Abstract
Boundaries occur naturally in kinetic equations, and boundary effects are crucial for dynamics of dilute gases governed by the Boltzmann equation. We develop a mathematical theory to study the time decay and continuity of Boltzmann solutions for four basic types of boundary conditions: in-flow, bounce-back reflection, specular reflection and diffuse reflection. We establish exponential decay in the L ∞ norm for hard potentials for general classes of smooth domains near an absolute Maxwellian. Moreover, in convex domains, we also establish continuity for these Boltzmann solutions away from the grazing set at the boundary. Our contribution is based on a new L 2 decay theory and its interplay with delicate L ∞ decay analysis for the linearized Boltzmann equation in the presence of many repeated interactions with the boundary.
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Arkeryd L.: On the strong L 1 trend to equilibrium for the Boltzmann equation’. Stud. Appl. Math. 87, 283–288 (1992)
Arkeryd L., Cercignani C.: A global existence theorem for the initial boundary value problem for the Boltzmann equation when the boundaries are not isothermal. Arch. Rational Mech. Anal. 125, 271–288 (1993)
Arkeryd, L., Esposito, R., Marra, R., Nouri, A.: Stability of the laminar solution of the Boltzmann equation for the Benard problem. Preprint 2007
Arkeryd L., Esposito R., Pulvirenti M.: The Boltzmann equation for weakly inhomogeneous data. Comm. Math. Phys. 111(3), 393–407 (1987)
Arkeryd L., Heintz A.: On the solvability and asymptotics of the Boltzmann equation in irregular domains. Comm. Partial Differ. Equ. 22(11–12), 2129–2152 (1997)
Beals R., Protopopescu V.: Abstract time-depedendent transport equations. J. Math. Anal. Appl. 212, 370–405 (1987)
Cercignani C.: The Boltzmann Equation and Its Application. Springer, Berlin (1988)
Cercignani C.: Equilibrium States and the trend to equlibrium in a gas according to the Boltzmann equation. Rend. Mat. Appl. 10, 77–95 (1990)
Cercignani C.: On the initial-boundary value problem for the Boltzmann equation. Arch. Ration. Mech. Anal. 116, 307–315 (1992)
Cannoe R., Cercignani C.: A trace theorem in kinetic theory. Appl. Math. Lett. 4, 63–67 (1991)
Cercignani C., Illner R., Pulvirenti M.: The Mathematical Theory of Dilute Gases. Springer, Berlin (1994)
Chernov N., Markarian R.: Chaotic Billiards. AMS, Providence (2006)
Deimling K.: Nonlinear Functional Analysis. Springer, Berlin (1988)
Desvillettes L.: Convergence to equilibrium in large time for Boltzmann and BGK equations. Arch. Ration. Mech. Anal. 110, 73–91 (1990)
Desvillettes L., Villani C.: On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation. Invent. Math. 159(2), 245–316 (2005)
Diperna R., Lions P.-L.: On the Cauchy problem for the Boltzmann equation. Ann. Math. 130, 321–366 (1989)
Diperna R., Lions P.-L.: Global weak solution of Vlasov-Maxwell systems. Comm. Pure Appl. Math. 42, 729–757 (1989)
Esposito, L., Guo, Y., Marra, R.: Phase transition of a Vlasov–Boltzamann system. Preprint 2009
Guo Y.: The Vlasov–Poisson–Boltzmann system near Maxwellians. Comm. Pure Appl. Math. 55(9), 1104–1135 (2002)
Guo Y.: The Vlasov–Maxwell–Boltzmann system near Maxwellians. Invent. Math. 153(3), 593–630 (2003)
Guo Y.: Singular solutions of the Vlasov-Maxwell system on a half line. Arch. Ration Mech. Anal. 131(3), 241–304 (1995)
Guo Y.: Regularity for the Vlasov equations in a half-space. Indiana Univ. Math. J. 43(1), 255–320 (1994)
