On a decay rate for 1D-viscous compressible barotropic fluid equations

The Navier-Stokes equations of a compressible barotropic fluid in 1D with zero velocity boundary conditions are considered. We study the case of large initial data in H ¹ as well as the mass force such that the stationary density is positive. The uniform lower bound for the density is proved. By constructing suitable Lyapunov functionals, decay rate estimates in L ²-norm and H ¹-norm are given. The decay rate is exponential if so the decay rate of the nonstationary part of the mass force is. The results are proved in the Eulerian coordinates for a wide class of increasing state functions including with any γ > 0 as well as functions of arbitrarily fast growth. We also extend the results for equations of a multicomponent compressible barotropic mixture (in the absence of chemical reactions). Received December 20, 2000; accepted February 27, 2001.     

On the regularity of the distribution of the maximum of one-parameter Gaussian processes

The main result in this paper states that if a one-parameter Gaussian process has C ^2k paths and satisfies a non-degeneracy condition, then the distribution of its maximum on a compact interval is of class C ^k . The methods leading to this theorem permit also to give bounds on the successive derivatives of the distribution of the maximum and to study their asymptotic behaviour as the level tends to infinity. Received: 14 May 1999 / Revised version: 18 October 1999 / Published online: 14 December 2000     

Lipschitz continuity of quantile functions on spaces of random variables

We prove that quantile functions on spaces of random variables satisfy the Lipschitz condition with constant 1 with respect to any norm on a subspace of a space of random variables that majorizes L^∞-norm. The considered random variables not necessarily belong to this Banach space. Bibliography: 3 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 351–358, 2008, pp. 253–258.     

Fractional integrals of periodic functions of several variables

In this paper it has been systematically studied the imbedding properties of fractional integral operators of periodic functions of several variables, and isomorphic properties of fractional integral operators in the spaces of Lipschitz continuous functions. It has also been proved that the space of fractional integration, the space of Lipschitz continuous functions and the Sobolev space are identical in L²-norm. Results obtained here are not true for fractional integrals (or Riesz potentials) in ℝ ⁿ . Supported by NNSFC     

On the uniqueness of time-periodic solutions to the Navier–Stokes equations in unbounded domains

We present a uniqueness theorem for time-periodic solutions to the Navier–Stokes equations in unbounded domains. Thus far, results on the uniqueness of time-periodic solutions to the Navier–Stokes equations in unbounded domain, roughly speaking, have only found that a small time-periodic L ⁿ -solution is unique within the class of solutions which have sufficiently small L ^∞(L ⁿ )-norm. In this paper, we show that a small time-periodic L ⁿ -solution is unique within the class of all time-periodic L ⁿ -solutions, which contains large solutions. We also consider the uniqueness of solutions in weak-L ⁿ space. The proof of the present uniqueness theorem is based on the method of dual equations.      

Global stability of solutions with discontinuous initial data containing vacuum states for the isentropic Euler equations

The global stability of Lipschitz continuous solutions with discontinuous initial data is established in a broad class of entropy solutions in L^￥L^\infty containing vacuum states. In particular, the uniqueness of Lipschitz solutions with discontinuous initial data is obtained in the broad class of entropy solutions in L^￥L^\infty .     

On logarithmic Sobolev inequalities for continuous time random walks on graphs

We establish modified logarithmic Sobolev inequalities for the path distributions of some continuous time random walks on graphs, including the simple examples of the discrete cube and the lattice ZZ ^d . Our approach is based on the Malliavin calculus on Poisson spaces developed by J. Picard and stochastic calculus. The inequalities we prove are well adapted to describe the tail behaviour of various functionals such as the graph distance in this setting. Received: 6 April 1998 / Revised version: 15 March 1999 / Published on line: 14 February 2000     

A Maximum Principle for an Explicit Finite Difference Scheme Approximating the Hodgkin-Huxley Model

