Summary
The asymptotic behaviour of solutions of the kinetic equations on the level of the Euler fluid dynamics is studied. In contrast to the previous studies in Sobolev spaces, the analysis is carried out in C0 (with respect to the space variable) setting. Thus, one needs to use only few Hilbert and initial layer terms, and consequently a lower smoothness is required for the solution of the Euler system.
Article PDF
Similar content being viewed by others
References
L. Arkeryd,On the Enskog equation in two space variables, Transport Theory Statist. Phys.,15 (5) (1986), pp. 673–691.
N. Bellomo -M. Lachowicz,Kinetic equation for dense gases. A review of physical and mathematical results, Internat. J. Modern Phys. B,1, n. 5 and 6 (1987), pp. 1193–1205.
R. Caflisch -G. Papanicolaou,The fluid dynamical limit of a nonlinear model Boltzmann equation, Comm. Pure Appl. Math.,32 (1979), pp. 589–616.
R. Caflisch,The fluid dynamic limit of the nonlinear Boltzmann equation, Comm. Pure Appl. Math.,33 (1980), pp. 651–666.
R.Caflisch,Fluid dynamics and the Boltzmann equation, inNonequilibrium Phenomena 1: The Boltzmann Equation, eds. L. J. Lebowitz and E. W. Montroll, North-Holland (1983).
C. Cercignani,Theory and Application of the Boltzmann Equation, Scottish Academic Press, Edinburgh (1975).
R. J. DiPerna -P. L. Lions,Solutions globales de l'équation de Boltzmann, C. R. Acad. Sci. Paris,306, s. 1 (1988), pp. 343–346.
J. H.Ferzigeŕ - H. G.Kaper,Mathematical Theory of Transport Processes in Gases, North-Holland (1972).
W.Fiszdon - M.Lachowicz - A.Palczewski,Existence problems of the nonlinear Boltzmann equation, inTrends and Applications of Pure Mathematics to Mechanics, eds. P. G. Ciarlet and M. Roseau, Springer-Verlag (1984), pp. 63–95.
H. Grad,Asymptotic equivalence of the Navier-Stokes and nonlinear Boltzmann equations, Proc. Symp. Appl. Nonlinear PDE in Math. Phys.,17 AMS, Providence (1965), pp. 154–183.
A. G. Heintz,On the solution of boundary value problems for the nonlinear Boltzmann equation in a bounded domain, Molecular Gas Dynamics, Rarefied Gas Aerodynamics,10 (1980), pp. 16–24 (in Russian).
A. G.Heintz,On the solution of initial-boundary problems for the non-linear Boltzmann equation in a bounded domain, Physical Gaskinetic, Rarafied Gas Aerodynamics, Issue11, Isdatel'stvo Leningradskovo Universiteta (1983), pp. 166–174 (in Russian).
M. Lachowicz,Initial layer and existence of a solution of the nonlinear Boltzmann equation, Proc. XV Internat. Symp. on Rarefied Gas Dynamics, Grado (1986), eds. V. Boffi and C. Cercignani, Teubner, Stuttgart (1986), pp. 150–159.
M. Lachowicz,Differentiability of the solution of a system of linear equations, Arch. Mech.,38, n. 1–2 (1986), pp. 127–141.
M. Lachowicz,On the initial layer and the existence theorem for the nonlinear Boltzmann equation, Math. Methods Appl. Sci.,9, n. 3 (1987), pp. 342–366.
M. Lachowicz,On the limit of the nonlinear Enskog equation corresponding with fluid dynamics, Arch. Rational Mech. Anal.,101, n. 2 (1988), pp. 179–194.
M. Lachowicz -M. Pulvirenti,A stochastic particle system modelling the Euler equation, Arch. Rational Mech. Anal.,109, n. 1 (1990), pp. 81–93.
A.Majda,Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer-Verlag (1984).
D. Morgenstern,Analytical studies related to the Maxwell-Boltzmann equation, Arch. Rational Mech. Anal.,4 (1955), pp. 533–555.
A. Palczewski,Exact and Chapman-Enskog solutions for the Carleman model, Math. Methods Appl. Sci.,6 (1984), pp. 417–432.
A. Palczewski,Spectral properties of the space nonhomogeneous linearized Boltzmann operator, Transport Theory Statist. Phys.,13, n. 3 and 4 (1984), pp. 409–430.
A. Ya. Povzner,The Boltzmann equation in the kinetic theory of gases, Amer. Math. Soc. Transl. (2),47 (1962), pp. 193–216.
C.Truesdell - R. G.Mancaster,Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas, Academic Press (1980).
S. Ukai -K. Asano,The Euler limit and initial layer of the nonlinear Boltzmann equation, Hokkaido Math. J.,12, n. 3, part 1 (October 1983), pp. 311–332.
S.Ukai - K.Asano,On th fluid dynamical limit of the Boltzmann equation, inRecent Topics in Nonlinear PDE, eds. M. Mimura and T. Nishida, North-Holland and Kinokuniya (1984), pp. 1–20.
Author information
Authors and Affiliations
Additional information
on leave from Wydział Matematyki, Uniwersytet Warszawski, Poland.
Rights and permissions
About this article
Cite this article
Lachowicz, M. On the asymptotic behaviour of solutions of nonlinear kinetic equations. Annali di Matematica pura ed applicata 160, 41–62 (1991). https://doi.org/10.1007/BF01764119
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01764119