DESIGNING AND DETAILING OF BUILDING SYSTEMS. MECHANICS IN CIVIL ENGINEERING

Secularitycondition of the kinetic Carleman system

Vestnik MGSU 7/2015
  • Vasil’eva Ol’ga Aleksandrovna - Moscow State University of Civil Engineering (National Research University) (MGSU) Candidate of Physical and Mathematical Sciences, Associate Professor, Department of Higher Mathematics, Moscow State University of Civil Engineering (National Research University) (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .
  • Dukhnovskiy Sergey Anatol’evich - Moscow State University of Civil Engineering (MGSU) postgraduate student, Department of Higher Mathematics, Moscow State University of Civil Engineering (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .

Pages 33-40

The kinetic theory of gases is considered as a collection of a large number of interacting particles. We consider the discrete kinetic model of one-dimensional gas consisting of identical monatomic molecules, which can have one of two speeds, namely, the Cauchy problem with periodic initial conditions for the system of the Carleman equation. This mathematical model has a number of properties of the Boltzmann equation. This system of equations is a quasi-linear hyperbolic system of partial differential equations. In general, there is no analytic solution for this system. Therefore, under some general assumptions we can find the finite-dimensional approximation of the solutions for the Carleman equation with small Knudsen numbers that allow us to study our problem on the widest scale. Moreover, we can find the secularity condition of the Carleman model. An approximation solution of the Carleman equation for non-periodic initial data will be found in the next article. There is an interesting problem of the existence of the shock waves connecting the pairs of equilibrium states. Here we have a catastrophe theory. It is assumed that the solutions of the Cauchy problem split into the superposition of weakly interacting solitons and decreasing dispersive waves. The Cauchy problem of the Carleman equation is studied for small perturbations of the equilibrium state whereby we have perturbed system. In order to construct the finite-dimensional approximation we use the Fourier method. Construction of finite-dimensional approximation allows doing theoretical studies of solutions for the Cauchy problem of the Carleman equation with small Knudsen numbers.

DOI: 10.22227/1997-0935.2015.7.33-40

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ON ESTIMATES OF THE LINEARIZED OPERATOR OF THE KINETIC CARLEMAN SYSTEM

Vestnik MGSU 9/2016
  • Dukhnovskiy Sergey Anatol’evich - Moscow State University of Civil Engineering (National Research University) (MGSU) postgraduate student, Department of Advanced Mathematics, Moscow State University of Civil Engineering (National Research University) (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .

Pages 7-14

In this article the author discusses the properties of linearized operator for the Carleman equation for unperturbed problem, i.e. without the perturbation operator. The solution of the Cauchy problem with periodic initial data is searched for small perturbations of the equilibrium state. The estimates are obtained using the Paley-Wiener theorem and the Laplace transformation. It is assumed that the solutions of the Cauchy problem split into the superposition of weakly interacting solutions and decreasing dispersive waves. The Carleman equation describes a combination of processes: relaxation and free movement. The aim of relaxation is to spread the particles in different directions. Such a system simulates some properties of the Boltzmann equation. The kinetic Carleman equation is a system of two nonlinear differential equations describing transportation processes and interaction of two classes of particles moving with the same speed in modulus in different directions on the line. This system belongs to the class of non-integrable equations which leads to important consequences. Namely, such a system can detect the irregular behavior of the solutions. The Carleman system occupies a special position with respect to other systems and allows us to prove the global existence theorem. In the works by Il’in the question of the stability of stationary but spatially inhomogeneous solutions of the Carleman system is posed. In the case of the discrete model the solution is stable in time for the homogeneous problem.

DOI: 10.22227/1997-0935.2016.9.7-14

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