On the Euler and Jacobi differential equations for variational problems with impulse parameters

We study the differential equation −(pu′)′ + qu = f with generalized coefficients for the case in which it is realized in the form of the Euler equation or the Jacobi equation for a variational problem with impulse parameters.     

Families of traveling impulses and fronts in some models with cross-diffusion

An analysis of traveling wave solutions of partial differential equation (PDE) systems with cross-diffusion is presented. The systems under study fall in a general class of the classical Keller–Segel models to describe chemotaxis. The analysis is conducted using the theory of the phase plane analysis of the corresponding wave systems without a priory restrictions on the boundary conditions of the initial PDE. Special attention is paid to families of traveling wave solutions. Conditions for existence of front–impulse, impulse–front, and front–front traveling wave solutions are formulated. In particular, the simplest mathematical model is presented that has an impulse–impulse solution; we also show that a non-isolated singular point in the ordinary differential equation (ODE) wave system implies existence of free-boundary fronts. The results can be used for construction and analysis of different mathematical models describing systems with chemotaxis.     

Exact Rate of Convergence of Some Approximation Schemes Associated to SDEs Driven by a Fractional Brownian Motion

In this article, we derive the exact rate of convergence of some approximation schemes associated to scalar stochastic differential equations driven by a fractional Brownian motion with Hurst index H. We consider two cases. If H>1/2, the exact rate of convergence of the Euler scheme is determined. We show that the error of the Euler scheme converges almost surely to a random variable, which in particular depends on the Malliavin derivative of the solution. This result extends those contained in J. Complex. 22(4), 459–474, 2006 and C.R. Acad. Sci. Paris, Ser. I 340(8), 611–614, 2005. When 1/6<H<1/2, the exact rate of convergence of the Crank-Nicholson scheme is determined for a particular equation. Here we show convergence in law of the error to a random variable, which depends on the solution of the equation and an independent Gaussian random variable.     

Conditional Viability for Impulse Differential Games

We introduce in this paper the concept of “impulse evolutionary game”. Examples of evolutionary games are usual differential games, differentiable games with history (path-dependent differential games), mutational differential games, etc. Impulse evolutionary systems and games cover in particular “hybrid systems” as well as “qualitative systems”. The conditional viability kernel of a constrained set (with a target) is the set of initial states such that for all strategies (regarded as continuous feedbacks) played by the second player, there exists a strategy of the first player such that the associated run starting from this initial state satisfies the constraints until it hits the target. This paper characterizes the concept of conditional viability kernel for “qualitative games” and of conditional valuation function for “qualitative games” maximinimizing an intertemporal criterion. The theorems obtained so far about viability/capturability issues for evolutionary systems, conditional viability for differential games and about impulse and hybrid systems are used to provide characterizations of conditional viability under impulse evolutionary games.     

The number of rooted Eulerian planar maps

In this paper we provide a solution of the functional equation unsolved in the paper, by the second author, "On functional equations arising from map enumerations" that appeared in Discrete Math, 123: 93-109 (1993). It is also the number of combinatorial distinct rooted general eulerian planar maps with the valency of root-vertex, the number of non-root vertices and non-root faces of the maps as three parameters. In particular, a result in the paper, by the same author, "On the number of eulerian planar map...     

Approximation of potential integral by radial bases for solutions of Helmholtz equation

For a Helmholtz equation Δu(x) + κ ² u(x) = f(x) in a region of R ^s , s ≥ 2, where Δ is the Laplace operator and κ = a + ib is a complex number with b ≥ 0, a particular solution is given by a potential integral. In this paper the potential integral is approximated by using radial bases with the order of approximation derived.      

Stability estimates for the solution to the problem of determining the kernel of a viscoelastic equation

For an integrodifferential equation corresponding to a two-dimensional viscoelastic problem, we study the problem of defining the spatial part of the kernel involved in the integral term of the equation. The support of the sought function is assumed to belong to a compact domain Ω. As information for solving this inverse problem, the traces of the solution to the direct Cauchy problem and its normal derivative are given for some finite time interval on the boundary of Ω. An important feature in the statement of the problem is the fact that the solution of the direct problem corresponds to the zero initial data and a force impulse in time localized on a fixed straight line disjoint with Ω. The main result of the article consists in obtaining a Lipschitz estimate for the conditional stability of the solution to the inverse problem under consideration.     

