Inviscid and viscous stationary waves of gas flow through contracting-expanding nozzles

In this work we study steady states of one-dimensional viscous isentropic compressible flows through a contracting-expanding nozzle. Treating the viscosity coefficient as a singular parameter, the steady-state problem can be viewed as a singularly perturbed system. For a contracting-expanding nozzle, a complete classification of steady states is given and the existence of viscous profiles is established via the geometric singular perturbation theory. Particularly interesting is the existence of a maximal sub-to-super transonic wave and its role in the formation of other complicated transonic waves consisting of a sub-to-super portion.     

Existence and uniqueness results for a nonlinear stationary system

We prove a few existence results of a solution for a static system with a coupling of thermoviscoelastic type. As this system involves L⊃ coupling terms we use the techniques of renormalized solutions for elliptic equations with L⊃ data. We also prove partial uniqueness results.AMS Subject Classification: 32D05, 74D99.     

Rarefied gas flow through a pipe of variable square cross section into vacuum

The kinetic S-model is used to study the steady rarefied gas flow through a long pipe of variable cross section joining two tanks with arbitrary differences in pressure and temperature. The kinetic equation is solved numerically by applying a second-order accurate conservative method on an unstructured mesh. The basic quantity to be computed is the gas flow rate through the pipe. The possibility of finding a solution based on the assumption of the plane cross sectional flow is also explored. The resulting solutions are compared with previously known results.     

Existence and uniqueness of solutions of stationary transport equations

We consider the abstract kinetic equation dψ /dx = –JLψ, x ∈ [0, τ ], in a Hilbert space H. It is supposed that J = J * = J^–1, L = L * ≥ 0, ker L = 0. The following theorem is proved: if JL is similar to a self-adjoint operator, then an associated boundary problem has a unique solution. We apply this theorem to the stationary equation of Brownian motion (sgn μ)|μ |α (∂ψ /∂x) (x,μ) = (∂²ψ /∂μ²) (x,μ), 0 < x < τ, μ ∈ ℝ. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence and uniqueness of traveling waves for non-monotone integral equations with applications

A class of integral equations without monotonicity is investigated. It is shown that there is a spreading speed c^∗>0 for such an integral equation, and that its limiting integral equation admits a unique traveling wave (up to translation) with speed c?c^∗ and no traveling wave with c<c^∗. These results are also applied to some nonlocal reaction-diffusion population models.     

Existence and uniqueness of traveling waves for non-monotone integral equations with application

This paper is concerned with the traveling waves in a class of non-monotone integral equations. First we establish the existence of traveling waves. The approach is based on the construction of two associated auxiliary monotone integral equations and a profile set in a suitable Banach space. Then we show that the traveling waves are unique up to translations under some reasonable assumptions. The exact asymptotic behavior of the profiles as ξ→−∞ and the existence of minimal wave speed are also obtained. Finally, we apply our results to an epidemic model with non-monotone “force of infection”.     

On non‐stationary viscous incompressible flow through a cascade of profiles

The paper deals with theoretical analysis of non‐stationary incompressible flow through a cascade of profiles. The initial‐boundary value problem for the Navier–Stokes system is formulated in a domain representing the exterior to an infinite row of profiles, periodically spaced in one direction. Then the problem is reformulated in a bounded domain of the form of one space period and completed by the Dirichlet boundary condition on the inlet and the profile, a suitable natural boundary condition on the outlet and periodic boundary conditions on artificial cuts. We present a weak formulation and prove the existence of a weak solution. Copyright © 2006 John Wiley & Sons, Ltd.     

Existence and uniqueness of splittings for stationary iterative methods with applications to alternating methods

Summary. Given a nonsingular matrix , and a matrix of the same order, under certain very mild conditions, there is a unique splitting , such that . Moreover, all properties of the splitting are derived directly from the iteration matrix . These results do not hold when the matrix is singular. In this case, given a matrix and a splitting such that , there are infinitely many other splittings corresponding to the same matrices and , and different splittings can have different properties. For instance, when is nonnegative, some of these splittings can be regular splittings, while others can be only weak splittings. Analogous results hold in the symmetric positive semidefinite case. Given a singular matrix , not for all iteration matrices there is a splitting corresponding to them. Necessary and sufficient conditions for the existence of such splittings are examined. As an illustration of the theory developed, the convergence of certain alternating iterations is analyzed. Different cases where the matrix is monotone, singular, and positive (semi)definite are studied. Received September 5, 1995 / Revised version received April 3, 1996     

Existence and uniqueness of solutions for the generalized linear complementarity problem

Cottle and Dantzig (Ref. 1) showed that the generalized linear complementarity problem has a solution for anyqR ^m ifM is a vertical blockP-matrix of type (m ₁,...,m _n ). They also extended known characterizations of squareP-matrices to vertical blockP-matrices.Here we show, using a technique similar to Murty's (Ref. 2), that there exists a unique solution for anyqR ^m if and only ifM is a vertical blockP-matrix of type (m ₁,...,m _n ). To obtain this characterization, we employ a generalization of Tucker's theorem (Ref. 3) and a generalization of a theorem initially introduced by Gale and Nikaido (Ref. 4).     

Existence and uniqueness of stationary Lévy-driven CARMA processes

Necessary and sufficient conditions for the existence of a strictly stationary solution of the equations defining a general Lévy-driven continuous-parameter ARMA process with index set R$\mathbb{R}$ are determined. Under these conditions the solution is shown to be unique and an explicit expression is given for the process as an integral with respect to the background driving Lévy process. The results generalize results obtained earlier for second-order processes and for processes defined by the Ornstein–Uhlenbeck equation.     

广义Tjon-Wu方程Cauchy问题强解的存在唯一性和稳态解的存在性

利用卷积的性质和schauder不动点原理等技巧,在L_(1,i)空间中证明了一类广义Tjon-Wu方程Cauchy问题强解的存在唯一性以及稳态解的存在性.     

Existence,uniqueness and stability of weak solutions of parabolic systems with discontinuous nonlinearities

This paper is devoted to the study of the initial value problem for parabolic systems with discontinuous nonlinearities from the viewpoint of the existence, uniqueness and stability of weak solutions. Also the relationship with the solutions of the corresponding differential inclusions is studied.      

Propagation of stationary waves in a compound deformable tube filled with a viscous liquid

The problem of modeling the motion of the blood in a large vessel, partly replaced by a prosthesis (or graft) is considered. It is assumed that the material of the vessel and prosthesis is represented by a Voigt linear viscoelastic model.S. M. Kirov Azerbaidzhan State University, Baku. Translated from Mekhanika Polimerov, No. 1, pp. 175–178, January–February, 1976.