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Nonlinear Functionals of Multi-D Discrete Velocity Boltzmann Equations

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Abstract

In this paper, we study nonlinear functionals measuring potential interactions and L 1-distance between two mild solutions for the multi-dimensional discrete velocity Boltzmann equations when the initial data are a small perturbation of a vacuum. We employ Bony's dispersion estimates to show that these functionals satisfy Lyapunov type estimates which are useful for the study of time-asymptotics and L 1-stability of mild solutions.

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References

  1. J.-T. Beale, Large-time behavior of the Broadwell model of a discrete velocity gas, Commun. Math. Phys. 102:217–235 (1985).

    Google Scholar 

  2. J.-T. Beale, Large-time behavior of discrete velocity Boltzmann equations, Commun. Math. Phys. 106:659–678 (1986).

    Google Scholar 

  3. J.-M. Bony, Solutions globales bornêes pour les modèles discrete de l'êquation de Boltzmann en dimension 1 d'espace, Actes Journêes E.D.P. St. Jean de Monts, No.XVI (1987).

  4. J.-M. Bony, Existence globale à donnêes de Cauchy petites pour modèles discrets de l'êquation de Boltzmann, Commun. Par. Diff. Eq. 16:533–545 (1991).

    Google Scholar 

  5. J.-M. Bony, Existence Globale et Diffusion pour les Modèles Discrets de la Cinêtique des Gaz, First European Congress of Mathematics, Vol.1 (1994), pp. 391–410.

    Google Scholar 

  6. A. Bressan, T.-P. Liu, and T. Yang, L1 stability estimates for n×n conservation laws, Arch. Ration. Mech. An. 149:1–22 (1999).

    Google Scholar 

  7. J. E. Broadwell, Shock structure in a simple discrete velocity gas, Phys. Fluids 7:1243–1247 (1964).

    Google Scholar 

  8. C. Cercignani, A remarkable estimate for the solutions of the Boltzmann equation, Appl. Math. Lett. 5:59–62 (1992).

    Google Scholar 

  9. C. Cercignani, Weak solutions of the Boltzmann equation and energy conservation, Appl. Math. Lett. 8:53–59 (1995). See also: C. CercignaniErrata, Appl. Math. Lett. 8:95–99 (1995).

    Google Scholar 

  10. C. Cercignani and R. Illner, Global weak solutions of the Boltzmann equation in a slab with diffusive boundary conditions, Arch. Ration. Mech. An. 134:1–16 (1996).

    Google Scholar 

  11. P. B. Dubovski and S.-Y. Ha, Existence, uniqueness, and stability for spatially inhomogeneous BeckerüDöring equations with diffusion and convection terms, submitted.

  12. R. Gatignol, Thêorie Cinêtique des Gaz à Rêpartition Discrète de Vitesses, Lectures Notes in Physics, Vol.36 (Springer, Berlin, 1975)

    Google Scholar 

  13. J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure. Appl. Math. 18:697–715 (1965).

    Google Scholar 

  14. S.-Y. Ha, L1 stability of one-dimensional Boltzmann equation with an inelastic collision, J. Differential Equations 190:621–642 (2003).

    Google Scholar 

  15. S.-Y. Ha, L1 stability of multi-dimensional discrete Boltzmann equations, to appear in Arch. Ration. Mech. An.

  16. S.-Y. Ha and A. E. Tzavaras, Lyapunov functionals and L1 stability of discrete Boltzmann equation, Commun. Math. Phys. 239:65–92 (2003).

    Google Scholar 

  17. R. Illner, Global existence results for discrete velocity models for the Boltzmann equation, J. Meca. Th. Appl. 1:611–622 (1982).

    Google Scholar 

  18. R. Illner, Examples of nonbounded solutions in discrete kinetic theory, J. Mêc. Thêor. Appl. 5:561–571 (1986).

    Google Scholar 

  19. R. Illner and T. Platkowski, Discrete velocity models of the Boltzmann equation: A survey on the mathematical aspects of the theory, SIAM Review, 30:000–000 (1988).

    Google Scholar 

  20. K. Illner and M. Shinbrot, Global existence for a rare gas in an infinite vacuum, Commun. Math. Phys. 95:217–226 (1984).

    Google Scholar 

  21. R. Illner and W. Wagner, A random discrete velocity model and approximation of the Boltzmann equation, J. Stat. Phys. 70:773–792 (1993).

    Google Scholar 

  22. S. Kawashima, Global solution of the initial value problem for a discrete velocity model of the Boltzmann equation, Proc. Japan Acad. 57:19–24 (1981).

    Google Scholar 

  23. S. Kawashima, Global Existence and Stability of Solutions for Discrete Velocity Models of the Boltzmann Equation, Recent Topics in Nonlinear PDE (Hiroshima, 1983), North-Holland Math Stud., Vol. 98 (North-Holland, Amsterdam, 1984), pp. 59–85.

    Google Scholar 

  24. T.-P. Liu and T. Yang, Well posedness theory for hyperbolic conservation laws, Comm. Pure Appl. Math. 52:1553–1586 (1999).

    Google Scholar 

  25. T. Nishida and M. Mimura, On the Broadwell model of the Boltzmann equation for a simple discrete velocity gas, Proc. Japan. Acad. 50:812–817 (1974).

    Google Scholar 

  26. M. Slemrod, Large time behavior of the Broadwell model of a discrete velocity gas with specular reflective boundary conditions, Arch. Rational Mech. Anal. 111:323–342 (1990).

  27. L. Tartar, Existence globale pour un système hyperbolique semi-linêaire de la thêorie cinêtique des gaz, Sêminaire GoulaouicüSchwartz, École Polytech., Palaiseau, No.1 (1976).

  28. L. Tartar, Some existence theorems for semilinear hyperbolic systems in one space variable, MRC technical summary report (University of Wisconsin, 1980).

  29. W. Wagner, Approximation of the Boltzmann equation by discrete velocity models, J. Stat. Phys. 78:1555–1570 (1995).

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Wisconsin-Madison, USA

    Mikhail Feldman & Seung-Yeal Ha

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  1. Mikhail Feldman
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  2. Seung-Yeal Ha
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Feldman, M., Ha, SY. Nonlinear Functionals of Multi-D Discrete Velocity Boltzmann Equations. Journal of Statistical Physics 114, 1015–1033 (2004). https://doi.org/10.1023/B:JOSS.0000012515.85916.2a

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  • DOI: https://doi.org/10.1023/B:JOSS.0000012515.85916.2a

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