October  2021, 14(5): 749-765. doi: 10.3934/krm.2021023

Heterogeneous discrete kinetic model and its diffusion limit

Department of Mathematical Sciences, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon, 34141, Rep. of Korea

* Corresponding author: Yong-Jung Kim

Received  November 2020 Revised  April 2021 Published  October 2021 Early access  June 2021

Fund Project: This research was supported by National Research Foundation of Korea (NRF-2017R1A2B2010398)

A revertible discrete velocity kinetic model is introduced when the environment is spatially heterogeneous. It is proved that the parabolic scale singular limit of the model exists and satisfies a new heterogeneous diffusion equation that depends on the diffusivity and the turning frequency together. An energy functional is introduced which takes into account spatial heterogeneity in the velocity field. The monotonicity of the energy functional is the key to obtain uniform estimates needed for the weak convergence proof. The Div-Curl lemma completes the strong convergence proof.

Citation: Ho-Youn Kim, Yong-Jung Kim, Hyun-Jin Lim. Heterogeneous discrete kinetic model and its diffusion limit. Kinetic & Related Models, 2021, 14 (5) : 749-765. doi: 10.3934/krm.2021023
References:
[1]

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Y.-J. Kim and H. Seo, Model for heterogeneous diffusion, SIAM J. Appl. Math., 81 (2021), 335-354.  doi: 10.1137/19M130087X.  Google Scholar

[13]

P. L. Lions and G. Toscani, Diffusive limit for finite velocity Boltzmann kinetic models, Rev. Mat. Iberoamericana, 13 (1997), 473-513.  doi: 10.4171/RMI/228.  Google Scholar

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B. Ph van Milligen, P. D. Bons, B. A. Carreras and R. Sánchez, On the applicability of Fick's law to diffusion in inhomogeneous systems, European Journal of Physics, 26 (2005), 913. doi: 10.1088/0143-0807/26/5/023.  Google Scholar

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H. G. OthmerS. R. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298.  doi: 10.1007/BF00277392.  Google Scholar

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T. Platkowski and R. Illner, Discrete velocity models of the Boltzmann equation: A survey on the mathematical aspects of the theory, SIAM Review, 30 (1988), 213-255.  doi: 10.1137/1030045.  Google Scholar

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A. Pulvirenti and G. Toscani, Fast diffusion as a limit of a two-velocity kinetic model, Rend. Circ. Mat. Palermo (2) Suppl., 45 (1996), 521-528.   Google Scholar

[20]

F. Salvarani, Diffusion limits for the initial-boundary value problem of the Goldstein-Taylor model, Rend. Sem. Mat. Univ. Politec. Torino, 57 (1999), 209-220.   Google Scholar

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F. Salvarani and G. Toscani, The diffusive limit of Carleman-type models in the range of very fast diffusion equations, J. Evol. Equ., 9 (2009), 67-80.  doi: 10.1007/s00028-009-0005-y.  Google Scholar

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F. Salvarani and J. L. Vázquez, The diffusive limit for Carleman type kinetic models, Nonlinearity, 18 (2005), 1223-1248.  doi: 10.1088/0951-7715/18/3/015.  Google Scholar

[23]

H. Seo and Y.-J. Kim, Biological invasion in a periodic environment, J. Math. Biol., submitted 2020. http://amath.kaist.ac.kr/papers/Kim/58.pdf Google Scholar

[24]

N. ShigesadaK. Kawasaki and E. Teramoto, Traveling periodic waves in heterogeneous environment, Theoret. Population Biol., 30 (1986), 143-160.  doi: 10.1016/0040-5809(86)90029-8.  Google Scholar

[25]

G. I. Taylor, Diffusion by continuous movements, Proc. London Math. Soc., 2 (1922), 196-212.  doi: 10.1112/plms/s2-20.1.196.  Google Scholar

[26]

M. Wereide, La diffusion dúne solution dont la concentration et la temperature sont variables, Ann. Physique, 2 (1914), 67-83.   Google Scholar

show all references

References:
[1]

J. E. Broadwell, Shock structure in a simple discrete velocity gas, Phys. of Fluids, 7 (1964), 1243-1247.  doi: 10.1063/1.1711368.  Google Scholar

[2]

T. Carleman, Problèmes Mathématiques dans la Théorie Cinétique de Gaz, Publ. Sci. Inst. Mittag-Leffler. 2 Almqvist & Wiksells Boktryckeri Ab, Uppsala, 1957.  Google Scholar

[3]

S. Chapman, On the Brownian displacements and thermal diffusion of grains suspended in a non-uniform fluid, Proc. Roy. Soc. Lond. A, 119 (1928), 34-54.   Google Scholar

[4]