Guo, Y., Jang, J., Jiang, N.: Acoustic limit of the Boltzmann equation with optimal scaling. Comm. Pure Appl. Math. (in press)
Glassey R.: The Cauchy Problems in Kinetic Theory. SIAM, Philadelphia (1996)
Glassey R., Strauss W.A.: Asymptotic stability of the relativistic Maxwellian. Publ. Res. Inst. Math. Sci. 29(2), 301–347 (1993)
Guo Y., Strauss W.A.: Instability of periodic BGK equilibria. Comm. Pure Appl. Math. 48(8), 861–894 (1995)
Grad, H.: Principles of the kinetic theory of gases. In: Handbuch der Physik, vol. XII, pp. 205–294. Springer, Berlin, 1958
Grad, H.: Asymptotic theory of the Boltzmann equation. II. Rarefied gas dynamics. In: Proceedings of the 3rd international Symposium, pp. 26–59, Paris, 1962
Guiraud J.P.: An H-theorem for a gas of rigid spheres in a bounded domain. In: Pichon, G. (eds) Theories cinetique classique et relativistes, pp. 29–58. CNRS, Paris (1975)
Hamdache K.: Initial boundary value problems for Boltzmann equation. Global existence of week solutions. Arch. Ration. Mech. Anal. 119, 309–353 (1992)
Hwang H.-J.: Regularity for the Vlasov–Poisson system in a convex domain. SIAM J. Math. Anal. 36(1), 121–171 (2004)
Hwang, H.-J., Velazquez, J.: Global existence for the Vlasov–Poisson system in bounded domain. Preprint 2007
Liu T.-P., Yu S.-H.: Initial-boundary value problem for one-dimensional wave solutions of the Boltzmann equation. Comm. Pure Appl. Math. 60(3), 295–356 (2007)
Liu T.-P., Yu S.-H.: Green’s function of Boltzmann equation, 3-D waves. Bull. Inst. Math. Acad. Sin. (N.S.) 1(1), 1–78 (2006)
Liu T.-P., Yu S.-H.: The Green’s function and large-time behavior of solutions for the one-dimensional Boltzmann equation. Comm. Pure Appl. Math. 57(12), 1543–1608 (2004)
Maslova N.B.: Nonlinear Evolution Equations, Kinetic Approach. World Scientific, Singapore (1993)
Mischler S.: On the initial boundary value problem for the Vlasov–Poisson–Boltzmann system. Commun. Math. Phys. 210, 447–466 (2000)
Masmoudi N., Saint-Raymond L.: From the Boltzmann equation to the Stokes–Fourier system in a bounded domain. Comm. Pure Appl. Math. 56(9), 1263–1293 (2003)
Shizuta Y.: On the classical solutions of the Boltzmann equation. Comm. Pure Appl. Math. 36, 705–754 (1983)
Shizuta Y., Asano K.: Global solutions of the Boltzmann equation in a bounded convex domain. Proc. Jpn. Acad. 53A, 3–5 (1977)
Strain R., Guo Y.: Exponential decay for soft potentials near Maxwellians. Arch. Ration. Mech. Anal. 187(2), 287–339 (2008)
Tabachnikov, S., Billiards, S.M.F. (1995)
Ukai, S.: Solutions of the Boltzmann equation. In: Pattern and Waves-Qualitative Analysis of Nonlinear Differential Equations, pp. 37–96, 1986
Ukai, S.: Private communications
Ukai S., Asano K.: On the initial boundary value problem of the linearized Boltzmann equation in an exterior domain. Proc. Jpn. Acad. 56, 12–17 (1980)
Villani, C.: Hypocoercivity. Memoir of AMS (in press)
Vidav I.: Spectra of perturbed semigroups with applications to transport theory. J. Math. Anal. Appl. 30, 264–279 (1970)
Yang T., Zhao H.-J.: A half-space problem for the Boltzmann equation with specular reflection boundary condition. Comm. Math. Phys. 255(3), 683–726 (2005)
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Guo, Y. Decay and Continuity of the Boltzmann Equation in Bounded Domains. Arch Rational Mech Anal 197, 713–809 (2010). https://doi.org/10.1007/s00205-009-0285-y
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DOI: https://doi.org/10.1007/s00205-009-0285-y