Abstract
Boundary effects are central to the dynamics of the dilute particles governed by the Boltzmann equation. In this paper, we study both the diffuse reflection and the specular reflection boundary value problems for the Boltzmann equation with a soft potential, in which the collision kernel is ruled by the inverse power law. For the diffuse reflection boundary condition, based on an L 2 argument and its interplay with intricate \({L^\infty}\) analysis for the linearized Boltzmann equation, we first establish the global existence and then obtain the exponential decay in \({L^\infty}\) space for the nonlinear Boltzmann equation in general classes of bounded domain. It turns out that the zero lower bound of the collision frequency and the singularity of the collision kernel lead to some new difficulties for achieving the a priori \({L^\infty}\) estimates and time decay rates of the solution. In the course of the proof, we capture some new properties of the probability integrals along the stochastic cycles and improve the \({L^2-L^\infty}\) theory to give a more direct approach to overcome those difficulties. As to the specular reflection condition, our key contribution is to develop a new time-velocity weighted \({L^\infty}\) theory so that we could deal with the greater difficulties stemming from the complicated velocity relations among the specular cycles and the zero lower bound of the collision frequency. From this new point, we are also able to prove that the solutions of the linearized Boltzmann equation tend to equilibrium exponentially in \({L^\infty}\) space with the aid of the L 2 theory and a bootstrap argument. These methods, in the latter case, can be applied to the Boltzmann equation with soft potential for all other types of boundary condition.
Article PDF
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
Avoid common mistakes on your manuscript.
References
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(3), 271–287 (1993)
Arkeryd L., Maslova N.: On diffuse reflection at the boundary for the Boltzmann equation and related equations. J. Stat. Phys. 77(5–6), 1051–1077 (1994)
Briant, M., Guo, Y.: Asymptotic stability of the Boltzmann equation with Maxwell boundary conditions. arXiv:1511.01305
Cercignani C.: On the initial-boundary value problem for the Boltzmann equation. Arch. Ration. Mech. Anal. 116(4), 307–315 (1992)
Cercignani C., Illner R., Pulvirenti M.: The Mathematical Theory of Dilute Gases. Springer, Berlin (1994)
Chapman, S., Colwing, T. G.: The Mathematical Theory of Non-uniform Gases, 3rd edn. Cambridge Math. Lib., Cambridge University Press, Cambridge, 1990
Desvillettes L.: Convergence to equilibrium in large time for Boltzmann and B.G.K. equations. Arch. Rational Mech. Anal. 110(1), 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.J., Lions P.-L.: On the Cauchy problem for Boltzmann equations: global existence and weak stability. Ann. Math. 130(2), 321–366 (1989)
DiPerna R.J., Lions P.-L.: Global weak solution of Vlasov-Maxwell systems. Commun. Pure Appl. Math. 42, 729–757 (1989)
Duan R.-J.: Global smooth dynamics of a fully ionized plasma with long-range collisions. Ann. Inst. H. Poincar Anal. Non Linaire 31(4), 751–778 (2014)
Duan R.-J., Liu S.-Q.: Stability of the rarefaction wave of the Vlasov-Poisson-Boltzmann system. SIAM J. Math. Anal. 47(5), 3585–3647 (2015)
Duan R.-J., Liu S.-Q., Yang T., Zhao H.-J.: Stability of the nonrelativistic Vlasov–Maxwell–Boltzmann system for angular non-cutoff potentials. Kinet. Relat. Models 6(1), 159–204 (2013)
Duan R.-J., Yang T., Zhao H.-J.: The Vlasov-Poisson-Boltzmann system in the whole space: the hard potential case. J. Differ. Equ. 252(12), 6356–6386 (2012)
Duan R.-J., Yang T., Zhao H.-J.: The Vlasov–Poisson–Boltzmann system for soft potentials. Math. Models Methods Appl. Sci. 23(6), 979–1028 (2013)
Esposito R., Guo Y., Marra R.: Phase transition in a Vlasov–Boltzmann binary mixture. Commun. Math. Phys. 296(1), 1–33 (2010)
Esposito R., Guo Y., Kim C., Marra R.: Non-isothermal boundary in the Boltzmann theory and Fourier law. Commun. Math. Phys. 323(1), 177–239 (2013)
Esposito, R., Guo, Y., Kim, C., Marra, R.: Stationary Solutions to the Boltzmann Equation in the Hydrodynamic Limit. arXiv:1502.05324
Esposito R., Lebowitz J. L., Marra R.: Hydrodynamic limit of the stationary Boltzmann equation in a slab. Commun. Math. Phys. 160, 49–80 (1994)
