Min-max game theory and nonstandard differential Riccati equations for abstract hyperbolic-like equations

We consider the abstract dynamical framework of Lasiecka and Triggiani (2000) [1, Chapter 9], which models a large variety of mixed PDE problems (see specific classes in the Introduction) with boundary or point control, all defined on a smooth, bounded domain Ω⊂Rⁿ, n arbitrary. This means that the input → solution map is bounded on natural function spaces. We then study min-max game theory problem over a finite time horizon. The solution is expressed in terms of a (positive, self-adjoint) time-dependent Riccati operator, solution of a non-standard differential Riccati equation, which expresses the optimal qualities in pointwise feedback form. In concrete PDE problems, both control and deterministic disturbance may be applied on the boundary, or as a Dirac measure at a point. The observation operator has some smoothing properties.     

Polynomial and exponential stability for one-dimensional problem in thermoelastic diffusion theory

We consider one-dimensional problem for the thermoelastic diffusion theory and we obtain polynomial decay estimates. Then we show that the solution decays exponentially to zero as time goes to infinity; that is, denoting by E(t) the first-order energy of the system, we show that positive constants C ₀ and c ₀ exist which satisfy E(t) ≤ C ₀ E(0)e ^{?c ₀ t} .     

Exponential stability in the theory of thermoelastic diffusion mixtures

In this article, we investigate the asymptotic behaviour of solutions to the one-dimensional initial-boundary value problem in the linear theory of thermoelastic diffusion mixtures recently developed by Aouadi [M. Aouadi, Qualitative results in the theory of thermoelastic diffusion mixtures, J. Thermal Stresses 33 (2010), pp. 595–615]. Our main result is to establish the necessary and sufficient conditions over the coefficients of the system to get the exponential stability of the corresponding semigroup.     

Existence of optimal controls for the diffusion equation

The existence is considered of a boundary control which drives a system governed by the one-dimensional diffusion equation from the zero state to a given final state, and at the same time minimizes a given functional. The problem is first modified to one in which the minimum is sought of a functional defined on a set of Radon measures. The existence of a minimizing measure is demonstrated, and it is shown that this measure may be approximated by a piecewise constant control. Finally, conditions are given under which a minimizing measurable control exists for the unmodified problem.     

Computation of multispecies diffusion rates

The Fick diffusion law is generally used in combustion theory. A shortcoming of this model is that it violates the momentum conservation law. The hydrodynamic theory of multispecies diffusion is free of this disadvantage. A highly effective method for determining an approximate solution to the multispecies fluid dynamics equations is proposed. The method is comparable to the Fick diffusion model in terms of computational costs.     

Symbolic dynamics and Arnold diffusion

We consider hyperbolic tori of three degrees of freedom initially hyperbolic Hamiltonian systems. We prove that if the stable and unstable manifold of a hyperbolic torus intersect transversaly, then there exists a hyperbolic invariant set near a homoclinic orbit on which the dynamics is conjugated to a Bernoulli shift. The proof is based on a new geometrico-dynamical feature of partially hyperbolic systems, the transversality-torsion phenomenon, which produces complete hyperbolicity from partial hyperbolicity. We deduce the existence of infinitely many hyperbolic periodic orbits near the given torus. The relevance of these results for the instability of near-integrable Hamiltonian systems is then discussed. For a given transition chain, we construct chain of hyperbolic periodic orbits. Then we easily prove the existence of periodic orbits of arbitrarily high period close to such chain using standard results on hyperbolic sets.     

Entropy solution theory for fractional degenerate convection–diffusion equations

We study a class of degenerate convection-diffusion equations with a fractional non-linear diffusion term. This class is a new, but natural, generalization of local degenerate convection-diffusion equations, and include anomalous diffusion equations, fractional conservation laws, fractional porous medium equations, and new fractional degenerate equations as special cases. We define weak entropy solutions and prove well-posedness under weak regularity assumptions on the solutions, e.g. uniqueness is obtained in the class of bounded integrable solutions. Then we introduce a new monotone conservative numerical scheme and prove convergence toward the entropy solution in the class of bounded integrable BV functions. The well-posedness results are then extended to non-local terms based on general Lévy operators, connections to some fully non-linear HJB equations are established, and finally, some numerical experiments are included to give the reader an idea about the qualitative behavior of solutions of these new equations.     

The characteristic finite difference streamline diffusion method for convection-dominated diffusion problems

In this paper, we consider the characteristic finite difference streamline diffusion method for two-dimensional convection-dominated diffusion problems. The scheme is combined the method of characteristics with the finite difference streamline diffusion (FDSD) method to create the characteristic FDSD (C-FDSD) procedures. Stability analysis and error estimate of the C-FDSD method are deduced. The scheme not only realizes the purpose of lowering the time-truncation error, using larger time step for solving the convection-dominated diffusion problems, but also keeps the favorable stability and high precision of the FDSD method. Finally, numerical experiments are presented to illustrate the availability of the scheme.     

