System Of Quasilinear Equations Research Articles

The nonlocal solvability conditions for a system with constant terms and coefficients of the variable t

We consider the Cauchy problem for a system of quasilinear differential equations with constantterms and coefficients of the variablet.We investigate the solvability of the Cauchy problem fora system of quasilinear differential equations with constant terms and coefficients of the variabletusing the additional argument method. A theorem on the existence and uniqueness of the localsolution of the Cauchy problem for a system of quasilinear differential equations with constantterms and coefficients of the variabletis formulated. We obtain sufficient conditions for theexistence and uniqueness of a nonlocal solution of the Cauchy problem in original coordinates fora system of quasilinear differential equations with constant terms and coefficients of the variablet.A theorem on the existence and uniqueness of the nonlocal solution of the Cauchy problem fora system of quasilinear differential equations with constant terms and coefficients of the variabletis formulated. A theorem on the existence and uniqueness of the nonlocal solution of the Cauchyproblem for a system of quasilinear differential equations with constant terms and coefficients ofthe variabletis proved. The proof of the nonlocal solvability of the Cauchy problem for a systemof quasilinear differential equations with constant terms and coefficients of the variabletrelies onglobal estimates.

Quasilinear system of equations with quadratic growth in unbounded domains

Quasilinear system of equations with quadratic growth in unbounded domains

Прямая и обратная динамическая задача пороупругости

In applied problems related to propagation of elastic waves, it is often necessary to take into account porosity, fluid saturation of the media, and the hydrodynamic background. Real geological media are multiphase, electrically conductive, fractured, porous, etc. When propagating, seismic waves dissipate due to the absorption of energy. In this paper, the wave propagation process occurs in terms of partial densities of phases, stress tensor, pore pressure, and velocities of the corresponding phases. In the first section, for completeness, the presentation presents a quasilinear system of equations of the poroelasticity theory [1-3]. In the second section, the corresponding linear system of equations of the poroelasticity theory for a homogeneous medium is obtained. In the third section, we construct a fundamental solution for the system of equations of the poroelasticity theory obtained in the second section. In the final section, the inverse poroelasticity problem of determining the distributed source in a half-space using additional information about the free surface mode is considered.

Hamiltonian Aspects of Three-Layer Stratified Fluids

The theory of three-layer density-stratified ideal fluids is examined with a view toward its generalization to the n-layer case. The focus is on structural properties, especially for the case of a rigid upper lid constraint. We show that the long-wave dispersionless limit is a system of quasi-linear equations that do not admit Riemann invariants. We equip the layer-averaged one-dimensional model with a natural Hamiltonian structure, obtained with a suitable reduction process from the continuous density stratification structure of the full two-dimensional equations proposed by Benjamin. For a laterally unbounded fluid between horizontal rigid boundaries, the paradox about the non-conservation of horizontal total momentum is revisited, and it is shown that the pressure imbalances causing it can be intensified by three-layer setups with respect to their two-layer counterparts. The generator of the x-translational symmetry in the n-layer setup is also identified by the appropriate Hamiltonian formalism. The Boussinesq limit and a family of special solutions recently introduced by de Melo Viríssimo and Milewski are also discussed.

The nonlocal solvability conditions for a system of quasilinear equations of the first order special right-hand sides

The Cauchy problem for a system of first-order quasilinear equations with special right-hand sides is considered. The study of solvability of this system in the original coordinates is based on the method of additional argument. It is proved that the local solution of such system exists and that its smoothness is not lower than the smoothness of the initial conditions. For system of two equations non-local solutions are considered that are continued by finite number of steps from the local solution. Sufficient conditions for the existence of such non-local solution are derived. The proof of the non-local resolvability of the system relies on original global estimates.

Constructive Estimation of the Lyapunov Function for Quadratic Nonlinear Systems

A system of perturbed equations of motion with quadratic nonlinearity is considered. New estimates of the Lyapunov function are established and two conclusions are formulated. New motion constraints are presented. For two coupled systems of equations, boundedness conditions for some of the variables are established. Practical motion constraints in given domains of initial and subsequent perturbations are established. A system of quasi-linear equations is considered as an example.