B. Choi and Y.-J. Kim, Diffusion of biological organisms: Fickian and Fokker–Planck type diffusions, SIAM J. Appl. Math., 79 (2019), 1501-1527.  doi: 10.1137/18M1163944.  Google Scholar

[5]

S.-H. Choi and Y.-J. Kim, Chemotactic traveling waves by the metric of food, SIAM J. Appl. Math., 75 (2015), 2268-2289.  doi: 10.1137/15100429X.  Google Scholar

[6]

A. Fick, On liquid diffusion, Phil. Mag., 10 (1855), 30-39.   Google Scholar

[7]

S. Goldstein, On diffusion by discontinuous movements and on the telegraph equation, Quart. J. Mech. Appl. Math., 4 (1951), 129-156.  doi: 10.1093/qjmam/4.2.129.  Google Scholar

[8]

T. Hillen, Existence theory for correlated random walks on bounded domains, Can. Appl. Math. Q., 18 (2010), 1-40.   Google Scholar

[9]

T. Hillen and H. G. Othmer, The diffusion limit of transport equations derived from velocity-jump processes, SIAM J. Appl. Math., 61 (2000), 751-775.  doi: 10.1137/S0036139999358167.  Google Scholar

[10]

T. Hillen and K. J. Painter, Transport and anisotropic diffusion models for movement in oriendted habitats, Dispersal, Individual Movement and Spatial Ecology, Lecture Notes in Math., Springer, Heidelberg, 2071 (2013), 177–222. doi: 10.1007/978-3-642-35497-7_7.  Google Scholar

[11]

M. Kac, A stochastic model related to the telegrapher's equation, Rocky Mountain J. Math., 4 (1974), 497-509.  doi: 10.1216/RMJ-1974-4-3-497.  Google Scholar

[12]

Y.-J. Kim and H. Seo, Model for heterogeneous diffusion, SIAM J. Appl. Math., 81 (2021), 335-354.  doi: 10.1137/19M130087X.  Google Scholar

[13]

P. L. Lions and G. Toscani, Diffusive limit for finite velocity Boltzmann kinetic models, Rev. Mat. Iberoamericana, 13 (1997), 473-513.  doi: 10.4171/RMI/228.  Google Scholar

[14]

B. Ph van Milligen, P. D. Bons, B. A. Carreras and R. Sánchez, On the applicability of Fick's law to diffusion in inhomogeneous systems, European Journal of Physics, 26 (2005), 913. doi: 10.1088/0143-0807/26/5/023.  Google Scholar

[15]

H. G. OthmerS. R. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298.  doi: 10.1007/BF00277392.  Google Scholar

[16]

H. G. Othmer and T. Hillen, The diffusion limit of transport equations II: Chemotaxis equations, SIAM J. Appl. Math., 62 (2002), 1222-1250.  doi: 10.1137/S0036139900382772.  Google Scholar

[17]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[18]

T. Platkowski and R. Illner, Discrete velocity models of the Boltzmann equation: A survey on the mathematical aspects of the theory, SIAM Review, 30 (1988), 213-255.  doi: 10.1137/1030045.  Google Scholar

[19]

A. Pulvirenti and G. Toscani, Fast diffusion as a limit of a two-velocity kinetic model, Rend. Circ. Mat. Palermo (2) Suppl., 45 (1996), 521-528.   Google Scholar

[20]

F. Salvarani, Diffusion limits for the initial-boundary value problem of the Goldstein-Taylor model, Rend. Sem. Mat. Univ. Politec. Torino, 57 (1999), 209-220.   Google Scholar

[21]

F. Salvarani and G. Toscani, The diffusive limit of Carleman-type models in the range of very fast diffusion equations, J. Evol. Equ., 9 (2009), 67-80.  doi: 10.1007/s00028-009-0005-y.  Google Scholar

[22]

F. Salvarani and J. L. Vázquez, The diffusive limit for Carleman type kinetic models, Nonlinearity, 18 (2005), 1223-1248.  doi: 10.1088/0951-7715/18/3/015.  Google Scholar

[23]

H. Seo and Y.-J. Kim, Biological invasion in a periodic environment, J. Math. Biol., submitted 2020. http://amath.kaist.ac.kr/papers/Kim/58.pdf Google Scholar

[24]

N. ShigesadaK. Kawasaki and E. Teramoto, Traveling periodic waves in heterogeneous environment, Theoret. Population Biol., 30 (1986), 143-160.  doi: 10.1016/0040-5809(86)90029-8.  Google Scholar

[25]

G. I. Taylor, Diffusion by continuous movements, Proc. London Math. Soc., 2 (1922), 196-212.  doi: 10.1112/plms/s2-20.1.196.  Google Scholar

[26]

M. Wereide, La diffusion dúne solution dont la concentration et la temperature sont variables, Ann. Physique, 2 (1914), 67-83.   Google Scholar

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