数学物理学报(英文版)

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Preface

数学物理学报(英文版). 2011 (6): 0-0. DOI: 10.1016/S0252-9602(11)60385-8

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This special issue of Acta Mathematica Scientia is dedicated to the 85th birthday of Peter D. Lax, one of the most prominent mathematicians of the past hundred years. Peter Lax has made important and seminal contributions in many areas of pure and applied mathematics, including the theory of integrable systems and solitons, the theory of shock waves in hyperbolic conservation laws, and the theory of scattering, with major applications to fluid dynamics and other branches of physics. He has also exerted a great influence in launching the field of scientific computation. Over the years, Peter Lax has been a great supporter of mathematics in China, having helped
numerous Chinese mathematicians. Peter’s great scientific achievements, as well as his significant contributions to pro-moting mathematics across the sciences, have profoundly influenced and inspired us. The papers herein, contributed by his former students, postdoctoral fellows, collabora-tors, and friends, stand as a modest testimony to their admiration for Peter’s scientific eminence and their lasting friendship for an exceptional advisor, mentor, collaborator, and colleague. On this special occasion of his 85th birthday, wewish Peter the very best in his scientific endeavors for years to come.

Standing Editors: Gui-Qiang G. Chen Constantine M. Dafermos Xiaqi Ding Tai-Ping Liu

Gui-Qiang G. Chen: chengq@maths.ox.ac.uk
Constantine Dafermos: Constantine Dafermos@brown.edu
Xiaqi Ding: xqding@amt.ac.cn
Tai-Ping Liu: liu@math.stanford.edu

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HUA CLASS ON REAL LINE AND TRICOMI PROBLEM

Xiaqi Ding

数学物理学报(英文版). 2011 (6): 2103-2106. DOI: 10.1016/S0252-9602(11)60386-X

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In this note, we study the Hua weak function class H on the real line. And then point out its connection with Tricomi ploblem of Lavrentyev equation.

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SOME EXACT SOLUTIONS OF 3-DIMENSIONAL ZERO-PRESSURE GAS DYNAMICS SYSTEM

K.T. Joseph and Manas R. Sahoo

数学物理学报(英文版). 2011 (6): 2107-2121. DOI: 10.1016/S0252-9602(11)60387-1

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The 3-dimensional zero-pressure gas dynamics system appears in the modeling for the large scale structure formation in the universe. The aim of this paper is to construct spherically symmetric solutions to the system. The radial component of the velocity and density satisfy a simpler one dimensional problem. First we construct explicit solutions of this one dimensional case with initial and boundary conditions. Then we get special radial solutions with different behaviours at the origin.

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EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS WITH NON-SEPARATED TYPE INTEGRAL BOUNDARY CONDITIONS

Bashir Ahmad, Juan J.Nieto, Ahmed Alsaedi

数学物理学报(英文版). 2011 (6): 2122-2130. DOI: 10.1016/S0252-9602(11)60388-3

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In this paper, we study a boundary value problem of nonlinear fractional dif-ferential equations of order q (1 < q 2) with non-separated integral boundary conditions. Some new existence and uniqueness results are obtained by using some standard fixed point theorems and Leray-Schauder degree theory. Some illustrative examples are also presented. We extend previous results even in the integer case q = 2.

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ON MULTI-DIMENSIONAL SONIC-SUBSONIC FLOW

黄飞敏, 王天怡, 王勇

数学物理学报(英文版). 2011 (6): 2131-2140. DOI: 10.1016/S0252-9602(11)60389-5

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In this paper, a compensated compactness framework is established for sonic-subsonic approximate solutions to the n-dimensional (n ≥ 2) Euler equations for steady irrotational flow that may contain stagnation points. This compactness framework holds provided that the approximate solutions are uniformly bounded and satisfy H⁻¹_loc (Ω) com-pactness conditions. As illustration, we show the existence of sonic-subsonic weak solution to n-dimensional (n ≥ 2) Euler equations for steady irrotational flow past obstacles or through an infinitely long nozzle. This is the first result concerning the sonic-subsonic limit for n-dimension (n ≥ 3).