Esposito R., Lebowitz J. L., Marra R.: The Navier–Stokes limit of stationary solutions of the nonlinear Boltzmann equation. J. Stat. Phys. 78, 389–412 (1995)
Golse F., Perthame B., Sulem C.: On a boundary layer problem for the nonlinear Boltzmann equation Arch. Ration. Mech. Anal. 103, 81–96 (1986)
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. 1963 Rarefied Gas Dynamics. Proceedings of the 3rd international Symposium, pp. 26–59, Paris, 1962
Guo Y.: Classical solutions to the Boltzmann equation for molecules with an angular cutoff. Arch. Ration. Mech. Anal. 169(4), 305–353 (2003)
Guo Y.: The Boltzmann equation in the whole space. Indiana Univ. Math. J. 53(4), 1081–1094 (2004)
Guo Y.: Boltzmann diffusive limit beyond the Navier–Stokes approximation. Commun. Pure. Appl. Math. 55(9), 0626–0687 (2006)
Guo Y.: Decay and continuity of the Boltzmann equation in bounded domains. Arch. Ration. Mech. Anal. 197(3), 713–809 (2010)
Guo, Y., Kim, C., Tonon, D., Trescases, A.: Regularity of the Boltzmann equation in convex domains, to appear in Invent. Math. doi:10.1007/s00222-016-0670-8
Guo Y., Kim C., Tonon D., Trescases A.: BV-regularity of the Boltzmann equation in non-convex domains. Arch. Ration. Mech. Anal. 220(3), 1045–1093 (2016)
Hamdache K.: Initial boundary value problems for Boltzmann equation: global existence of week solutions. Arch. Ration. Mech. Anal. 119(4), 309–353 (1992)
Kim C.: Formation and propagation of discontinuity for Boltzmann equation in non-convex domains. Commun. Math. Phys. 308(3), 641–701 (2011)
Kuo H.-W., Liu T.-P., Tsai L.-C.: Equilibrating effects of boundary and collision in rarefied gases. Commun. Math. Phys. 328(2), 421–480 (2014)
Liu T.-P., Yu S.-H.: Invariant manifolds for steady Boltzmann flows and applications. Arch. Ration. Mech. Anal. 209(3), 869–997 (2013)
Liu T.-P., Yu S.-H.: The Green’s function and large-time behavior of solutions for the one-dimensional Boltzmann equation. Commun. Pure Appl. Math. 57(12), 1543–1608 (2004)
Liu T.-P., Yu S.-H.: Initial-boundary value problem for one-dimensional wave solutions of the Boltzmann equation. Commun. Pure Appl. Math. 60(3), 295–356 (2007)
Liu T.-P., Yu S.-H.: Boltzmann equation, boundary effects. Discr. Contin. Dyn. Syst. 24(1), 145–157 (2009)
Masmoudi N., Saint-Raymond L.: From the Boltzmann equation to the Stokes-Fourier system in a bounded domain. Commun. Pure Appl. Math. 56(9), 1263–1293 (2003)
Mischler S.: On the initial boundary value problem for the Vlasov–Poisson–Boltzmann system. Commun. Math. Phys. 210(2), 447–466 (2000)
Shizuta Y., Asano K.: Global solutions of the Boltzmann equation in a bounded convex domain. Proc. Jpn Acad. Ser. A Math. Sci. 53(1), 3–5 (1977)
Sone, Y.: Molecular Gas Dynamics. Theory, Techniques, and Applications, Modeling and Simulation in Science, Engineering and Technology. Birkhäuser Boston, Inc., Boston, MA, 2007
Strain R. M.: Asymptotic stability of the relativistic Boltzmann equation for the soft potentials. Commun. Math. Phys. 300(2), 529–597 (2010)
Strain R.M., Guo Y.: Almost exponential decay near Maxwellian. Commun. Partial Differ. Equ. 31, 417–429 (2006)
Strain R.M., Guo Y.: Exponential decay for soft potentials near Maxwellian. Arch. Ration. Mech. Anal. 187(2), 287–339 (2008)
Ukai S.: Solutions of the Boltzmann equations In: pattern and Waves–Qualitative Analysis of Nonlinear Differential Equations. Stud. Math. Appl. 18, 37–96 (1986)
Ukai S., Yang T.: The Boltzmann equation in the space \({L^2\cap L^\infty_\beta}\): global and time-periodic solutions. Anal. Appl. (Singap.) 4, 263–310 (2006)
Vidav I.: Spectra of perturbed semigroups with applications to transport theory. J. Math. Anal. Appl. 30, 264–279 (1970)
Villani, C.: Hypocoercivity, Mem. Am. Math. Soc. 202(950), iv+141 (2009)
Yang T., Zhao H.-J.: A half-space problem for the Boltzmann equation with specular reflection boundary condition. Commun. Math. Phys. 255(3), 683–726 (2005)
Yu S.-H.: Stochastic formulation for the initial-boundary value problems of the Boltzmann equation. Arch. Ration. Mech. Anal. 192(2), 217–274 (2009)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by L. Saint-Raymond
Rights and permissions
About this article
Cite this article
Liu, S., Yang, X. The Initial Boundary Value Problem for the Boltzmann Equation with Soft Potential. Arch Rational Mech Anal 223, 463–541 (2017). https://doi.org/10.1007/s00205-016-1038-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-016-1038-3