We analyze an explicit finite difference scheme for the general form of the Hodgkin-Huxley model, which is a nonlinear partial differential equation coupled to a set of ODEs. The system of equations describes propagation of an electrical signal in excitable cells. We prove that the numerical solution is bounded in the L^∞-norm and L² converges to a unique solution. The L^∞-bound, which is the key point of our analysis, is proved by showing that the discrete solutions are invariant in a physically relevant bounded region. For the convergence proof we use the compactness method. AMS subject classification (2000) 65F20     

An Ito formula for domain-valued processes driven by stochastic flows

 We consider a natural class of stochastic processes taking values in the space of smoothly bounded domains in ℝ ⁿ with compact closure. These processes are generated by stochastic flows on ℝ ⁿ which are obtained as the solutions of stochastic differential equations on ℝ ⁿ . We establish an Ito formula for smooth domain functionals, applied to processes in this class. Received: 2 March 2001 / Revised version: 10 January 2002 / Published online: 22 August 2002     

Stable windings on hyperbolic surfaces

Let ℳ be a geometrically finite hyperbolic surface with infinite volume, having at least one cusp. We obtain the limit law under the Patterson-Sullivan measure on T ¹ℳ of the windings of the geodesics of ℳ around the cusps. This limit law is stable with parameter 2δ− 1, where δ is the Hausdorff dimension of the limit set of the subgroup Γ of M?bius isometries associated with ℳ. The normalization is t ^{−1/(2δ−1)}, for geodesics of length t. Our method relies on a precise comparison between geodesics and diffusion paths, for which we need to approach the Patterson-Sullivan measure mentioned above by measures that are regular along the stable leaves. Received: 8 October 1999 / Revised version: 2 June 2000 / Published online: 21 December 2000     

On the existence of square-integrable solutions for systems with weak nonlinearity

The paper considers the problem of structural stability of systems under disturbance of coefficients having small L ²(ℝ)-norm. We derive conditions which guarantee that for every solution of the perturbed system there exists a solution of the original system which is close to the former in L ²(ℝ)-norm.     

Error estimate for finite volume scheme

We study the convergence of a finite volume scheme for the linear advection equation with a Lipschitz divergence-free speed in R ^d . We prove a h ^1/2-error estimate in the L ^∞(0,t;L ¹)-norm for BV data. This result was expected from numerical experiments and is optimal.     

On the noncommutative residue and the heat trace expansion on conic manifolds

 Given a cone pseudodifferential operator P we give a full asymptotic expansion as t → 0 ⁺ of the trace Tr Pe ^-tA , where A is an elliptic cone differential operator for which the resolvent exists on a suitable region of the complex plane. Our expansion contains log t and new (log t)² terms whose coefficients are given explicitly by means of residue traces. Cone operators are contained in some natural algebras of pseudodifferential operators on which unique trace functionals can be defined. As a consequence of our explicit heat trace expansion, we recover all these trace functionals. Received: 12 November 2001 / Revised version: 26 June 2002 Mathematics Subject Classification (2000): Primary 58J35; Secondary 35C20, 58J42     

The Karp complexity of unstable classes

A class K of structures is controlled if, for all cardinals λ, the relation of L _∞,λ-equivalence partitions K into a set of equivalence classes (as opposed to a proper class). We prove that the class of doubly transitive linear orders is controlled, while any pseudo-elementary class with the ω-independence property is not controlled. Received: 23 September 1998 / Revised version: 6 July 1999 / Published online: 21 December 2000     

Local boundedness of quasi-minimizers of integral functionals with variable exponent anisotropic growth and applications

The local boundedness of local quasi-minimizers of integral functionals with variable exponent anisotropic ${\overrightarrow{p}(x)}The local boundedness of local quasi-minimizers of integral functionals with variable exponent anisotropic p^?(x){\overrightarrow{p}(x)} growth under suitable assumptions is proved. Based on this result, the global boundedness and the Lipschitz continuity of weak solutions of Dirichlet or Neumann boundary value problems for the p^?(x){\overrightarrow{p}(x)}-Laplace type equations are obtained.     