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INVARIANCE AND STABILITY OF THE PROFILE EQUATIONS OF GEOMETRIC OPTICS

Guy Métivier, Jeffrey Rauch

数学物理学报(英文版). 2011 (6): 2141-2158. DOI: 10.1016/S0252-9602(11)60390-1

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The profile equations of geometric optics are described in a form invariant un-der the natural transformations of first order systems of partial differential equations. This allows us to prove that various strategies for computing profile equations are equivalent. We prove that if L generates an evolution on L² the same is true of the profile equations. We prove that the characteristic polynomial of the profile equations is the localization of the characteristic polynomial of the background operator at (y, dΦ(y)) where Φ is the bak-
ground phase. We prove that the propagation cones of the profile equations are subsets of
the propagation cones of the background operator.

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TWO PHASE COMPRESSIBLE FLOW IN POROUS MEDIA

应隆安

数学物理学报(英文版). 2011 (6): 2159-2168. DOI: 10.1016/S0252-9602(11)60391-3

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We study the mathematical model of two phase compressible flows through porous media. Under the condition that the compressibility of rock, oil, and water is small, we prove that the initial-boundary value problem of the nonlinear system of equations admits a weak solution.

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COMPRESSIBLE NON-ISENTROPIC BIPOLAR NAVIER-STOKES-POISSON SYSTEM IN R³

肖玲, 李海梁, 杨彤, 邹晨

数学物理学报(英文版). 2011 (6): 2169-2194. DOI: 10.1016/S0252-9602(11)60392-5

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The compressible non-isentropic bipolar Navier–Stokes–Poisson (BNSP) sys-tem is investigated in R³ in the present paper, and the optimal time decay rates of global strong solution are shown. For initial data being a perturbation of equilibrium state in H^l(R³)∩˙B ^−s_1,1(R³) for l ≥4 and s ∈ (0, 1], it is shown that the density and temperature for each charged particle (like electron or ion) decay at the same optimal rate (1 + t)^−3/4 , but the momentum for each particle decays at the optimal rate (1 + t)^−1/4−s/2 which is slower than the rate (1+t^{)−3/4−s/2} for the compressible Navier–Stokes (NS) equations [19] for same initial data. However, the total momentum tends to the constant state at the rate (1+t)^−3/4as well, due to the interplay interaction of charge particles which counteracts the influence of electric field.

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THE EARLY INFLUENCE OF PETER LAX ON COMPUTATIONAL HYDRODYNAMICS AND AN APPLICATION OF LAX-FRIEDRICHS AND#br# LAX-WENDROFF ON TRIANGULAR GRIDS IN LAGRANGIAN COORDINATES

Richard Liska, Mikhail Shashkov, Burton Wendroff

数学物理学报(英文版). 2011 (6): 2195-2202. DOI: 10.1016/S0252-9602(11)60393-7

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We give a brief discussion of some of the contributions of Peter Lax to Com-putational Fluid Dynamics. These include the Lax-Friedrichs and Lax-Wendroff numerical schemes. We also mention his collaboration in the 1983 HLL Riemann solver. We de-velop two-dimensional Lax-Friedrichs and Lax-Wendroff schemes for the Lagrangian form of the Euler equations on triangular grids. We apply a composite scheme that uses a Lax-Friedrichs time step as a dissipative filter after several Lax-Wendroff time steps. Numerical results for Noh’s infinite strength shock problem, the Sedov blast wave problem, and the Saltzman piston problem are presented.

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TIME AND NORM OPTIMAL CONTROLS: A SURVEY OF RECENT RESULTS AND OPEN PROBLEMS

H. O. Fattorini

数学物理学报(英文版). 2011 (6): 2203-2218. DOI: 10.1016/S0252-9602(11)60394-9

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We present in this paper a survey of recent results on the relation between time and norm optimality for linear systems and the infinite dimensional version of Pontryagin’s maximum principle. In particular, we discuss optimality (or nonoptimality) of singular controls satisfying the maximum principle and smoothness of the costate in function of smoothness of the target.

