontinuity and shock wave is established under some smallness conditions. To do this, we first construct a new viscous contact wave such that the momentum equation is satisfied exactly and then determine the shift of the viscous shock wave. By using them
together with an inequality concerning the heat kernel in the half space, we obtain the desired a priori estimates. The proof is based on the elementary energy method by the anti-derivative argument.

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FREE BOUNDARY VALUE PROBLEM OF ONE DIMENSIONAL TWO-PHASE LIQUID-GAS MODEL

王振, 张卉

数学物理学报(英文版). 2012 (1): 413-432. DOI: 10.1016/S0252-9602(12)60026-5

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In this paper, we study a free boundary value problem for two-phase liquid-gas model with mass-dependent viscosity coeffcient when both the initial liquid and gas masses connect to vacuum continuously. The gas is assumed to be polytropic whereas the liquid is treated as an incompressible fluid. We give the proof of the global existence and uniqueness of weak solutions when β ∈ (0,1), which have improved the result of Evje and
Karlsen, and we obtain the regularity of the solutions by energy method.

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THE STRESS-ENERGY TENSOR AND POHOZAEV’S IDENTITY FOR SYSTEMS

N. D. Alikakos, A. C. Faliagas

数学物理学报(英文版). 2012 (1): 433-439. DOI: 10.1016/S0252-9602(12)60027-7

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Utilizing stress-energy tensors which allow for a divergence-free formulation, we establish Pohozaev’s identity for certain classes of quasilinear systems with variational structure.

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