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Existence and stability analysis for the Carleman kinetic system

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Abstract

A global existence theorem for the discrete Carleman system in the Sobolev class W 1,2 is proved by the Leray-Schauder topological degree method, which was not previously applied to discrete kinetic equations. The instability of the nonequilibrium steady flow on a bounded interval is established in the linear approximation.

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References

  1. U. M. Sultangazin, Nonlinear Discrete Velocity Models of the Boltzmann Equation (Nauka, Alma-Ata, 1985) [in Russian].

    Google Scholar 

  2. R. Monaco and L. Preziosi, Fluid Dynamic Applications of the Discrete Boltzmann Equation (World Scientific, Singapore, 1991).

    MATH  Google Scholar 

  3. J.-L. Lions, Quelques méthodes de résolution des problémes aux limites non linéires (Dunod, Paris, 1969; Mir, Moscow, 1972).

    Google Scholar 

  4. V. V. Vedenyapin, “Differential Forms in Spaces without a Norm: A Theorem on the Uniqueness of Boltzmann’s H-Function,” Usp. Mat. Nauk 43(1), 159–179 (1988).

    MathSciNet  Google Scholar 

  5. B. A. Dubrovin, A. T. Fomenko, and S. P. Novikov, Modern Geometry: Methods and Applications (Springer-Verlag, New York, 1984–1990; Editorial, Kiev, 2001).

    Google Scholar 

  6. D. H. Sattinger, Topics in Stability and Bifurcation Theory (Springer-Verlag, Berlin, 1972).

    Google Scholar 

  7. R. A. Adams, Sobolev Spaces (Academic, New York, 1975).

    MATH  Google Scholar 

  8. M. A. Krasnosel’skii, Topological Methods in the Theory of Nonlinear Integral Equations (GITTL, Moscow, 1956; Pergamon, New York, 1964).

    Google Scholar 

  9. L. Nirenberg, Functional Analysis (Courant Institute, New York Univ., New York, 1975; Mir, Moscow, 1977).

    Google Scholar 

  10. A. V. Bobylev, Exact and Approximate Methods in the Theory of Nonlinear Boltzmann and Landau Kinetic Equations (Inst. Prikl. Mat. Akad. Nauk SSSR, Moscow, 1987) [in Russian].

    Google Scholar 

  11. P. Hartman, Ordinary Differential Equations (Wiley, New York, 1973).

    MATH  Google Scholar 

  12. E. L. Ince, Ordinary Differential Equations (Dover, New York, 1927; ONTI, Kharkov, 1939).

    MATH  Google Scholar 

  13. V. V. Aristov, “Solution to the Boltzmann Equation at Small Knudsen Numbers,” Zh. Vychisl. Mat. Mat. Fiz. 44, 1127–1140 (2004) [Comput. Math. Math. Phys. 44, 1069–1081 (2004)].

    MATH  MathSciNet  Google Scholar 

  14. C. Cercignani, Theory and Application of the Boltzmann Equation (Elsevier, New York, 1975; Mir, Moscow, 1978).

    MATH  Google Scholar 

  15. M. N. Kogan, Rarefied Gas Dynamics (Nauka, Moscow, 1967; Plenum, New York, 1969).

    Google Scholar 

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Authors and Affiliations

  1. Dorodnicyn Computing Center, Russian Academy of Sciences, ul. Vavilova 40, Moscow, 119991, Russia

    O. V. Ilyin

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  1. O. V. Ilyin
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Correspondence to O. V. Ilyin.

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Original Russian Text © O.V. Ilyin, 2007, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2007, Vol. 47, No. 12, pp. 2076–2087.

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Ilyin, O.V. Existence and stability analysis for the Carleman kinetic system. Comput. Math. and Math. Phys. 47, 1990–2001 (2007). https://doi.org/10.1134/S0965542507120093

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