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AN ASYMPTOTIC PRESERVING SCHEME FOR THE VLASOV-POISSON-FOKKER-PLANCK SYSTEM IN THE HIGH FIELD REGIME

Shi Jin, Li Wang

数学物理学报(英文版). 2011 (6): 2219-2232. DOI: 10.1016/S0252-9602(11)60395-0

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The Vlasov-Poisson-Fokker-Planck system under the high field scaling de-scribes the Brownian motion of a large system of particles in a surrounding bath where both collision and field effects (electrical or gravitational) are dominant. Numerically solving this system becomes challenging due to the stiff collision term and stiff nonlinear transport term with respect to the high field. We present a class of Asymptotic-Preserving scheme which is efficient in the high field regime, namely, large time steps and coarse meshes can be used, yet the high field limit is still captured. The idea is to combine the two stiff terms and treat them implicitly. Thanks to the linearity of the collision term, using the discretization described in [Jin S, Yan B. J. Comp. Phys., 2011, 230: 6420-6437] we only need to invert a symmetric matrix. This method can be easily extended to higher dimensions. The method is shown to be positive, stable, mass and asymptotic preserv-ing. Numerical experiments validate its efficiency in both kinetic and high field regimes including mixing regimes.

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A RIEMANN-HILBERT PROBLEM IN A RIEMANN SURFACE

Spyridon Kamvissis

数学物理学报(英文版). 2011 (6): 2233-2246. DOI: 10.1016/S0252-9602(11)60396-2

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One of the inspirations behind Peter Lax’s interest in dispersive integrable systems, as the small dispersion parameter goes to zero, comes from systems of ODEs discretizing 1-dimensional compressible gas dynamics [17]. For example, an understanding of the asymptotic behavior of the Toda lattice in different regimes has been able to shed light on some of von Neumann’s conjectures concerning the validity of the approximation of PDEs by dispersive systems of ODEs.

Back in the 1990s several authors have worked on the long time asymptotics of the Toda lattice [2, 7, 8, 19]. Initially the method used was the method of Lax and Levermore [16], reducing the asymptotic problem to the solution of a minimization problem with constraints (an “equilibrium measure” problem). Later, it was found that the asymptotic method of Deift and Zhou (analysis of the associated Riemann-Hilbert factorization problem in the complex plane) could apply to previously intractable problems and also produce more detailed information.

Recently, together with Gerald Teschl, we have revisited the Toda lattice; instead of solu-tions in a constant or steplike constant background that were considered in the 1990s we have been able to study solutions in a periodic background.

Two features are worth noting here. First, the associated Riemann-Hilbert factorization problem naturally lies in a hyperelliptic Riemann surface. We thus generalize the Deift-Zhou “nonlinear stationary phase method” to surfaces of nonzero genus. Second, we illus-trate the important fact that very often even when applying the powerful Riemann-Hilbert method, a Lax-Levermore problem is still underlying and understanding it is crucial in the analysis and the proofs of the Deift-Zhou method!

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NEW METHOD FOR IMPROVED CALCULATIONS OF UNSTEADY COMPLEX FLOWS IN LARGE ARTERIES

A. Cheer, Harry A. Dwyer, T. Kim

数学物理学报(英文版). 2011 (6): 2247-2264. DOI: 10.1016/S0252-9602(11)60397-4

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Using an improved computational fluid dynamics (CFD) method developed for highly unsteady three-dimensional flows, numerical simulations for oscillating flow cy-cles and detailed unsteady simulations of the flow and forces on the aortic vessels at the iliac bifurcation, for both healthy and diseased patients, are analyzed. Improvements in computational efficiency and acceleration in convergence are achieved by calculating both an unsteady pressure gradient which is due to fluid acceleration and a good global pressure field correction based on mass flow for the pressure Poisson equation. Applications of the enhanced method to oscillatory flow in curved pipes yield an order of magnitude increase in speed and efficiency, thus allowing the study of more complex flow problems such as flow through the mammalian abdominal aorta at the iliac arteries bifurcation.