The generalized solution of the system of quasilinear equations

The generalized solution of the system of quasilinear equations

Обобщенное решение системы квазилинейных уравнений

Обобщенное решение системы квазилинейных уравнений

Hyperbolic submodels of an incompressible viscoelastic Maxwell medium

A two-dimensional motion of an incompressible viscoelastic Maxwell continuum is considered. The system of quasilinear equations describing this motion has both real and complex characteristics. A class of effectively one-dimensionalmotions is analyzed for which the original system of equations is decomposed into a hyperbolic subsystem and a quadrature. The properties of the hyperbolic submodels obtained depend on the choice of the invariant derivative in the rheological relation. When one chooses the Jaumann corotational derivative as the invariant derivative, the equations of the submodel remain quasilinear. They can be represented in the form of conservation laws, which allows one to analyze discontinuous solutions to these equations. When one chooses the upper or lower convected derivative, the equations of one-dimensional hyperbolic submodels turn out to be linear. The problem of shear motion between parallel plates and the problem of interaction between the stress field that does not depend on one of the coordinates and a transverse shear flow with initially constant vorticity are studied in detail. It is established that a plane Couette flow in the model with the corotational derivative is unstable in the linear approximation in the class of shear flows if the Weissenberg number is greater than one. The development of small perturbations gives rise to discontinuities in tangential velocities and stresses. The hysteresis phenomenon is observed when the Weissenberg number successively increases and decreases while passing through a critical value. The Couette flow in models with the upper and lower convected derivative remains stable with respect to one-dimensional perturbations.

Lagged diffusivity fixed point iteration for solving steady-state reaction diffusion problems

The paper concerns with the computational algorithms for a steady-state reaction diffusion problem. A lagged diffusivity iterative algorithm is proposed for solving resulting system of quasilinear equations from a finite difference discretization. The convergence of the algorithm is discussed and the numerical results show the efficiency of this algorithm.

Justification of the stabilization method for a mathematical model of charge transport in semiconductors

An initial boundary value problem for a quasilinear system of equations is studied and effectively applied to numerically determine, by the stabilization method, stationary solutions of a hydrodynamic model describing the motion of electrons in the silicon transistor MESFET (metal semiconductor field effect transistor).

Local-in-time well-posedness of a regularized mathematical model for silicon MESFET

We prove the local-in-time well-posedness of the initial boundary value problem for a system of quasilinear equations. This system is used for finding numerical stationary solutions of the hydrodynamical model of charge transport in the silicon MESFET (metal semiconductor field effect transistor). The initial boundary value problem has the following peculiarities: the quasilinear system is not a Cauchy-Kovalevskaya-type system; the boundary is a non-smooth curve and has angular points; nonlinearity of the problem is mainly connected with squares of gradients of the unknown functions. By using a special representation for the solution of a model problem we reduce the original problem to an integro-differential system. The local-in-time existence of a weakened generalized solution of this system is then proved by the fixed-point argument.

Classical solvability of a problem with moving boundaries for a hyperbolic system of quasilinear equations

Using the method of characteristics and the method of contracting mappings, we establish the local classical solvability of a problem for a hyperbolic system of quasilinear first-order equations with moving boundaries and nonlinear boundary conditions. Under additional assumptions on the monotonicity and sign constancy of initial data and a restriction on the growth of the right-hand sides of the system, we formulate sufficient conditions for the global classical solvability of the problem.

On the global continuous solvability of the mixed problem for one-dimensional hyperbolic systems of quasilinear equations

We consider a hyperbolic system of quasilinear equations written in Riemann invariants for the case of one spatial variable. For this system, we obtain sufficient conditions for the global generalized continuous solvability of the mixed problem in the class of functions monotone with respect to x for arbitrary t and with respect to t for x = 0. In contrast to earlier studies, we assume that the boundary conditions may depend not only on time but also on the unknown functions.

