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

[2]

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

[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. 

[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.

[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.

[6]

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

[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.

[8]

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

[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.

[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.

[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.

[12]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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. 

[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.

[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.

[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

[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.

[25]

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

[26]

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

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.

[2]

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

[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. 

[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.

[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.

[6]

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

[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.

[8]

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

[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.

[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.

[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.

[12]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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. 

[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.

[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.

[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

[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.

[25]

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

[26]

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

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