To analyze the large forces which can exist on stent graft of patients with abdominal aor-tic aneurysm (AAA) disease, a complete derivation of the force equations is presented. The accelerated numerical algorithm and the force equations derived are used to calculate flow and forces for two individuals whose geometry is obtained from CT data and whose respective blood pressure measurements are obtained experimentally. Although the use of endovascular stent grafts in diseased patients can alter vessel geometries, the physical characteristics of stents are still very different when compared to native blood vessels of healthy subjects. The geometry for the AAA stent graph patient studied in this investi-gation induced flows that resulted in large forces that are primarily caused by the blood pressure. These forces are also directly related to the flow cross-sectional area and the an-gle of the iliac arteries relative to the main descending aorta. Furthermore, the fluid flow is significantly disturbed in the diseased patient with large flow recirculation and stagnant regions which are not present for healthy subjects.

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THE LARGE TIME GENERIC FORM OF THE SOLUTION TO HAMILTON-JACOBI EQUATIONS

王靖华, 温海瑞, 赵引川

数学物理学报(英文版). 2011 (6): 2265-2277. DOI: 10.1016/S0252-9602(11)60398-6

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We use Hopf-Lax formula to study local regularity of solution to Hamilton-Jacobi (HJ) equations of multi-dimensional space variables with convex Hamiltonian. Then we give the large time generic form of the solution to HJ equation, i.e. for most initial data there exists a constant T > 0, which depends only on the Hamiltonian and initial datum, for t > T the solution of the IVP (1.1) is smooth except for a smooth n-dimensional hypersurface, across which Du(x, t) is discontinuous. And we show that the hypersurface tends asymptotically to a given hypersurface with rate t^−1/4 .

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CONCENTRATION OF BLOCH EIGENSTATES IN THE PRESENCE OF GAUGE AT THE SEMI-CLASSICAL LIMIT

Gershon Wolansky

数学物理学报(英文版). 2011 (6): 2278-2284. DOI: 10.1016/S0252-9602(11)60399-8

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We prove a concentration result of a Bloch eigenstate in a periodic channel under a constant gauge. In the semi-classical limit h ! 0 these eigenstates concentrate near a maximizer of the scalar potential of the associated Schr¨odinger operator, provided the constant gauge converges to a critical value from above. This is in contrast with the ground states which concentrate for any gauge in this limit near a minimizer of the scalar potential.

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AN EXTENSION OF THE HARDY-LITTLEWOOD-PÓLYA INEQUALITY

Congming Li, John Villavert

数学物理学报(英文版). 2011 (6): 2285-2288. DOI: 10.1016/S0252-9602(11)60400-1

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STABILITY OF EQUILIBRIA OF NEMATIC LIQUID CRYSTALLINE POLYMERS

Hong Zhou, Hongyun Wang

数学物理学报(英文版). 2011 (6): 2289-2304. DOI: 10.1016/S0252-9602(11)60401-3

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We provide an analytical study on the stability of equilibria of rigid rodlike nematic liquid crystalline polymers (LCPs) governed by the Smoluchowski equation with the Maier-Saupe intermolecular potential. We simplify the expression of the free energy of an orientational distribution function of rodlike LCP molecules by properly selecting a coordinate system and then investigate its stability with respect to perturbations of orientational probability density. By computing the Hessian matrix explicitly, we are able to prove the hysteresis phenomenon of nematic LCPs: when the normalized polymer concentration b is below a critical value b* (6.7314863965), the only equilibrium state is isotropic and it is stable; when b* < b < 15/2, two anisotropic (prolate) equilibrium states occur together with a stable isotropic equilibrium state. Here the more aligned prolate state is stable whereas the less aligned prolate state is unstable. When b > 15/2, there are three equilibrium states: a stable prolate state, an unstable isotropic state and an unstable oblate state.

