Abstract
In this paper we study a class of solutions of the Boltzmann equation which have the form f (x, v, t) = g (v − L (t) x, t) where L (t) = A (I + tA)−1 with the matrix A describing a shear flow or a dilatation or a combination of both. These solutions are known as homoenergetic solutions. We prove the existence of homoenergetic solutions for a large class of initial data. For different choices for the matrix A and for different homogeneities of the collision kernel, we characterize the long time asymptotics of the velocity distribution for the corresponding homoenergetic solutions. For a large class of choices of A we then prove rigorously, in the case of Maxwell molecules, the existence of self-similar solutions of the Boltzmann equation. The latter are non Maxwellian velocity distributions and describe far-from-equilibrium flows. For Maxwell molecules we obtain exact formulas for the H-function for some of these flows. These formulas show that in some cases, despite being very far from equilibrium, the relationship between density, temperature and entropy is exactly the same as in the equilibrium case. We make conjectures about the asymptotics of homoenergetic solutions that do not have self-similar profiles.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bobylev A.V.: The method of the Fourier transform in the theory of the Boltzmann equation for Maxwell molecules. (Russian) Dokl. Akad. Nauk. SSSR 225, 1041–1044 (1975)
Bobylev A.V.: A class of invariant solutions of the Boltzmann equation. (Russian). Dokl. Akad. Nauk SSSR 231, 571–574 (1976)
Bobylev A.V., Caraffini G.L., Spiga G.: On group invariant solutions of the Boltzmann equation. J. Math. Phys. 37, 2787–2795 (1996)
Bobylev A.V., Gamba I.M., Panferov V.: Moment inequalities and high-energy tails for the Boltzmann equations with inelastic interactions. J. Stat. Phys. 116(5–6), 1651–1682 (2004)
Cercignani C.: Mathematical Methods in Kinetic Theory. Plenum Press, New York (1969)
Cercignani C.: Existence of homoenergetic affine flows for the Boltzmann equation. Arch. Ration. Mech. Anal. 105(4), 377–387 (1989)
Cercignani C.: Shear flow of a granular material. J. Stat. Phys. 102(5), 1407–1415 (2001)
Cercignani C.: The Boltzmann equation approach to the shear flow of a granular material. Philos. Trans. R. Soc. 360, 437–451 (2002)
Cercignani C., Illner R., Pulvirenti M.: The Mathematical Theory of Dilute Gases. Springer, Berlin (1994)
Dayal K., James R.D.: Nonequilibrium molecular dynamics for bulk materials and nanostructures. J. Mech. Phys. Solids 58, 145–163 (2010)
Dayal K., James R.D.: Design of viscometers corresponding to a universal molecular simulation method. J. Fluid Mech. 691, 461–486 (2012)
Escobedo M., Mischler S., Rodriguez Ricard M.: On self-similarity and stationary problem for fragmentation and coagulation models. Ann. Inst. Henri Poincaré (C) Anal. Non Linéaire 22(1), 99–125 (2005)
Escobedo M., Velázquez J.J.L.: On the theory of weak turbulence for the nonlinear Schrödinger equation. Mem. AMS 238, 1124 (2015)
Galkin V.S.: On a class of solutions of Grad’s moment equation. PMM 22(3), 386–389 (1958) (Russian version PMM 20, 445–446 1956)
Galkin V.S.: One-dimensional unsteady solution of the equation for the kinetic moments of a monatomic gas. PMM 28(1), 186–188 (1964)
Galkin V.S.: Exact solutions of the kinetic-moment equations of a mixture of monatomic gases. Fluid Dyn. (Izv. AN SSSR) 1(5), 41–50 (1966)
Gamba I.M., Panferov V., Villani C.: On the Boltzmann equation for diffusively excited granular media. Commun. Math. Phys. 246(3), 503–541 (2004)
Garzó V., Santos A.: Kinetic Theory of Gases in Shear Flows: Nonlinear Transport. Kluwer Academic Publishers, Dordrecht (2003)
Ikenberry E., Truesdell C.: On the pressures and the flux of energy in a gas according to Maxwell’s kinetic theory, I. J. Ration. Mech. Anal. 5(1), 1–54 (1956)
James, R.D., Nota, A., Velázquez, J.J.L.: Long time asymptotics for homoenergetic solutions of the Boltzmann equation. Collision-dominated case. (in preparation)
James, R.D., Nota, A., Velázquez, J.J.L.: Long time asymptotics for homoenergetic solutions of the Boltzmann equation. Hyperbolic-dominated case. (in preparation)
Kato T.: Perturbation Theory for Linear Operators Classics in Mathematics. Springer, Berlin (1976)
Kierkels A., Velázquez J.J.L.: On the transfer of energy towards infinity in the theory of weak turbulence for the nonlinear Schrödinger equation. J. Stat. Phys. 159, 668–712 (2015)
Niethammer B., Velázquez J.J.L.: Self-similar solutions with fat tails for Smoluchowski’s coagulation equation with locally bounded kernels. Commun. Math. Phys. 318(2), 505–532 (2013)
Niethammer B., Throm S., Velázquez J.J.L.: Self-similar solutions with fat tails for Smoluchowski’s coagulation equation with singular kernels. Ann. Inst. Henri Poincaré (C) Nonlinear Anal. 33(5), 1223–1257 (2016)
Nikol’skii A.A.: On a general class of uniform motions of continuous media and rarefied gas. Sov. Eng. J. 5(6), 757–760 (1965)
Nikol’skii A.A.: Three-dimensional homogeneous expansion–contraction of a rarefied gas with power-law interaction functions. DAN SSSR 151(3), 522–524 (1963)
Rudin W.: Functional Analysis. McGraw-Hill, New York (1973)
Truesdell C.: On the pressures and flux of energy in a gas according to Maxwell’s kinetic theory, II. J. Ration. Mech. Anal. 5, 55–128 (1956)
Truesdell C., Muncaster R.G.: Fundamentals of Maxwell’s Kinetic Theory of a Simple Monatomic Gas. Academic Press, Cambridge (1980)
Villani, C.: A review of mathematical topics in collisional kinetic theory. Hand-Book of Mathematical Fluid Dynamics vol. 1, pp. 71–305. North-Holland, Amsterdam 2002
Villani C.: Topics in Optimal Transportation, vol. 58 of Graduate Studies in Mathematics. American Mathematical Society, Providence (2003)
Acknowledgements
We thank Stefan Müller, who motivated us to study this problem, indulged us in useful discussions and made suggestions on the topic. The work of R.D.J. was supported byONR(N00014-14-1-0714), AFOSR(FA9550-15-1-0207), NSF (DMREF-1629026), and the MURI program(FA9550-18-1-0095, FA9550-16-1-0566). A.N. and J.J.L.V. acknowledge support through the CRC 1060 Themathematics of emergent effects of the University of Bonn that is funded through the German Science Foundation (DFG).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Communicated by C. Mouhot
Rights and permissions
About this article
Cite this article
James, R.D., Nota, A. & Velázquez, J.J.L. Self-Similar Profiles for Homoenergetic Solutions of the Boltzmann Equation: Particle Velocity Distribution and Entropy. Arch Rational Mech Anal 231, 787–843 (2019). https://doi.org/10.1007/s00205-018-1289-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-018-1289-2