On the Problem of Evolution of an Isolated Liquid Mass

The paper is concerned with the problem of stability of equilibrium figures of a uniformly rotating, viscous, incompressible, self-gravitating liquid subjected to capillary forces at the boundary. It is shown that a rotationally symmetric equilibrium figure F is exponentially stable if the functional G defined on the set of domains close to F and satisfying the conditions of volume invariance (||=|F|) and the barycenter position attains its minimum for =F. The proof is based on the direct analysis of the corresponding evolution problem with initial data close to the regime of a rigid rotation.     

L q-Estimates for a Solution to the Problem about the Evolution of an Isolated Amount of a Fluid

We construct the L _q-theory of the nonstationary free boundary problem for the Navier–Stokes equations under the action of surface tension forces. Although the main theorem is proved for some finite time interval, it can be used for extending the solution to a larger interval since the smoothness of the solution is preserved for t < 0. Bibliography: 10 titles.     

The Problem on Evolution of a Self-Gravitating Isolated Fluid Mass That is not Subject to the Surface Tension Forces

We establish the solvability of the nonstationary free boundary problem for the Navier–Stokes equations which describes the evolution of a self-gravitating isolated fluid mass that is not subject to the surface tension forces. Bibliography: 12 titles.     

Dissipative Solvability of an Alpha Model of Fluid Flow with Memory

Computational Mathematics and Mathematical Physics - The solvability, in some sense, of the initial–boundary value problem for an alpha model of a fluid with memory is analyzed. Namely, the...     

Integrability of the Problem of the Motion of a Cylinder and a Vortex in an Ideal Fluid

In this paper, we obtain a nonlinear Poisson structure and two first integrals in the problem of the plane motion of a circular cylinder and n point vortices in an ideal fluid. This problem is a priori not Hamiltonian; specifically, in the case n= 1 (i.e., in the problem of the interaction of a cylinder with a vortex) it is integrable.     

On the Steady Solutions to a Model of Compressible Heat Conducting Fluid in Two Space Dimensions

We consider steady compressible Navier-Stokes-Fourier system in a bounded two-dimensional domain with the pressure law p（e,θ） - qθ＋eln^α（1＋e）. For the heat flux q ~ -（1＋θ^m） △θwe show the existence of a weak solution provided α〉max{1,1/m}, m 〉0. This improves the recent result from [1].     

On Solvability of an Initial-Boundary Value Problem for a Viscoelasticity Model with Fractional Derivatives

We establish the existence and uniqueness (the latter only in the plane case) of a weak solution to an initial-boundary value problem for the system of the equations of motion of a viscoelastic fluid, namely, for the anti-Zener model whose constitutive law contains fractional derivatives. We use the approximation of this problem by a sequence of regularized Navier–Stokes systems and passage to the limit.     

Global Unique Solvability of an Initial-Boundary Value Problem for the One-Dimensional Barotropic Equations of Binary Mixtures of Viscous Compressible Fluids

Journal of Applied and Industrial Mathematics - We consider the equations of a multivelocity model of a binary mixture of viscous compressible fluids (the two-fluid medium) in the case of...     

Classical Solvability of the First Initial Boundary-Value Problem for a Nonlinear Strongly Degenerate Parabolic Equation

We prove the existence of a classical solution global in time for the first initial boundary-value problem for a nonlinear strongly degenerate parabolic equation.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 10, pp. 1299 – 1320, October, 2004.     

Nontrivial Attractors in a Model Related to the Three-Body Quantum Problem

A model of a quantum mechanical system related to the three-body problem is studied. The model is defined in terms of a symmetric pseudo-differential operator (PDO) with unbounded symbol. The entire family of self-adjoint extensions of this operator is studied using harmonic analysis. A regularization procedure for this PDO is introduced and the spectral properties of the operators obtained in this way are investigated. The limit behavior of the regularized operators when the regularization parameter is removed is analyzed and a nontrivial attractor is exhibited.     

Nonuniqueness of a Solution to the Problem on Motion of a Rigid Body in a Viscous Incompressible Fluid

This paper is devoted to the problem on motion of a rigid body in a viscous incompressible fluid. It is proved that there exist at least two weak solutions of this problem if collisions of the body with the boundary of the flow domain are allowed. These solutions have different behavior of the body after the collision. Namely, for the first solution, the body goes away from the boundary after the collision. In the second solution, the body and the boundary remain in contact. Bibliography 15 titles.To Vsevolod Alekseevich Solonnikov on the occasion of his jubilee__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 306, 2003, pp. 199–209.     