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NONEXISTENCE OF PSEUDO-SELF-SIMILAR SOLUTIONS TO INCOMPRESSIBLE EULER EQUATIONS

Maria Schonbek

数学物理学报(英文版). 2011 (6): 2305-2312. DOI: 10.1016/S0252-9602(11)60402-5

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In this paper we study a generalization of self-similar solutions. We show that just as for the solutions to the Navier-Stokes equations these supposedly singular solution reduce to the zero solution.

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THE RIEMANN PROBLEM FOR THE NONLINEAR DEGENERATE WAVE EQUATIONS

刘小民, 王振

数学物理学报(英文版). 2011 (6): 2313-2322. DOI: 10.1016/S0252-9602(11)60403-7

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In this paper we consider the Riemann problem for the nonlinear degenerate wave equations. This problem has been studied by Sun and Sheng, however the so-called degenerate shock solutions did not satisfy the R-H condition. In the present paper, the Riemann solutions of twelve regions in the v -u plane are completely constructed by the Liu-entropy condition. Our Riemann solutions are very different to that one obtained by Sun and Sheng in some regions.

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EFFECTIVE DIFFUSION AND EFFECTIVE DRAG COEFFICIENT OF A BROWNIAN PARTICLE IN A PERIODIC POTENTIAL

Hongyun Wang

数学物理学报(英文版). 2011 (6): 2323-2342. DOI: 10.1016/S0252-9602(11)60404-9

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We study the stochastic motion of a Brownian particle driven by a constant force over a static periodic potential. We show that both the effective diffusion and the effective drag coefficient are mathematically well-defined and we derive analytic expressions for these two quantities. We then investigate the asymptotic behaviors of the effective diffusion and the effective drag coefficient, respectively, for small driving force and for large driving force. In the case of small driving force, the effective diffusion is reduced from its Brownian value by a factor that increases exponentially with the amplitude of the potential. The effective drag coefficient is increased by approximately the same factor. As a result, the Einstein relation between the diffusion coefficient and the drag coefficient is approximately valid when the driving force is small. For moderately large driving force, both the effective diffusion and the effective drag coefficient are increased from their Brownian values, and the Einstein relation breaks down. In the limit of very large driving force, both the effective diffusion and the effective drag coefficient converge to their Brownian values and the Einstein relation is once again valid.

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ON THE MODIFIED NONLINEAR SCHRÖDINGER EQUATION IN THE SEMICLASSICAL LIMIT: SUPERSONIC, SUBSONIC, AND TRANSSONIC BEHAVIOR

Jeffery C.DiFranco, Peter D.Miller, Benson K.Muite

数学物理学报(英文版). 2011 (6): 2343-2377. DOI: 10.1016/S0252-9602(11)60405-0

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The purpose of this paper is to present a comparison between the modified nonlinear Schr¨odinger (MNLS) equation and the focusing and defocusing variants of the (unmodified) nonlinear Schr¨odinger (NLS) equation in the semiclassical limit. We describe aspects of the limiting dynamics and discuss how the nature of the dynamics is evident theoretically through inverse-scattering and noncommutative steepest descent methods. The main message is that, depending on initial data, the MNLS equation can behave either like the defocusing NLS equation, like the focusing NLS equation (in both cases the analogy is asymptotically accurate in the semiclassical limit when the NLS equation is posed with appropriately modified initial data), or like an interesting mixture of the two. In the latter case, we identify a feature of the dynamics analogous to a sonic line in gas dynamics, a free boundary separating subsonic flow from supersonic flow.

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A NONLINEAR LAVRENTIEV-BITSADZE MIXED TYPE EQUATION

陈恕行

数学物理学报(英文版). 2011 (6): 2378-2388. DOI: 10.1016/S0252-9602(11)60406-2

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In this paper the Tricomi problem for a nonlinear mixed type equation is studied. The coefficients of themixed type equation are discontinuous on the line, where the equation changes its type. The existence of solution to this problem is proved. The method
developed in this paper can be applied to study more difficult problems for nonlinear mixed type equations arising in gas dynamics.

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