Maximal regularity via reverse Hölder inequalities for elliptic systems of n-Laplace type involving measures

In this note, we consider the regularity of solutions of the nonlinear elliptic systems of n-Laplacian type involving measures, and prove that the gradients of the solutions are in the weak Lebesgue space L ^n,∞. We also obtain the a priori global and local estimates for the L ^n,∞-norm of the gradients of the solutions without using BMO-estimates. The proofs are based on a new lemma on the higher integrability of functions.     

Averaging of the Dirichlet problem for a special hyperbolic Kirchhoff equation

We prove a statement on the averaging of a hyperbolic initial-boundary-value problem in which the coefficient of the Laplace operator depends on the space L ²-norm of the gradient of the solution. The existence of the solution of this problem was studied by Pokhozhaev. In a space domain in ℝⁿ, n ≥ 3, we consider an arbitrary perforation whose asymptotic behavior in a sense of capacities is described by the Cioranesku-Murat hypothesis. The possibility of averaging is proved under the assumption of certain additional smoothness of the solutions of the limiting hyperbolic problem with a certain stationary capacitory potential. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 2, pp. 236–249, February, 2006.     

Super-Brownian motion with reflecting historical paths

We consider super-Brownian motion whose historical paths reflect from each other, unlike those of the usual historical super-Brownian motion. We prove tightness for the family of distributions corresponding to a sequence of discrete approximations but we leave the problem of uniqueness of the limit open. We prove a few results about path behavior for processes under any limit distribution. In particular, we show that for any γ > 0, a “typical” increment of a reflecting historical path over a small time interval Δt is not greater than (Δt)^3/4−γ. Received: 16 March 2000 / Revised version: 26 February 2001 / Published online: 9 October 2001     

Convex duality and the Skorokhod Problem. I

The solution to the Skorokhod Problem defines a deterministic mapping, referred to as the Skorokhod Map, that takes unconstrained paths to paths that are confined to live within a given domain G⊂ℝ ⁿ . Given a set of allowed constraint directions for each point of ∂G and a path ψ, the solution to the Skorokhod Problem defines the constrained version φ of ψ, where the constraining force acts along one of the given boundary directions using the “least effort” required to keep φ in G. The Skorokhod Map is one of the main tools used in the analysis and construction of constrained deterministic and stochastic processes. When the Skorokhod Map is sufficiently regular, and in particular when it is Lipschitz continuous on path space, the study of many problems involving these constrained processes is greatly simplified. We focus on the case where the domain G is a convex polyhedron, with a constant and possibly oblique constraint direction specified on each face of G, and with a corresponding cone of constraint directions at the intersection of faces. The main results to date for problems of this type were obtained by Harrison and Reiman [22] using contraction mapping techniques. In this paper we discuss why such techniques are limited to a class of Skorokhod Problems that is a slight generalization of the class originally considered in [22]. We then consider an alternative approach to proving regularity of the Skorokhod Map developed in [13]. In this approach, Lipschitz continuity of the map is proved by showing the existence of a convex set that satisfies a set of conditions defined in terms of the data of the Skorokhod Problem. We first show how the geometric condition of [13] can be reformulated using convex duality. The reformulated condition is much easier to verify and, moreover, allows one to develop a general qualitative theory of the Skorokhod Map. An additional contribution of the paper is a new set of methods for the construction of solutions to the Skorokhod Problem. These methods are applied in the second part of this paper [17] to particular classes of Skorokhod Problems. Received: 17 April 1998 / Revised version: 8 January 1999     

Amalgamation properties and finite models in Ln-theories

 Djordjević [Dj 1] proved that under natural technical assumptions, if a complete L ⁿ -theory is stable and has amalgamation over sets, then it has arbitrarily large finite models. We extend his study and prove the existence of arbitrarily large finite models for classes of models of L ⁿ -theories (maybe omitting types) under weaker amalgamation properties. In particular our analysis covers the case of vector spaces. Received: 24 July 2000 / Published online: 20 December 2001