Measure-valued Solutions of the Euler Equations for Ideal Compressible Polytropic Fluids

The existence of global measure-valued solutions to the Euler equations describing the motion of an ideal compressible and heat conducting fluid is proved. The motion is considered in a bounded domain Ω⊂ℝ³ with impermeable boundary. The solution is a limit of an approximate solution obtained by adding the sixth-order elliptic operator in the equation of momentum.     

Small vibrations of an elastic conductor in a magnetic field

In this paper we study the motion of an elastic conducting wire in a magnetic field. The motion of the conductor induces a current in the wire (Faraday's law) which, in turn produces a force on the wire. We consider the linear equation obtained by linearizing the resulting equations of motion about an equilibrium solution. This is a hyperbolic partial differential equation with a non-local term. We prove existence and uniqueness of a weak solution of an initial–boundary value problem for this equation. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

The transversal homoclinic solutions and chaos for stochastic ordinary differential equations

We consider the persistence of a transversal homoclinic solution and chaotic motion for ordinary differential equations with a homoclinic solution to a hyperbolic equilibrium under an unbounded random forcing driven by a Brownian force. By Lyapunov–Schmidt reduction, the persistence of transversal homoclinic solution is reduced to find the zeros of some bifurcation functions defined between two finite spaces. It is shown that, for almost all sample paths of the Brownian motion, the perturbed system exhibits chaos.     

Propagation of disturbances in a flexible extensible filament under longitudinal accelerated motions in a fluid

We seek a solution of the linearized equation of motion of a flexible extensible filament in a fluid in the form of an expansion in eigenfunctions of a boundary-value problem. For a uniformly accelerated motion and for motion accelerated according to a hyperbolic tangent law we find the exact solutions. For other forms of accelerated motion we propose a numerical solution of the initial inhomogeneous problem. We carry out an analysis of the solutions obtained. It is found that the first peak of the tension depends only weakly on the resistance of the fluid, but strongly on the acceleration parameters. The natural vibrations damp out more rapidly both as the resistance increases and as the acceleration increases. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 24, 1993, pp. 104–110.     

Asymptotic distributions of solutions of ordinary differential equations with wide band noise inputs: approximate invariant measures

Systems of Wick stochastic differential equations are studied. Using an estimate on the Wick product we apply Picard iteration to prove a general existence and uniqueness theorem for systems of Wick stochastic differential equations. We also show the solution is stable with respect to perturbations of the noise. This result is used to show that the solution of a linear system of Wick stochastic differential equations driven by smoothed Brownian motion tends to the solution of the corresponding It equation as the smoothed process tends to Brownian motion     

Shell dynamics in the presence of a central body

We construct an idealized spherically symmetric relativistic model of an exploding object in the framework of the theory of surface layers in general relativity and match a Vaidya solution for a radially radiating star to another Vaidya solution through a thin spherical shell. We reduce the equations of motion and the radiation density of the Vaidya solution given by the matching conditions to a first-order system and analyze the general characteristics of the motion. We use a post-Newtonian approximation to find the equation of motion of a spherically symmetric radiating shell moving in a central gravitational potential.     

On the slow motion of a self-propelled rigid body in a viscous incompressible fluid

In this paper we study the Stokes approximation of the self-propelled motion of a rigid body in a viscous liquid that fills all the three-dimensional space exterior to the body. We prove the existence and uniqueness of strong solution to the coupled systems of equations describing the motion of the system body-liquid, for any time and any regular distribution of velocity on the boundary of the body. For the corresponding stationary problem we derive L^p-estimates for the solution in terms of the data. Finally, we prove that every steady solution is attainable as the limit, when t→∞, of an unsteady self-propelled solution which starts from rest.     

Motion planning for a piezo-actuated structure modeling a wingsail

This contribution presents the modeling and the motion planning of a flexible interconnected beam structure representing a wingsail. The structure is equipped with spatially distributed embedded actuators. The solution of the motion planning provides a feedforward control law to realize a desired spatial-temporal out-of-plane deflection trajectory. For this, a systematic flatness-based methodology is proposed, that allows for an efficient numerical solution exploiting, e.g., finite element approximations. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the mechanics of helical flows in an ideal incompressible nonviscous continuous medium

We find a general solution to the problem on the motion in an incompressible continuous medium occupying at any time a whole domain D ? R ³ under the conditions that D is an axially symmetric cylinder and the motion is described by the Euler equation together with the continuity equation for an incompressible medium and belongs to the class of helical flows (according to I.S. Gromeka’s terminology), in which sreamlines coincide with vortex lines. This class is constructed by the method of transformation of the geometric structure of a vector field. The solution is characterized in Theorem 2 in the end of the paper.     

