Global small solutions of the cauchy problem for a semilinear system of equations of thermoelasticity

2000 ◽  
Vol 52 (4) ◽  
pp. 499-512 ◽  
Author(s):  
O. M. Botsenyuk
Keyword(s):  
Cauchy Problem ◽  
System Of Equations ◽  
The Cauchy Problem

scholarly journals Remarks on the Systems of Semilinear Fractional Rayleigh-Stokes Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Le Dinh Long
Keyword(s):  
Cauchy Problem ◽  
Fourier Analysis ◽  
Stokes Equation ◽  
Stokes Equations ◽  
Newtonian Fluids ◽  
System Of Equations ◽  
Main Technique ◽  
The Cauchy Problem ◽  
The Stability

In this paper, we study the Cauchy problem for a system of Rayleigh-Stokes equations. In this system of equations, we use derivatives in the classical Riemann-Liouville sense. This system has many applications in some non-Newtonian fluids. We obtained results for the existence, uniqueness, and frequency of the solution. We discuss the stability of the solutions and find the solution spaces. Our main technique is to use the Banach mapping theorem combined with some techniques in Fourier analysis.

scholarly journals THE CAUCHY PROBLEM FOR THE SYSTEM OF EQUATIONS OF THERMOELASTICITY IN E^n

2014 ◽  
Vol 15 (1) ◽  
Author(s):  
Ikbol E. Niyozov ◽  
O. I. Makhmudov
Keyword(s):  
Cauchy Problem ◽  
Bounded Domain ◽  
Analytical Continuation ◽  
System Of Equations ◽  
The Cauchy Problem

ABSTRACT: In this paper we consider the problem of analytical continuation of solutions to the system of equations of thermoelasticity in a bounded domain from their values and values of their strains on a part of the boundary of this domain, i.e., we study the Cauchy problem. ABSTRAK: Di dalam kajian ini, kami menyelidiki masalah keselanjaran analitik bagi penyelesaian-penyelesaian terhadap sistem persamaan-persamaan termoelastik di dalam domain bersempadan berdasarkan nilai-nilainya dan nilai tegasannya bagi sebahagian daripada sempadan domain tersebut, iaitu kami mengkaji masalah Cauchy.

scholarly journals Flow stability and correctness of the Cauchy problem for a two-speed medium model with different phase pressures

2021 ◽  
Vol 52 ◽  
pp. 66-77
Author(s):  
Alexander Evgenievich Kroshilin ◽  
Mikhail Evgenievich Kroshilin
Keyword(s):  
Cauchy Problem ◽  
Fluid Model ◽  
Stationary Solutions ◽  
System Of Equations ◽  
Two Phase ◽  
System A ◽  
The Cauchy Problem ◽  
The Difference ◽  
Two Fluid Model ◽  
The Stability

At present, to describe the two-velocity flow of a dispersed mixture, as a rule, a two-fluid model is used with equal pressure of the phases of the medium and different velocities of the phases. The corresponding system of equations without special, postulated, stabilizing terms is non-hyperbolic. This can lead to difficulties in finding a solution. Recently, it has been proposed to use similar models more widely, but with different pressures of the phases of the medium. Such models allow one to take into account new physical effects associated with different phase pressures and often provide hyperbolicity of the corresponding system of equations. This article analyzes the influence of the difference in the pressure of the phases of the medium on the properties of the system: the importance of the corresponding new effects, the hyperbolicity of the system of equations, the stability of its stationary solutions, and the correctness of the corresponding Cauchy problem are investigated. Three systems are considered. The first, simplest model system is based on the well-known non-hyperbolic system, which has been modernized. It is shown that the Cauchy problem for the modified system is formally correct, but the practical possibility of using the calculation results obtained from the solution of this system should be investigated in each specific case, and depends on the calculated step and duration of the process under study. The techniques worked out to solve the first simplest system were used for other systems. As the second system, a model of the flow of a two-phase medium with different phase pressures and two momentum equations is considered. We will assume the phases are barotropic. Let us postulate an equation relating the pressure in the phases. It is proved that this system is always hyperbolic. The stability of its stationary solutions is investigated. Relationships are derived that make it possible to determine under what conditions, due to instability, the obtained solutions are unreliable. The properties of this system are compared with the system of two-speed flow of a dispersed mixture with equal pressure of the phases of the medium. As a third system, a two-pressure model describing bubble pulsations is considered. We will assume the phases are barotropic. Conditions are determined when the system is non-hyperbolic and the Cauchy problem is incorrect. It is investigated for what conditions the ill-posedness of the Cauchy problem leads to the unreliability of the solution, and under what conditions the ill-posedness of the Cauchy problem does not lead to the unreliability of the solution.

scholarly journals NUMERICAL SOLUTION OF THE GODUNOV - SULTANGAZIN SYSTEM OF EQUATIONS. PERIODIC CASE

Vestnik MGSU ◽  
2016 ◽  
pp. 27-35 ◽  
Author(s):  
Ol’ga Aleksandrovna Vasil’eva
Keyword(s):  
Cauchy Problem ◽  
Numerical Investigation ◽  
Initial Conditions ◽  
Periodic Case ◽  
Problem Solution ◽  
System Of Equations ◽  
Stabilization Time ◽  
System Solution ◽  
The Cauchy Problem ◽  
Quasi Linear System

The Cauchy problem of the Godunov - Sultangazin system of equations with periodic initial conditions is considered in the article. The Godunov - Sultangazin system of equations is a model problem of the kinetic theory of gases. It is a discrete kinetic model of one-dimensional gas consisting of identical monatomic molecules. The molecules can have one of three speeds. So, there are three groups of molecules. The molecules of the first two groups have the speeds equal in values and opposite in directions. The molecules of the third group have zero speed. The considered mathematical model has a number of properties of Boltzmann equation. This system of the equations is a quasi-linear system of partial differential equations. There is no analytic solution for this problem in the general case. So, numerical investigation of the Cauchy problem of the Godunov - Sultangazin system is very important. The finite-difference method of the first order is used for numerical investigation of the Cauchy problem of the Godunov - Sultangazin system of equations. The paper presents and discusses the results of numerical investigation of the Cauchy problem for the studied system solution with periodic initial condition. The dependence of the time of stabilization of the Cauchy problem solution of Godunov - Sultangazin system of equations from the decreasing parameter of system are obtained. The paper presents the dependence of time of energy exchange from the decreasing parameter. The solution stabilization to the equilibrium state is obtained. The stabilization time of Godunov - Sultangazin system solution is compared to the stabilization time of Carleman system solution in periodic case. The results of numerical investigation are in good agreement with the theoretical results obtained previously.

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