Lattice points on circles and discrete velocity models for the boltzmann equation

SIAM Journal on Mathematical Analysis

Volumn 37, Issue 6, 2006, Pages 1903-1922

  Fainsilber, Laura ^a   Kurlberg, P A R ^b   Wennberg, Bernt ^a  

^a CHALMERS UNIVERSITY OF TECHNOLOGY   (Sweden)

^b ROYAL INSTITUTE OF TECHNOLOGY   (Sweden)

Author keywords

Boltzmann equation; Discrete velocity model; Distribution of Gaussian primes; Multiplicative functions

Indexed keywords

APPROXIMATION THEORY; INTEGRAL EQUATIONS; MATHEMATICAL MODELS; MAXWELL EQUATIONS; NUMERICAL METHODS;

BOLTZMANN EQUATION; DISCRETE VELOCITY MODEL; GAUSSIAN PRIMES; MULTIPLICATIVE FUNCTIONS;

VELOCITY MEASUREMENT;

EID: 33750971724     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/040618916     Document Type: Article

References (26)

1. A. V. BOBYLEV AND N. BERNHOFF, Discrete velocity models and dynamical systems, in Lecture Notes on the Discretization of the Boltzmann Equation, N. Bellomo and R. Gatignol, eds., World Scientific, River Edge, NJ, 2003, pp. 203-222.
2. A. V. BOBYLEV, A. PALCZEWSKI, AND J. SCHNEIDER, On approximation of the Boltzmann equation by discrete velocity models, C. R. Acad. Sci. Paris Sér. I Math., 320 (1995), pp. 639-644.
3. A. PALCZEWSKI, J. SCHNEIDER, A BOBYLEV, A consistency result for a discrete-velocity model of the Boltzmann equation, SIAM J. Numer. Anal., 34 (1997), pp. 1865-1883.
4. A. BOBYLEV AND C. CERCIGNANI, Discrete velocity models without nonphysical invariants, J. Statist. Phys., 97 (1999), pp. 677-686.
5. T. CARLEMAN, Problèmes mathématiques dans la théorie cinétique des gaz, Almqvist & Wiksell, Uppsala, Sweden, 1957.
6. J. CILLERUELO, The distribution of lattice points on circles, J. Number Theory, 43 (1993), pp. 198-202.
7. C. CERCIGNANI, R. ILLNER, M. PULVIRENTI, The Mathematical Theory of Dilute Gases, Springer-Verlag, New York, 1994.
8. L. DESVILLETTES AND S. MISCHLER, About the splitting algorithm for Boltzmann and B.G.K. equations, Math. Models Methods Appl. Sci., 6 (1996), pp. 1079-1101.
9. L. DESVILLETTES AND B. WENNBERG, Regularity of solutions to the spatially homogeneous Boltzmann equation without cutoff, Comm. Partial Differential Equations, 29 (2003), pp. 133-156.
10. W. DUKE, Hyperbolic distribution problems and half-integral weight Maass forms, Invent. Math., 92 (1988), pp. 73-90.
11. P. ERDOS AND R. R. HALL, On the angular distribution of Gaussian integers with fixed norm, Discrete Math., 200 (1999), pp. 87-94. Paul Erdo′s memorial collection.
12. L. FAINSILBER, P. KURLBERG, AND B. WENNBERG, The Boltzmann equation on a two dimensional lattice, theoretical and numerical results, Transport. Theory Statist. Phys., to appear.
13. E. GROSSWALD, Representations of Integers as Sums of Squares, Springer-Verlag, New York, 1985.
14. E. P. GOLUBEVA AND O. M. FOMENKO, Asymptotic distribution of lattice points on the three-dimensional sphere, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 160 (1987), pp. 54-71, 297.
15. H. HALBERSTAM AND H.-E. RICHERT, On a result of R. R. Hall, J. Number Theory, 11 (1979). pp. 76-89.
16. G. H. HARDY AND E. M. WRIGHT, An Introduction to the Theory of Numbers, 3rd ed., Clarendon Press, Oxford, 1954.
17. V. PANFEROV AND A. HEINTZ, A new consistent discrete-velocity model for the Boltzmann equation, Math. Methods Appl. Sci., 25 (2002), pp. 571-593.
18. H. IWANIEC, Fourier coefficients of modular forms of half-integral weight, Invent. Math., 87 (1987), pp. 385-401.
19. I. KÁTAI AND I. KÖRNYEI, On the distribution of lattice points on circles, Ann. Univ. Sci. Budapest. Eötvös Sect. Math., 19 (1977), pp. 87-91.
20. I. KUBILYUS, The distribution of Gaussian primes in sectors and contours, Leningrad. Gos. Univ. Uč. Zap. Ser. Mat. Nauk, 137 (1950), pp. 40-52.
21. A. PALCZEWSKI AND J. SCHNEIDER, Existence, stability, and convergence of solutions of discrete velocity models to the Boltzmann equation, J. Statist. Phys., 91 (1998), pp. 307-326.
22. C. POMMERENKE, Über die Gleichverteilung von Gitterpunkten auf m-dimensionalen Ellipsoiden, Acta Arith., 5 (1959), pp. 227-257.
23. F. ROGIER AND J. SCHNEIDER, A direct method for solving the Boltzmann equation, Transport Theory Statist. Phys., 23 (1994), pp. 313-338.
24. V. V. VEDENYAPIN, Velocity inductive construction for mixtures, Transport Theory Statist. Phys., 28 (1999), pp. 727-742.
25. V. V. VEDENYAPIN, S. A. AMOSOV, AND L. TOSCANO, Invariants for Hamiltonians and kinetic equations, Russian Math. Surveys, 54 (1999), pp. 1056-1057.
26. E. WIRSING, Das asymptotische Verhalten von Summen über multiplikative Funktionen, Math. Ann., 143 (1961), pp. 75-102.