Abstract
In this article we present an alternative formulation of the spatially homogeneous Boltzmann equation. Rewriting the weak form of the equation with shifted test functions and using Fourier techniques, it turns out that the transformed problem contains only a three-fold integral.
Explicit formulas for the transformed collision kernel are presented in the case of VHS models for hard and soft potentials. For isotropic Maxwellian molecules, a classical result by Bobylev is recovered, too.
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Alexandre, R.: Around 3D Boltzmann non linear operator without angular cutoff, a new formulation. M2AN Math. Model. Numer. Anal. 34(3), 575–590 (2000)
Alexandre, R., Desvillettes, L., Villani, C., Wennberg, B.: Entropy dissipation and long-range interactions. Arch. Ration. Mech. Anal. 152(4), 327–355 (2000)
Alexandre, R., El Safadi, M.: Littlewood-Paley theory and regularity issues in Boltzmann homogeneous equations, I: non-cutoff case and Maxwellian molecules. Math. Models Methods Appl. Sci. 15(6), 907–920 (2005)
Arkeryd, L.: Intermolecular forces of infinite range and the Boltzmann equation. Arch. Ration. Mech. Anal. 77(1), 11–21 (1981)
Bobylev, A.V.: Fourier transform method in the theory of the Boltzmann equation for Maxwell molecules. Dokl. Akad. Nauk SSSR 225, 1041–1044 (1975)
Bobylev, A.V., Rjasanow, S.: Difference scheme for the Boltzmann equation based on fast Fourier transform. Eur. J. Mech. B/Fluids 16(2), 293–306 (1997)
Bobylev, A.V., Rjasanow, S.: Fast deterministic method of solving the Boltzmann equation for hard spheres. Eur. J. Mech. B/Fluids 18(5), 869–887 (1999)
Bouchut, F., Desvillettes, L.: A proof of the smoothing properties of the positive part of Boltzmann’s kernel. Rev. Mat. Iberoam. 14(1), 47–61 (1998)
Desvillettes, L.: About the use of the Fourier transform for the Boltzmann equation. In: Summer School on “Methods and Models of Kinetic Theory” (M&MKT 2002). Riv. Mat. Univ. Parma (7) 2, 1–99 (2003)
Desvillettes, L., Wennberg, B.: Smoothness of the solution of the spatially homogeneous Boltzmann equation without cutoff. Commun. Partial Differ. Equ. 29(1–2), 133–155 (2004)
Duduchava, R., Rjasanow, S.: Mapping properties of the Boltzmann collision operator. Integral Equ. Oper. Theory 52(1), 61–84 (2005)
Gelfand, I.M., Shilov, G.E.: Generalized Functions. Academic Press, New York (1968)
Ibragimov, I., Rjasanow, S.: Numerical solution of the Boltzmann equation on the uniform grid. Computing 69(2), 163–186 (2002)
Karwowski, A.: Grad’s 13 moment equations in a modified form. J. Stat. Phys. doi: 10.1007/s10955-006-9216-6 (2007)
Lions, P.-L.: Compactness in Boltzmann’s equation via Fourier integral operators and applications, I. J. Math. Kyoto Univ. 34(2), 391–427 (1994)
Lions, P.-L.: Compactness in Boltzmann’s equation via Fourier integral operators and applications, II. J. Math. Kyoto Univ. 34(2), 429–461 (1994)
Mouhot, C., Pareschi, L.: Fast methods for the Boltzmann collision integral. C. R. Math. Acad. Sci. Paris 339(1), 71–76 (2004)
Mouhot, C., Villani, C.: Regularity theory for the spatially homogeneous Boltzmann equation with cut-off. Arch. Ration. Mech. Anal. 173(2), 169–212 (2004)
Pareschi, L., Perthame, B.: A Fourier spectral method for homogeneous Boltzmann equations. Transp. Theory Stat. Phys. 25, 369–382 (1996)
Pareschi, L., Russo, G.: Numerical solution of the Boltzmann equation, I: spectrally accurate approximation of the collision operator. SIAM J. Numer. Anal. 37(4), 1217–1245 (2000)
Pareschi, L., Russo, G.: On the stability of spectral methods for the homogeneous Boltzmann equation. Transp. Theory Stat. Phys. 29(3-5), 431–447 (2000)
Pareschi, L., Toscani, G., Villani, C.: Spectral methods for the non cut-off Boltzmann equation and numerical grazing collision limit. Numer. Math. 93(3), 527–548 (2003)
Taylor, M.E.: Partial Differential Equations—Basic Theory. Springer, New York (1996)
Villani, C.: On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations. Arch. Ration. Mech. Anal. 143(3), 273–307 (1998)
Villani, C.: Regularity estimates via the entropy dissipation for the spatially homogeneous Boltzmann equation without cut-off. Rev. Mat. Iberoam. 15(2), 335–352 (1999)
Wennberg, B.: Regularity in the Boltzmann equation and the radon transform. Commun. Partial Differ. Equ. 19(11/12), 2057–2074 (1994)
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Kirsch, R., Rjasanow, S. A Weak Formulation of the Boltzmann Equation Based on the Fourier Transform. J Stat Phys 129, 483–492 (2007). https://doi.org/10.1007/s10955-007-9374-1
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DOI: https://doi.org/10.1007/s10955-007-9374-1