Hermite G 1 rational spline motion of degree six

Applying geometric interpolation techniques to motion construction has many advantages, e.g., the parameterization is chosen automatically and the obtained rational motion is of the lowest possible degree. In this paper a G ¹ Hermite rational spline motion of degree six is presented. An explicit solution of nonlinear equations that determine the spherical part of the motion is derived. Particular emphasis is placed on the construction of the translational part of the motion. Since the center trajectory is a G ¹ continuous for an arbitrary choice of lengths of tangent vectors, additional free parameters are obtained, which are used to minimize particular energy functionals. Thenumerical examples provide an evidence that the obtained motions have nice shapes.     

On the potential in non‐Gaussian chain polymer models

In this paper, we investigate the potential for a class of non‐Gaussian processes so‐called generalized grey Brownian motion. We obtain a closed analytic form for the potential as an integral of the M‐Wright functions and the Green function. In particular, we recover the special cases of Brownian motion and fractional Brownian motion. In addition, we give the connection to a fractional partial differential equation and its the fundamental solution.     

Continuous conjugation of special nonisentropic one-dimensional gas motions

The exact partially invariant solution of equations of motion of a compressible fluid describing the collapse of particles to a point and an instantaneous source from the point in a one-dimensional nonisentropic motion is cut off by the characteristics and glued into a continuous solution of a one-dimensional submodel in a finite domain. The possibility of a continuous periodic nonisentropic motion of a compressible fluid in a bounded domain under the action of a piston is shown.     

Corrigendum to “Third-order partial differential equations arising in the impulsive motion of a flat plate” [Commun Nonlinear Sci Numer Simulat 2009;14:2629–36]

We correct an oversight in our recently published paper Van Gorder and Vajravelu [Third-order partial differential equations arising in the impulsive motion of a flat plate. Commun Nonlinear Sci Numer Simulat 2009;14:2629–36], regarding the correct form of the boundary condition in Stokes’ first problem for the impulsive motion of an infinite flat plate. We present the corrected solution to the flow problem.     

Convergence of delay differential equations driven by fractional Brownian motion

In this note, we prove an existence and uniqueness result of solution for stochastic differential delay equations with hereditary drift driven by a fractional Brownian motion with Hurst parameter H > 1/2. Then, we show that, when the delay goes to zero, the solutions to these equations converge, almost surely and in L ^p , to the solution for the equation without delay. The stochastic integral with respect to the fractional Brownian motion is a pathwise Riemann–Stieltjes integral.     

Reflected BSDE Driven by a Lévy Process

In this paper, we study the reflected solution of one-dimensional backward stochastic differential equation driven by Teugels martingales and an independent Brownian motion. We prove the existence and uniqueness of the solution using a penalization method combined with Snell envelope theory.      

Pathwise uniqueness for perturbed versions of Brownian motion and reflected Brownian motion

Any solution of the functional equation

The self-propulsion of a body with moving internal masses in a viscous fluid

An investigation of the characteristics of motion of a rigid body with variable internal mass distribution in a viscous fluid is carried out on the basis of a joint numerical solution of the Navier — Stokes equations and equations of motion for a rigid body. A nonstationary three-dimensional solution to the problem is found. The motion of a sphere and a drop-shaped body in a viscous fluid in a gravitational field, which is caused by the motion of internal material points, is explored. The possibility of self-propulsion of a body in an arbitrary given direction is shown.     

Brownian fluctuations in space-time with applications to vibrations of rods

In this paper Brownian fluctuations in space-time are considered. Time is assumed to run alternately forward and backward, the alternance being marked by a Poisson process with rate λ. It is shown that the law of this motion is a solution of a fourth-order partial differential equation. Furthermore the law of this movement in the presence of an absorbing barrier is derived. The equation ruling the movement analysed, when λ = 0 and is submitted to the change t' = −it, reduces to the equation of vibrations of rods. This fact is exploited to obtain the solution of boundary value problems concerning the equation of vibrating beams by means of Brownian motion techniques.     

Théorème limite pour une équation différentielle à coefficient aléatoire à mémoire longue

We consider an ordinary differential equation depending on a small parameter and with a long-range random coefficient. We establish that the solution of this ordinary differential equation converges to the solution of a stochastic differential equation driven by a fractional Brownian motion. The index of the fractional Brownian motion depends on the asymptotic behavior of the covariance function of the random coefficient. The proof of the convergence uses the T. Lyons theory of “rough paths”. To cite this article: R. Marty, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Asymptotic stability and associated problems of dynamics of falling rigid body

We consider two problems from the rigid body dynamics and use new methods of stability and asymptotic behavior analysis for their solution. The first problem deals with motion of a rigid body in an unbounded volume of ideal fluid with zero vorticity. The second problem, having similar asymptotic behavior, is concerned with motion of a sleigh on an inclined plane. The equations of motion for the second problem are non-holonomic and exhibit some new features not typical for Hamiltonian systems. A comprehensive survey of references is given and new problems connected with falling motion of heavy bodies in fluid are proposed.