Abstract.
We study the L1 stability of classical solutions to the Boltzmann equation for a hard-sphere model, when initial datum is a small perturbation of a vacuum, and tends to zero exponentially fast at infinity in the phase space. For this, we introduce nonlinear functionals measuring potential interactions between particles with different velocities and L1 distance between classical solutions. We use pointwise estimates for a solution and the gain term of a collision operator to control the time-evolution of nonlinear functionals.
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Arkeryd, L.: Stability in L1 for the spatially homogeneous Boltzmann equation. Arch. Rational Mech. Anal. 103, 151–168 (1988)
Arkeryd, L.: Loeb solutions of the Boltzmann equation. Arch. Rational Mech. Anal. 86, 85–97 (1984)
Bellomo, N., Palczewski, A. & Toscani, G.: Mathematical topics in nonlinear kinetic theory. World Scientific Publishing Co., Singapore, 1988
Bellomo, N., Toscani, G.: On the Cauchy problem for the nonlinear Boltzmann equation: global existence, uniqueness and asymptotic behavior. J. Math. Phys. 26, 334–338 (1985)
Bellomo, N., Toscani, G.: Global existence, uniqueness and stability of the nonlinear Boltzmann equation with almost general gas-particle interaction potential. Proceedings of the conference commemorating the 1st centeninal of the Circolo Matematico di Palermo (Palermo, 1984). Rend. Circ. Mat. Palermo (2) Suppl. 8, 419–433 (1984)
Bony, J.-M.: Existence globale et diffusion pour les modèles discrets de la cinétique des gaz. First European Congress of Mathematics, 391–410 (1994)
Bony, J.-M.: Existence globale à données de Cauchy petites pour modèles discrets de l’équation de Boltzmann. Comm. Partial Differential Equations 16, 533–545 (1991)
Bony, J.-M.: Solutions globales bornées pour les modèles discrete de l’équation de Boltzmann en dimension 1 d’espace. Actes Journees E.D.P.St. Jean de Monts, no XVI (1987)
Bressan, A., Liu, T.-P., Yang, T.: L1 stability estimates for n × n conservation laws. Arch. Rational Mech. Anal. 149, 1–22 (1999)
Cercignani, C., Illner, R., Pulvirenti, M.: The mathematical theory of dilute gases. Applied Mathematical Sciences, 106, Springer-Verlag, New York, 1994
DiPerna, R., Lions, P.-L.: On the Cauchy problem for the Boltzmann equation: Global existence and weak stability results. Annals of Math. 130, 1189–1214 (1990)
Feldman, M., Ha, S.-Y.: Nonlinear functional and multi-dimensional discrete velocity Boltzmann equation. J. Stat. Phys. 114, 1015–1033 (2004)
Glassey, R.-T.: The Cauchy problem in kinetic theory. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA., 1996
Glikson, A.: On the existence of general solutions of the initial-value problem for the nonlinear Boltzmann equation with a cut-off. Arch. Rational Mech. Anal. 45, 35–46 (1972)
Glimm, J.: Solutions in the large for nonlinear hyperbolic systems of equations. Comm. Pure. Appl. Math. 18, 697–715 (1965)
Ha, S.-Y.: Lyapunov functionals for the Enskog-Boltzmann equation. To appear in Indiana Univ. Math. J.
Ha, S.-Y.: L1-stability of the Boltzmann equation for Maxwellian molecules. Submitted
Ha, S.-Y.: L1 stability of multi-dimensional discrete Boltzmann equations. Arch. Rational Mech. Anal. 171, 25–42 (2004)
Ha, S.-Y.: L1 stability of one-dimensional Boltzmann equation with an inelastic collision. Journal of differential equations 190, 621–642 (2003)
Ha, S.-Y., Tzavaras, A.E.: Lyapunov functionals and L1 stability of discrete Boltzmann equation. Commun. Math. Phys. 239, 65–92 (2003)
Hamdache, K.: Existence in the large and asymptotic behavior for the Boltzmann equation. Japan. J. Appl. Math. 2, 1–15 (1984)
Illner, R., Shinbrot, M.: Global existence for a rare gas in an infinite vacuum. Commun. Math. Phys. 95, 217–226 (1984)
Kaniel, S., Shinbrot, M.: The Boltzmann equation 1: Uniqueness and local existence. Commun. Math. Phys. 58, 65–84 (1978)
Liu, T.-P., Yang, T.: Well posedness theory for hyperbolic conservation laws. Comm. Pure Appl. Math. 52, 1553–1586 (1999)
Polewczak, J.: Classical solution of the nonlinear Boltzmann equation in all R3: asymptotitc behavior of solutions. J. Stat. Phys. 50, 611–632 (1988)
Tartar, L.: Some existence theorems for semilinear hyperbolic systems in one-space variable. MRC Technical Summary Report, University of Wisconsin, 1980
Toscani, G.: Global solution of the initial value problem for the Boltzmann equaiton near a local Maxwellian. Arch. Rational Mech. Anal. 102, 231–241 (1988)
Toscani, G.: H-theorem and asymptotic trend of the solution for a rarefied gas in a vacuum. Arch. Rational Mech. Anal. 100, 1–12 (1987)
Toscani, G.: On the nonlinear Boltzmann equation in unbounded domains. Arch. Rational Mech. Anal. 95, 37–49 (1986)
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Communicated by T.-P. Liu
Dedicated to Marshall Slemrod on the occasion of his 60th birthday
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Ha, SY. L1 Stability of the Boltzmann Equation for the Hard-Sphere Model. Arch. Rational Mech. Anal. 173, 279–296 (2004). https://doi.org/10.1007/s00205-004-0321-x
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DOI: https://doi.org/10.1007/s00205-004-0321-x