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Self-Similar Profiles for Homoenergetic Solutions of the Boltzmann Equation: Particle Velocity Distribution and Entropy

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In this paper we study a class of solutions of the Boltzmann equation which have the form f (x, v, t) = g (vL (t) x, t) where L (t) =  A (ItA)−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.

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References

  1. 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)

    ADS  MathSciNet  Google Scholar 

  2. Bobylev A.V.: A class of invariant solutions of the Boltzmann equation. (Russian). Dokl. Akad. Nauk SSSR 231, 571–574 (1976)

    ADS  MathSciNet  Google Scholar 

  3. Bobylev A.V., Caraffini G.L., Spiga G.: On group invariant solutions of the Boltzmann equation. J. Math. Phys. 37, 2787–2795 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. 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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Cercignani C.: Mathematical Methods in Kinetic Theory. Plenum Press, New York (1969)

    Book  MATH  Google Scholar 

  6. Cercignani C.: Existence of homoenergetic affine flows for the Boltzmann equation. Arch. Ration. Mech. Anal. 105(4), 377–387 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cercignani C.: Shear flow of a granular material. J. Stat. Phys. 102(5), 1407–1415 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cercignani C.: The Boltzmann equation approach to the shear flow of a granular material. Philos. Trans. R. Soc. 360, 437–451 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Cercignani C., Illner R., Pulvirenti M.: The Mathematical Theory of Dilute Gases. Springer, Berlin (1994)

    Book  MATH  Google Scholar 

  10. Dayal K., James R.D.: Nonequilibrium molecular dynamics for bulk materials and nanostructures. J. Mech. Phys. Solids 58, 145–163 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Dayal K., James R.D.: Design of viscometers corresponding to a universal molecular simulation method. J. Fluid Mech. 691, 461–486 (2012)

    Article  ADS  MATH  Google Scholar 

  12. 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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Escobedo M., Velázquez J.J.L.: On the theory of weak turbulence for the nonlinear Schrödinger equation. Mem. AMS 238, 1124 (2015)

    Google Scholar 

  14. 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)

    MathSciNet  Google Scholar 

  15. Galkin V.S.: One-dimensional unsteady solution of the equation for the kinetic moments of a monatomic gas. PMM 28(1), 186–188 (1964)

    MathSciNet  Google Scholar 

  16. 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)

    Google Scholar 

  17. Gamba I.M., Panferov V., Villani C.: On the Boltzmann equation for diffusively excited granular media. Commun. Math. Phys. 246(3), 503–541 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Garzó V., Santos A.: Kinetic Theory of Gases in Shear Flows: Nonlinear Transport. Kluwer Academic Publishers, Dordrecht (2003)

    Book  MATH  Google Scholar 

  19. 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)

    MathSciNet  MATH  Google Scholar 

  20. James, R.D., Nota, A., Velázquez, J.J.L.: Long time asymptotics for homoenergetic solutions of the Boltzmann equation. Collision-dominated case. (in preparation)

  21. James, R.D., Nota, A., Velázquez, J.J.L.: Long time asymptotics for homoenergetic solutions of the Boltzmann equation. Hyperbolic-dominated case. (in preparation)

  22. Kato T.: Perturbation Theory for Linear Operators Classics in Mathematics. Springer, Berlin (1976)

    Google Scholar 

  23. 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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. 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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. 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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. 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)

    Google Scholar 

  27. 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)

    Google Scholar 

  28. Rudin W.: Functional Analysis. McGraw-Hill, New York (1973)

    MATH  Google Scholar 

  29. 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)

    MathSciNet  MATH  Google Scholar 

  30. Truesdell C., Muncaster R.G.: Fundamentals of Maxwell’s Kinetic Theory of a Simple Monatomic Gas. Academic Press, Cambridge (1980)

    Google Scholar 

  31. 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

  32. Villani C.: Topics in Optimal Transportation, vol. 58 of Graduate Studies in Mathematics. American Mathematical Society, Providence (2003)

    Google Scholar 

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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).

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Correspondence to Alessia Nota.

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Communicated by C. Mouhot

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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

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