Monotone perturbations of the laplacian inL 1(R N )

The semilinear perturbation of Poisson’s equation (E): −Δu+β(u)∋f, where β is a maximal monotone graph inR, has been investigated by Ph. Bénilan, H. Brézis and M. Crandall forf∈L ¹(R ^N ),N≧1, under the assumptions 0∈β(0) ifN≧3 and 0∈β(0) ∩ Int β(R) ifN=1,2. We discuss in this paper the solvability and well-posedness of (E) in terms of any maximal monotone graph β. In particular, if β takes only positive values andN≧3 we prove that no solution exists; ifN=2 we give necessary and sufficient conditions on β andf for (E) to be solvable in a natural sense.     

The Value Function of Singularly Perturbed Control Systems

The limit as ɛ→ 0 of the value function of a singularly perturbed optimal control problem is characterized. Under general conditions it is shown that limit value functions exist and solve in a viscosity sense a Hamilton—Jacobi equation. The Hamiltonian of this equation is generated by an infinite horizon optimization on the fast time scale. In particular, the limit Hamiltonian and the limit Hamilton—Jacobi equation are applicable in cases where the reduction of order, namely setting ɛ = 0 , does not yield an optimal behavior. Accepted 18 November 1999     

Oscillatory Periodic Solutions of Nonlinear Second Order Ordinary Differential Equations

In this paper the existence results of oscillatory periodic solutions are obtained for a second order ordinary differential equation -u″（t） = f（t, u（t））, where f ： R^2 → R is a continuous odd function and is 2π-periodic in t. The discussion is based on the fixed point index theory in cones.     

Global existence of strong solutions to the one-dimensional full model for phase transitions in thermoviscoelastic materials

This paper is devoted to the analysis of a one-dimensional model for phase transition phenomena in thermoviscoelastic materials. The corresponding parabolic-hyperbolic PDE system features a strongly nonlinear internal energy balance equation, governing the evolution of the absolute temperature ϑ, an evolution equation for the phase change parameter χ, including constraints on the phase variable, and a hyperbolic stress-strain relation for the displacement variable u. The main novelty of the model is that the equations for χ and u are coupled in such a way as to take into account the fact that the properties of the viscous and of the elastic parts influence the phase transition phenomenon in different ways. However, this brings about an elliptic degeneracy in the equation for u which needs to be carefully handled. First, we prove a global well-posedness result for the related initial-boundary value problem. Secondly, we address the long-time behavior of the solutions in a simplified situation. We prove that the ω-limit set of the solution trajectories is nonempty, connected and compact in a suitable topology, and that its elements solve the steady state system associated with the evolution problem. Dedicated to Jürgen Sprekels on the occasion of his 60th birthday     

A Pair of Explicitly Solvable Singular Stochastic Control Problems

We consider a general model of singular stochastic control with infinite time horizon and we prove a ``verification theorem' under the assumption that the Hamilton—Jacobi—Bellman (HJB) equation has a C <sup>2</sup> solution. In the one-dimensional case, under the assumption that the HJB equation has a solution in W <sub>loc</sub> <sup>2,p</sup>(R) with , we prove a very general ``verification theorem' by employing the generalized Meyer—Ito change of variables formula with local times. In what follows, we consider two special cases which we explicitly solve. These are the formal equivalent of the one-dimensional infinite time horizon LQG problem and a simple example with radial symmetry in an arbitrary Euclidean space. The value function of either of these problems is C ² and is expressed in terms of special functions, and, in particular, the confluent hypergeometric function and the modified Bessel function of the first kind, respectively. Accepted 21 February 1997     

Stochastic 2-D Navier—Stokes Equation

   Abstract. In this paper we prove the existence and uniqueness of strong solutions for the stochastic Navier—Stokes equation in bounded and unbounded domains. These solutions are stochastic analogs of the classical Lions—Prodi solutions to the deterministic Navier—Stokes equation. Local monotonicity of the nonlinearity is exploited to obtain the solutions in a given probability space and this significantly improves the earlier techniques for obtaining strong solutions, which depended on pathwise solutions to the Navier—Stokes martingale problem where the probability space is also obtained as a part of the solution.     