Geodesic restrictions for the Casimir operator

We consider the hyperbolic Casimir operator C defined on the tangent sphere bundle SY of a compact hyperbolic Riemann surface Y. We prove a non-trivial bound on the L²-norm of the restriction of eigenfunctions of C to certain natural hypersurfaces in SY. The result that we obtain goes beyond known (sharp) local bounds of L. Hörmander.     

Solutions for a fractional diffusion equation with noninteger dimensions

We investigate a fractional diffusion equation with a nonlocal reaction term by using the Green function approach. We also consider a modified spatial operator in order to cover situations characterized by a noninteger dimension. The results show a nonusual spreading of the initial condition which can be connected to a rich class of anomalous diffusive processes.     

Solutions for a generalized fractional anomalous diffusion equation

In this paper, we investigate the solutions for a generalized fractional diffusion equation that extends some known diffusion equations by taking a spatial time-dependent diffusion coefficient and an external force into account, which subjects to the natural boundaries and the generic initial condition. We obtain explicit analytical expressions for the probability distribution and study the relation between our solutions and those obtained within the maximum entropy principle by using the Tsallis entropy.     

A variable nonlinear splitting algorithm for reaction diffusion systems with self‐ and cross‐ diffusion

Self‐ and cross‐diffusion are important nonlinear spatial derivative terms that are included into biological models of predator–prey interactions. Self‐diffusion models overcrowding effects, while cross‐diffusion incorporates the response of one species in light of the concentration of another. In this paper, a novel nonlinear operator splitting method is presented that directly incorporates both self‐ and cross‐diffusion into a computational efficient design. The numerical analysis guarantees the accuracy and demonstrates appropriate criteria for stability. Numerical experiments display its efficiency and accuracy.     

On the reservoir technique convergence for nonlinear hyperbolic conservation laws - Part I

This paper is devoted to the analysis of flux schemes coupled with the reservoir technique for approximating hyperbolic equations and linear hyperbolic systems of conservation laws [F. Alouges, F. De Vuyst, G. Le Coq, E. Lorin, The reservoir scheme for systems of conservation laws, in: Finite Volumes for Complex Applications, III, Porquerolles, 2002, Lab. Anal. Topol. Probab. CNRS, Marseille, 2002, pp. 247-254 (electronic); F. Alouges, F. De Vuyst, G. Le Coq, E. Lorin, Un procédé de réduction de la diffusion numérique des schémas à différence de flux d'ordre un pour les systèmes hyperboliques non linéaires, C. R. Math. Acad. Sci. Paris 335 (7) (2002) 627-632; F. Alouges, F. De Vuyst, G. Le Coq, E. Lorin, The reservoir technique: A way to make Godunov-type schemes zero or very low diffusive. Application to Colella-Glaz, Eur. J. Mech. B Fluids 27 (6) (2008)]. We prove the long time convergence of the reservoir technique and its TVD property for some specific but still general configurations. Proofs are based on a precise study of the treatment by the reservoir technique of shock and rarefaction waves.     

Refining diffusion approximations for queues

This note illustrates the need to refine diffusion approximations for queues. Diffusion approximations are developed in several different ways for the mean waiting time in a GI/G/1 queue, yielding different results, all of which fail obvious consistency checks with bounds and exact values.     

反应扩散方程的奇摄动

§ 1　 IntroductionThe nonlinearsingularly perturbed problem is a very attractive subjectof study in theinternational academic circles[1 ] .During the past decade many approximate methods havebeen developed and refined,including the method of average,boundary layer method,matched asymptotic expansions,and multiple scales.Recently,many scholars,for example,Bohé[2 ] ,Butuzov and Smurov[3] ,O Malley[4] ,Butuzov,Nefedov and Schneider[5] ,Kelley[6]and so on did a great deal of work.Mo consider…     

Chance and necessity in sociological theory

This paper proposes a mathematical model of financial markets as networks. The model examines the effect of network structure on market behavior (price volatility and trading volume). In the model, investors are arrayed in various network configurations through which they gather information to make trading decisions. The basic network considered is a chain graph with two parameters, number of investors (n) and the length of time in which information is transmitted (k). Closed‐form expressions for price volatility and expected trading volume are provided. The model is generalized to more complex networks, focusing on the hub‐and‐spoke network. The network configurations analyzed do not represent the real (and unknown) communication network among investors, but predictions from the model are consistent with price and volume patterns observed in sociological and economic research on financial markets. The main result is that network structure alone influences price volatility and expected trading volume even though investors are homogeneous and the information introduced into the system is unbiased and random. This result suggests that the structure of the real communication network among investors may influence market behavior.