Weak Solutions to the Equations of Motion of Viscous Compressible Reacting Binary Mixtures: Uniqueness and Lipschitz-Continuous Dependence on Data

1. An extensive literature is devoted to systems of quasilinear equations of motion of viscous compressible media (see [1, 2]). One of the problems of current interest is the development of the theory of global (with respect to time and data) weak solutions, including questions of their existence, uniqueness, and continuous dependence on data (such as initial data, the coefficients of the equations, the constant terms, etc.). For the system of equations of one-dimensional motion of a viscous heat-conducting gas, the last two questions were studied in [3–6] (see also [7, 8]). For more complicated systems of equations for reacting gases (in other words, in combustion problems) weak solutions were studied in [9, 10]. Quite recently, in [11], the existence of weak solutions was obtained for an even more complicated system of equations, namely, for a reacting binary gas mixture. Note that regular solutions of problems concerning binary mixtures were studied earlier in [12, 13]. The present paper is devoted to the questions of uniqueness of these weak solutions and of their continuous dependence on data. The system of quasilinear equations of one-dimensional motion of a viscous compressible reacting binary gas mixture is of the form

Construction of piecewise-analytical gas flows joined through shock waves near the axis or center of symmetry

Analytical solutions of a quasilinear system of equations with partial derivatives are constructed in the case where the initial data for different functions are specified on different surfaces and the resultant problem has singularities of the form u/x and w/x. Conditions for existence and uniqueness of a solution in the form of formal power series for the problem posed and sufficient conditions for convergence of the series are indicated. A generalization of the problem considered is given. Results of the study are used to solve the problem of the focussing of a compression wave generated by a piston moving smoothly in a quiescent gas: a solution for t=0, including determination of the piston trajectory, and a solution for t 0. The solution of this problem is a generalization to the case of two independent variable self-similar Sedov's solutions.

A linear theory for Ekman's boundary layer according to a triple deck

The problem of the interaction between Ekman's classical boundary layer and that induced by a thermally nonhomogeneous site shows that the latter is strongly linked to the order of magnitude of the horizontal scale of the site. Our purpose is the analysis of the local interaction equations (Boussinesq equations) starting from a triple-deck model. This analysis yields a system of quasi-linear equations for the viscous lower deck. The linear theory of this system shows that the thermal non-homogeneity has a significant influence on the Ekman boundary layer flow owing to the interactive nature of the triple-deck structure. The numerical solution of the quasi-linear system confirms to a large extent this influence. The numerical results are given in graph form.

Nonlinear dynamics of inelastic failure of damaged layered systems

A method of solution is proposed and processes of the formation and development of a failure zone in piecewise-homogeneous rod during impact loading were analyzed. Such properties as being damageable and healing of defects as well as compliance of a barrier are taken into account for the first time. Deformation of the barrier is taken into account by means of a model of a Winkler foundation. The fulfillment of the known failure criterion at the failure front is assumed. The solution is reduced to a system of quasi-linear equations which are hyperbolic in the narrow sense, the right side of which contains integral terms of the hereditary type. As a result of numerical realization for the case of a bilayer rod, when the region contacting the barrier is damaged, it was found that the place and time of formation of the failure zone and time of complete failure depend substantially on compliance of the barrier as well as nonlinearity of the material.

A method for obtaining evolution equations for nonlinear waves in a random medium

Most of the methods used to derive deterministic equations governing the evolution of linear waves in random media are based on the mean field approach. For a given linear system of equations with random coefficients this approach results in an approximate deterministic equation (or system) for the mean (averaged over the set of all realizations) field. The coefficients of such equations are associated with the statistical moments of the random coefficients of the initial system. Sometimes similar approaches are also applied to nonlinear problems. However, recently, in a number of examples, it was shown that the mean field approach in nonlinear problems may give the wrong results. The error is related to the infinite growth of the root-mean-square phase fluctuations due to fluctuations of the wave velocity in an inhomogeneous medium. To overcome this effect the idea of eliminating the unbounded growth of phase fluctuations by using an appropriate coordinate transformation was proposed resulting in a method called the “mean waveform method”. In the present paper we extend the idea of the mean waveform method in order to develop a systematic approach which enables the construction of an approximate deterministic evolution equation for a given quasi-hyperbolic and quasi-linear system of equations with weak nonlinearity and stationary random coefficients. The applicability of the proposed approach is demonstrated by means of an example.

Solution of inverse problems for a system of quasilinear equations of heat conduction in a self-similar regime

Explicit solutions are obtained for inverse problems for a system of heat-conduction equations in a self-similar regime. The thermophysical characteristic being sought depend on the temperature distribution.