A fast matrix–vector multiplication method for solving the radiosity equation

A “fast matrix–vector multiplication method” is proposed for iteratively solving discretizations of the radiosity equation (I — К)u = E. The method is illustrated by applying it to a discretization based on the centroid collocation method. A convergence analysis is given for this discretization, yielding a discretized linear system (I — K _n )u _n = E _n. The main contribution of the paper is the presentation of a fast method for evaluating multiplications K_n v, avoiding the need to evaluate K_n explicitly and using fewer than O(n ²) operations. A detailed numerical example concludes the paper, and it illustrates that there is a large speedup when compared to a direct approach to discretization and solution of the radiosity equation. The paper is restricted to the surface S being unoccluded, a restriction to be removed in a later paper. This revised version was published online in June 2006 with corrections to the Cover Date.     

Solution of the spectral problem for the curl and Stokes operators with periodic boundary conditions

In this paper, we establish relations between eigenvalues and eigenfunctions of the curl operator and Stokes operator (with periodic boundary conditions). These relations show that the curl operator is the square root of the Stokes operator with ν = 1. The multiplicity of the zero eigenvalue of the curl operator is infinite. The space L ₂(Q, 2π) is decomposed into a direct sum of eigenspaces of the operator curl. For any complex number λ, the equation rot u + λu = f and the Stokes equation −ν(Δv + λ ²v) + ∇p = f, div v = 0, are solved. Bibliography: 15 titles. Dedicated to the memory of Olga Aleksandrovna Ladyzhenskaya __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 318, 2004, pp. 246–276.     

Attractors and Long Time Behavior of von Karman Thermoelastic Plates

This paper undertakes a study of asymptotic behavior of solutions corresponding to von Karman thermoelastic plates. A distinct feature of the work is that the model considered has no added dissipation—particularly mechanical dissipation typically added to plate equation when long time-behavior is considered. Thus, the model consists of undamped oscillatory plate equation strongly coupled with heat equation. Nevertheless we are able to show that the ultimate (asymptotic) behavior of the von Karman evolution is described by finite dimensional global attractor. In addition, the obtained estimate for the dimension and the size of the attractor are independent of the rotational inertia parameter γ and heat/thermal capacity κ, where the former is known to change the character of dynamics from hyperbolic (γ>0) to parabolic like (γ=0). Other properties of attractors such as additional smoothness and upper-semicontinuity with respect to parameters γ and κ are also established. The main ingredients of the proofs are (i) sharp regularity of Airy’s stress function, and (ii) newly developed (Chueshov and Lasiecka in Memoirs of AMS, in press) “compensated” compactness methods applicable to non-compact dynamics. I. Lasiecka’s research partially supported by the NSF Grant DMS-0104305 and ARO Grant DAAD19-02-10179.     

Sharp existence results for self-similar solutions of semilinear wave equations

The existence of self-similar and asymptotically self-similar solutions of the nonlinear wave equation with or in R ³×R ⁺ for small Cauchy data is proven if . A counterexample is given which shows that the lower bound on α is sharp. Received April 1999 – Accepted September 1999     

Un Nouveau ℓ (K) Qui Possede la Propriete de Grothendieck

Using the continuum hypothesis, we construct a compact spaceK such that ℓ(K) possesses the Grothendieck property, but such that the unit ball of ℓ(K)′ does not containβ N, and hence, in particular, such thatl ^∞(N) is neither a subspace nor quotient of ℓ(K). In particular,K does not contain a convergent sequence but does not containβ N.      

Explicit form of characteristic numbers of a periodic problem

A periodic problem for a linear differential equation of the second order is reduced to a periodic problem for a differential equation of the first order, but with deviation argument. We indicate the cases when the characteristic numbers are determined explicitly. This paper is the continuation of investigations commenced in “Differents. Uravneniya,” 44 (4) (2008).