Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type

We establish the existence and stability of multidimensional transonic shocks for the Euler equations for steady potential compressible fluids. The Euler equations, consisting of the conservation law of mass and the Bernoulli law for the velocity, can be written as a second-order, nonlinear equation of mixed elliptic-hyperbolic type for the velocity potential. The transonic shock problem can be formulated into the following free boundary problem: The free boundary is the location of the transonic shock which divides the two regions of smooth flow, and the equation is hyperbolic in the upstream region where the smooth perturbed flow is supersonic. We develop a nonlinear approach to deal with such a free boundary problem in order to solve the transonic shock problem. Our results indicate that there exists a unique solution of the free boundary problem such that the equation is always elliptic in the downstream region and the free boundary is smooth, provided that the hyperbolic phase is close to a uniform flow. We prove that the free boundary is stable under the steady perturbation of the hyperbolic phase. We also establish the existence and stability of multidimensional transonic shocks near spherical or circular transonic shocks.

     

Computation of efficient solutions of discretely distributed stochastic optimization problems

In engineering and economics often a certain vectorx of inputs or decisions must be chosen, subject to some constraints, such that the expected costs (or loss) arising from the deviation between the outputA() x of a stochastic linear systemxA()x and a desired stochastic target vectorb() are minimal. Hence, one has the following stochastic linear optimization problem minimizeF(x)=Eu(A()x b()) s.t.xD, (1) whereu is a convex loss function on ^m , (A(), b()) is a random (m,n + 1)-matrix, E denotes the expectation operator andD is a convex subset of ⁿ . Concrete problems of this type are e.g. stochastic linear programs with recourse, error minimization and optimal design problems, acid rain abatement methods, problems in scenario analysis and non-least square regression analysis.Solving (1), the loss functionu should be exactly known. However, in practice mostly there is some uncertainty in assigning appropriate penalty costs to the deviation between the outputA ()x and the targetb(). For finding in this situation solutions hedging against uncertainty a set of so-called efficient points of (1) is defined and a numerical procedure for determining these compromise solutions is derived. Several applications are discussed.     

Asymptotic limit for condensate solutions in the Abelian Chern-Simons Higgs model II

In this paper we show that the maximal condensate solutions in the Abelian Chern-Simons Higgs model converge to in higher norms, where is a harmonic map.

     

Synergistic effects in mixed wormlike micelles of dimeric and single-chain cationic surfactants at high ionic strength

The mixed micellization between the cationic gemini surfactant [ C₁₂H₂₅( CH₃)₂N⁺( C₂H₄) N⁺( CH₃)₂ C₁₂H₂₅•2Br^-] and the cationic cetyltrimethylammonium bromide (CTAB) in 150 mM KBr solutions has been investigated. The variation of the cmc of the mixtures, measured by surface tension experiments, with composition revealed synergism in micelle formation. T-Jump and light scattering experiments performed in the vicinity of the crossover volume fraction showed the existence of two micellar populations, possibly linear and toroidal micelles. Rheological and dynamic light scattering experiments allowed to fully characterize the linear viscoelasticity of the mixtures. These measurements revealed synergistic gains in viscoelastic properties with a maximum of the stress-relaxation time around the equimolar composition. These effects are ascribed to a progressive intermicellar crosslinking resulting from a continuous increase of the end-cap energy with the 12-2-12 content in the mixture. Received: 18 November 2002 / Accepted: 8 April 2003 / Published online: 27 May 2003 RID="a" ID="a"e-mail: candau@fresnel.u-strasbg.fr     

Pearling instabilities in water-dispersed copolymer cylinders with charged brushes

We investigate the structural behavior of a poly(styrene)-block-poly(acrylic acid) diblock copolymer which forms hexagonally-packed PS cylinders (C-phase) in the melt state. The water dispersion of this structure provides hairy cylinders which comprise a PAA swollen cylindrical brush with a height h tunable via its degree of ionization and the ionic strength in the solution, and a water-free, PS cylindrical core of constant radius R_C. Such system constitutes an out-of-equilibrium frustrated model system: the selective swelling of the PAA brush results in a frustration of the interface curvature, which the ratio h/R_C allows to quantify. Upon heating at a temperature higher than the glass transition temperature of the PS core, the glassiness of the core is relieved and the mechanical constraints arising from the selective swelling of the structure can be relaxed: the cylinders undergo a cylinder-to-sphere transition upon annealing at high temperature, when above a frustration threshold h/R_C 1.8. Thanks to a careful mapping of the transition diagram, an undulating cylindrical morphology (UC) is identified between unchanged cylinders ( h/R_C 1.8) and spheres ( h/R_C 2.0), which appears to result from a Rayleigh-like pearling instability of the copolymer cylinders.     

Equilibrium points of the tilted perfect fluid Bianchi VI<Subscript><Emphasis Type="Italic">h</Emphasis></Subscript> state space

We present the full set of evolution equations for the spatially homogeneous cosmologies of type VI_h filled with a tilted perfect fluid and we provide the corresponding equilibrium points of the resulting dynamical state space. It is found that only when the group parameter satisfies h > –1 a self-similar solution exists. In particular we show that for h > – there exists a self-similar equilibrium point provided that whereas for h < – the state parameter belongs to the interval (1, . This family of new exact self-similar solutions belongs to the subclass n = 0 having non-zero vorticity. In both cases the equilibrium points have a six-dimensional stable manifold and may act as future attractors at least for the models satisfying n = 0. Also we give the exact form of the self-similar metrics in terms of the state and group parameter. As an illustrative example we provide the explicit form of the corresponding self-similar radiation model ( = ), parametrised by the group parameter h. Finally we show that there are no tilted self-similar models of type III and irrotational models of type VI_h.     

轴对称薄圆环壳方程级数解收敛性的研究

本文研究轴对称薄圆环壳齐次方程级数解的收敛性,证明了每种环壳都有在全域上收敛的级数解。     

On the existence of self-similar solutions of the equations governing unsteady flow through a porous medium

In this paper the conditions for the existence of self-similar solutions of the equations governing unsteady flows through a porous medium are presented and discussed. The first two sections deal with the case of non-Newtonian fluids of power-law behavior; the third section analyzes the case of non-Darcy gas flows. The boundary and initial conditions occuring currently in a large class of fluid mechanics problems, of practical interest in engineering, are considered.     

Positons of the modified Korteweg de Vries equation

The concept of positons, i.e. certain multiparametric solutions of the Korteweg de Vries equation with new properties, is extended to the modified Korteweg de Vries equation. It is shown that the essential features of positons carry over to this case; the collision of positons, the solitary-wave-positon interaction and simple generalizations are discussed in detail. Suggestions for future research and possible applications of the present work are sketched.     

修正的BBM方程新的精确解

应用进一步修正的简单方程法对修正的 Benjamin -Bona -Mahoney (mBBM )方程进行求解,给出了mBBM方程新的精确类孤波解,取定某些参数值,便可得到精确孤波解.这种方法也可用于寻找其它常系数以及变系数非线性发展方程(组)的精确解,具有一定的普适性.