Approximation of the perturbation equations of a quasi-linear hyperbolic system in the neighborhood of a bicharacteristic

In 1956 Whitham gave a nonlinear theory for computing the intensity of an acoustic pulse of an arbitrary shape. The theory has been used very successfully in computing the intensity of the sonic bang produced by a supersonic plane. [4.] derived an approximate quasi-linear equation for the propagation of a short wave in a compressible medium. These two methods are essentially nonlinear approximations of the perturbation equations of the system of gas-dynamic equations in the neighborhood of a bicharacteristic curve (or rays) for weak unsteady disturbances superimposed on a given steady solution. In this paper we have derived an approximate quasi-linear equation which is an approximation of perturbation equations in the neighborhood of a bicharacteristic curve for a weak pulse governed by a general system of first order quasi-linear partial differential equations in m + 1 independent variables (t, x₁,…, x_m) and derived Gubkin's result as a particular case when the system of equations consists of the equations of an unsteady motion of a compressible gas. We have also discussed the form of the approximate equation describing the waves propagating upsteam in an arbitrary multidimensional transonic flow.     

Time decay of the solution to the Cauchy problem for a three-dimensional model of nonsimple thermoelasticity

This paper is devoted to the investigation of the solution to the Cauchy problem for a system of partial differential equations describing thermoelasticity of nonsimple materials in a three-dimensional space. The model of linear dynamical thermoelasticity of nonsimple materials is considered as the system of partial differential equations of fourth order. In this paper, we proposed a convenient evolutionary method of approach to the system of equations of nonsimple thermoelasticity. We proved the L^p−L^q time decay estimates for the solution to the Cauchy problem for linear thermoelasticity of nonsimple materials.     

First-Order Equations of Motion in the Supersymmetric Yang–Mills Theory with a Scalar Multiplet

We propose a system of first-order equations of motion all solutions of which are solutions of a system of second-order equations of motion for the supersymmetric Yang–Mills theory with a scalar multiplet. We find N = 1 transformations under which the systems of first- and second-order equations of motion are invariant.     

On a certain infinite system of semilinear evolution equations

Summary We consider a certain infinite system of semilinear evolution equations in a Banach space. There are proved the existence and uniqueness of solutions of the Cauchy problem for the above system. These results involve, as a particular case, a system of integro-differential evolution equations with functional arguments.     

The kinetics equations in a multiplying medium with free boundaries

We study a mathematical model of neutron multiplication in a slab ??, by taking into account temperature feedback effects and considering one group of delayed neutrons. The thickness 2a of ?? is time dependent because of temperature variations due to the energy released by fissions. Starting from a quite detailed picture of the physical phenomena occurring in ??, we derive a system of three coupled ordinary differential equations for the total number of neutrons F? = F?(t), for the total number of precursors ? = ?(t), and for the half-thickness of ??, a = a(t). We finally examine some stability properties of such a system of ordinary differential equations.     

Ergodicity of a Galerkin approximation of three-dimensional magnetohydrodynamics system forced by a degenerate noise

Magnetohydrodynamics system consists of a coupling of the Navier-Stokes and Maxwell's equations and is most useful in studying the motion of electrically conducting fluids. We prove the existence of a unique invariant, and consequently ergodic, measure for the Galerkin approximation system of the three-dimensional magnetohydrodynamics system. The proof is inspired by those of [E. Weinan and J.C. Mattingly, Ergodicity for the Navier-Stokes equation with degenerate random forcing: Finitedimensional approximation, Comm. Pure Appl. Math. LIV (2001), pp. 1386–1402; M. Romito, Ergodicity of the finite dimensional approximation of the 3D Navier-Stokes equations forced by a degenerate noise, J. Stat. Phys. 114 (2004), pp. 155–177] on the Navier-Stokes equations; however, computations involve significantly more complications due to the coupling of the velocity field equations with those of magnetic field that consists of four non-linear terms.     

Homogenization of a fluid problem with a free boundary

The stationary Stokes equations with a free boundary are studied in a perforated domain. The perforation consists of a periodic array of cylinders of size and distance O(ε). The free boundary is given as the graph of a function on a two‐dimensional perforated domain. We derive equations for the two‐scale limit of solutions. The limiting equation is a free boundary system. It involves a nonlinear eliptic operator corresponding to the nonlinear mean‐curvature expression in the original equations. Depending on the equation for the contact angle, the pressure is in general unbounded. © 2000 John Wiley & Sons, Inc.     

Independence principles in the equations of state of a deformable material

We consider the principles of coordinate, rotational, and initial independence of the equations of state for a deformable material and the theorem on the existence of elasticity potential connected with them. We show that the well-known axiomatic substantiation and mathematical representation of these principles in “rational continuum mechanics” as well as the proof of the theorem are erroneous. A correct proof of the principles and theorem is presented for the most general case (a stressed anisotropic body under the action of an arbitrary tensor field) without applying any axioms. On this basis, we eliminated the dependence on an arbitrary initial state and the corresponding accumulated strain from the system of equations of state of a deformable material. The obtained forms of equations are convenient for constructing and analyzing the equations of local influence of initial stresses on physical fields of different nature. Finally, these equations represent governing equations for the problems of nondestructive testing of inhomogeneous three-dimensional stress fields and for theoretical-and-experimental investigation of the nonlinear equations of state.     

On the state stability of a system of integro-differential equations of nonstationary aeroelasticity

We consider a process of nonstationary aeroelasticity, which is described by a system of integro-dijferential equations that cannot be solved for the derivative. We formulate necessary and sufficient conditions, in terms of Lyapunov functionals, for the exponential stability of such a system with respect to the metric of an infinite Hilbert space. A formula is given for the total derivative of a Lyapunov functional via the initial equations of motion.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 70, pp. 112–121, 1990.     