Solvability of a Thermoviscoelastic Model of the Motion of Solutions of Polymers Satisfying the Objectivity Principle

Mathematical Notes - The existence of weak solutions of the initial boundary-value problem for a mathematical model describing the motion of weakly concentrated aqueous solutions of polymers is...     

Stability of Unique Solvability in an Ill-Posed Dirichlet Problem

Suppose that Ω ? ∝ ⁿ is a compact domain with Lipschitz boundary ?Ω which is the closure of its interior Ω₀. Consider functions ? _i , τ _i : Ω→ ∝ belonging to the space L _q (Ω) for q ∈ (1, +∞] and a locally Holder mapping F: Ω × ∝ → ∝ such that F(·, 0) ≡ 0 on Ω. Consider two quasilinear inhomogeneous Dirichlet problems $$\left\{ {_{u = \tau _i \quad \quad \quad \quad \quad \quad \quad \;\;\,\operatorname{on} \;\partial \Omega ,}^{\Delta u_i = F(x,\;u_i ) + \varphi _i (x)\quad \operatorname{on} \;\Omega _0 ,} } \right.\quad \quad i = 1,\;2.$$ In this paper, we study the following problem: Under certain conditions on the function F generally not ensuring either the uniqueness or the existence of solutions in these problems, estimate the deviation of the solutions u _i (assuming that they exist) from each other in the uniform metric, using additional information about the solutions u _i . Here we assume that the solutions are continuous, although their continuity is a consequence of the constraints imposed on F, τ_i, ?_i. For the additional information on the solutions u _i , i = 1, 2 we take their values on the grid; in particular, we show that if their values are identical on some finite grid, then these functions coincide on Ω.     

Determination of a Constitutive Friction Law Using an Elastic-Plastic Half-Space Model

Friction influences metal forming processes both in economic and technical terms. A precise understanding of friction is inevitable as friction restricts the potential of the product design. Friction depends on the occurring contact loads which is especially true for sheet-bulk metal forming (SBMF) as the incorporated contact loads appear in a very broad range. Numerical simulations, which are verified experimentally, are carried out to analyse contact interactions which typically appear in SBMF. On the one hand the multi-scale character of rough surfaces requires a very fine resolution of the contact area, on the other hand a large contact area is necessary to be representative. A half-space model is chosen for the contact analysis, because it only depends on the two-dimensional surface boundary which consumes less computing capacity than the finite element method (FEM) with its three-dimensional volume approach for the same surface resolution and area. The outcome is a constitutive friction law (CFL) consisting of two equations. The law is implemented into the framework of FEM to see the impact in a typical SBMF-process which aims to form metal sheet into cups with integrated gearing teeth. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

双层浅水波模型柯西问题的经典解

主要研究浅水理论中双层浅水波模型柯西问题的经典解.在适当的初值条件下,得到整体经典解存在唯一性的充要条件.同时对单层浅水波模型也得到相应结果.     

On the Domain Dependence of Solutions to the Compressible Navier-Stokes Equations of an Isothermal Fluid

The aim of this paper is to study the behaviour of the variational solutions to the Navier-Stokes equations describing viscous compressible isothermal fluids with nonlinear stress tensors in a sequence of domains $\{\varOmega_{n}\} _{n=1}^{\infty}$ . The sequence converges in sense of the Sobolev-Orlicz capacity to domain Ω. We prove that the solutions of the equations in Ω _n converge to a solution of the respective equations in Ω. Moreover, The result can be applied to generalization of the existence result.     

Unique Solvability of the Inverse Problem with a Time-Dependent Boundary Condition for a Sorption Model

A mixed initial boundary-value problem is considered for nonequilibrium sorption dynamics with inner-diffusion kinetics. The problem allows for convection and longitudinal diffusion and has a time-dependent boundary condition. This condition contains the time derivative of a solution component and constitutes the balance equation for the absorbed mixture near the output boundary of the sorption region—inside the diffusion barrier. Bounds on the solution of the direct problem are obtained: nonnegativity of the solution and its first time derivatives, and boundedness of the solution by known functions. The inverse problem of estimating the nonlinear system parameter—the sorption isotherm—is considered and a uniqueness theorem is proved.