Three-webs defined by a system of ordinary differential equations

We consider a three-web W(1, n, 1) formed by two n-parametric family of curves and one-parameter family of hypersurfaces on a smooth (n + 1)-dimensional manifold. For such webs, the family of adapted frames is defined and the structure equations are found, and geometric objects arising in the third-order differential neighborhood are investigated. It is showed that every system of ordinary differential equations uniquely defines a three-web W(1, n, 1). Thus, there is a possibility to describe some properties of a system of ordinary differential equations in terms of the corresponding three-web W(1, n, 1). In particular, autonomous systems of ordinary differential equations are characterized.     

The first-approximation stability of a nonstationary system with delay

We study the exponential stability of a nonlinear system of differential equations with constant delay such that the right-hand side of one of its subsystems contains the multiplier e ^t . We obtain a sufficient condition for the first-approximation stability of this system.     

State estimation for a dynamical system described by a linear equation with unknown parameters

We investigate the state estimation problem for a dynamical system described by a linear operator equation with unknown parameters in a Hilbert space. In the case of quadratic restrictions on the unknown parameters, we propose formulas for a priori mean-square minimax estimators and a posteriori linear minimax estimators. A criterion for the finiteness of the minimax error is formulated. As an example, the main results are applied to a system of linear algebraic-differential equations with constant coefficients.     

Solution of a mixed boundary-value problem of thermodiffusion for layered bodies of canonical shape

We propose a method of solving coupled thermodiffusion problems for layered bodies of canonical shape. The method is based on separating the coupled system of equations and boundary conditions into independent boundary-value problems. The solution contains arbitrary functions of time determined from a system of second-order integral equations of convolution type obtained as a result of satisfying the boundary conditions.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 32, 1990, pp. 87–91.     

On the resolution and optimization of a system of fuzzy relational equations with sup-T composition

This paper provides a thorough investigation on the resolution of a finite system of fuzzy relational equations with sup-T composition, where T is a continuous triangular norm. When such a system is consistent, although we know that the solution set can be characterized by a maximum solution and finitely many minimal solutions, it is still a challenging task to find all minimal solutions in an efficient manner. Using the representation theorem of continuous triangular norms, we show that the systems of sup-T equations can be divided into two categories depending on the involved triangular norm. When the triangular norm is Archimedean, the minimal solutions correspond one-to-one to the irredundant coverings of a set covering problem. When it is non-Archimedean, they only correspond to a subset of constrained irredundant coverings of a set covering problem. We then show that the problem of minimizing a linear objective function subject to a system of sup-T equations can be reduced into a 0–1 integer programming problem in polynomial time. This work generalizes most, if not all, known results and provides a unified framework to deal with the problem of resolution and optimization of a system of sup-T equations. Further generalizations and related issues are also included for discussion.     

The Cauchy Problem on a Characteristic Cone for the Einstein Equations in Arbitrary Dimensions

We derive explicit formulae for a set of constraints for the Einstein equations on a null hypersurface, in arbitrary space–time dimensions n + 1 ≥ 3. We solve these constraints and show that they provide necessary and sufficient conditions so that a spacetime solution of the Cauchy problem on a characteristic cone for the hyperbolic system of the reduced Einstein equations in wave-map gauge also satisfies the full Einstein equations. We prove a geometric uniqueness theorem for this Cauchy problem in the vacuum case.     

Global weak solutions of a three-dimensional flow model for a multi-species mixture in porous media

We consider an initial boundary value problem for a non-linear differential system consisting of one equation of parabolic type coupled with a n × n semi-linear hyperbolic system of first order. This system of equations describes the compressible miscible displacement of n + 1 chemical species in a porous medium, in the absence of diffusion and dispersion. We assume the viscosity of the fluid mixture to be constant. We prove, in three space dimensions, the existence of a global weak solution with non-smooth initial data for the concentration. The proof is based on the artificial viscosity method together with a compensated compactness argument. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

A posteriori error estimates for a coupled wave system with a local damping

We study a finite element method applied to a system of coupled wave equations in a bounded smooth domain in \mathbbR^d {\mathbb{R}^d} , d = 1, 2, 3, associated with a locally distributed damping function. We start with a spatially continuous finite element formulation allowing jump discontinuities in time. This approach yields, L ₂(L ₂) and L _∞(L ₂), a posteriori error estimates in terms of weighted residuals of the system. The proof of the a posteriori error estimates is based on the strong stability estimates for the corresponding adjoint equations. Optimal convergence rates are derived upon the maximal available regularity of the exact solution and justified through numerical examples. Bibliography: 14 titles. Illustrations: 4 figures.     

The Cauchy problem for a system of two partial differential equations of infinite order

Starting from the generalized scheme of separation of variables, we propose a new effective method of constructing the solution of the Cauchy problem for a system of two partial differential equations, in general of infinite order with respect to the spatial variable. We consider the example of the Cauchy problem for the system of Lamé equations in the case of a two-dimensional strain.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 35, 1992, pp. 204–210.     

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.     

Symmetries of shallow water equations on a rotating plane

We consider a system of nonlinear differential equations which describes the spatial motion of an ideal incompressible fluid on a rotating plane in the shallow water approximation and a more general system of the theory of long waves which takes into account the specifics of shear flows. Using the group analysis methods, we calculate the 9-dimensional Lie algebras of infinitesimal operators admissible by the models. We establish an isomorphism of these Lie algebras with a known Lie algebra of operators admissible by the system of equations for the two-dimensional isentropic motions of a polytropic gas with the adiabatic exponent γ = 2. The nontrivial symmetries of the models under consideration enable us to carry out the group generation of the solutions. The class of stationary solutions to the equations of rotating shallow water transforms into a new class of periodic solutions.