On the vanishing viscosity limit of a chemotaxis model

Hua Chen; Jian-Meng Li; Kelei Wang

doi:10.3934/dcds.2020101

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March 2020, 40(3): 1963-1987. doi: 10.3934/dcds.2020101

On the vanishing viscosity limit of a chemotaxis model

Hua Chen , Jian-Meng Li and Kelei Wang ^,

School of Mathematics and Statistics, Wuhan University, Computational Science Hubei Key Laboratory, Wuhan University, Wuhan 430072, China

^* Corresponding author: Kelei Wang

Received September 2019 Published December 2019

Fund Project: H. Chen and K. Wang were supported by the NSFC grant no. 11631011.

A vanishing viscosity problem for the Patlak-Keller-Segel model is studied in this paper. This is a parabolic-parabolic system in a bounded domain $ \Omega\subset \mathbb{R}^n $, with a vanishing viscosity $ \varepsilon\to 0 $. We show that if the initial value lies in $ W^{1, p} $ with $ p>\max\{2, n\} $, then there exists a unique solution $ (u_\varepsilon, v_\varepsilon) $ with its lifespan independent of $ \varepsilon $. Furthermore, as $ \varepsilon\rightarrow 0 $, $ (u_\varepsilon, v_\varepsilon) $ converges to the solution $ (u, v) $ of the limiting system in a suitable sense.

Keywords: Chemotaxis, vanishing viscosity, parabolic-parabolic system, parabolic-ODE system.

Mathematics Subject Classification: Primary: 35K51, 35Q92; Secondary: 35M12.

Citation: Hua Chen, Jian-Meng Li, Kelei Wang. On the vanishing viscosity limit of a chemotaxis model. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1963-1987. doi: 10.3934/dcds.2020101

References:

[1]	M. Bramanti, L. Brandolini, E. Lanconelli and F. Uguzzoni, Non-divergence equations structured on Hörmander vector fields: Heat kernels and Harnack inequalities, Mem. Amer. Math. Soc., 204 (2010). doi: 10.1090/S0065-9266-09-00605-X.
[2]	L. Corrias, B. Perthame and H. Zaag, A chemotaxis model motivated by angiogenesis, C. R. Math. Acad. Sci. Paris, 336 (2003), 141-146. doi: 10.1016/S1631-073X(02)00008-0.
[3]	L. Corrias, B. Perthame and H. Zaag, Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math., 72 (2004), 1-28. doi: 10.1007/s00032-003-0026-x.
[4]	C. Deng and T. Li, Well-posedness of a 3D parabolic-hyperbolic Keller-Segel system in the Sobolev space framework, J. Differential Equations, 257 (2014), 1311-1332. doi: 10.1016/j.jde.2014.05.014.
[5]	E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.
[6]	L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998.
[7]	E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.
[8]	O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R.I. 1968.
[9]	D. Li, T. Li and K. Zhao, On a hyperbolic-parabolic system modeling chemotaxis, Math. Models Methods Appl. Sci., 21 (2011), 1631-1650. doi: 10.1142/S0218202511005519.
[10]	P. Li, Geometric Analysis, Cambridge Studies in Advanced Mathematics, 134. Cambridge University Press, Cambridge, 2012. doi: 10.1017/CBO9781139105798.
[11]	T. Li, R. H. Pan and K. Zhao, Global dynamics of a hyperbolic-parabolic model arising from chemotaxis, SIAM J. Appl. Math., 72 (2012), 417-443. doi: 10.1137/110829453.
[12]	G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302.
[13]	H. G. Othmer and A. Stevens, Aggregation, blowup, and collapse: The ABCs of taxis in reinforced random walks, SIAM J. Appl. Math., 57 (1997), 1044-1081. doi: 10.1137/S0036139995288976.
[14]	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.
[15]	H. Y. Peng, H. Y. Wen and C. J. Zhu, Global well-posedness and zero diffusion limit of classical solutions to 3D conservation laws arising in chemotaxis, Z. Angew. Math. Phys., 65 (2014), 1167-1188. doi: 10.1007/s00033-013-0378-1.
[16]	T. Suzuki, Free Energy and Self-Interacting Particles, Progress in Nonlinear Differential Equations and their Applications, 62. Birkhäuser Boston, Inc., Boston, MA, 2005. doi: 10.1007/0-8176-4436-9.
[17]	M. E. Taylor, Partial Differential Equations III: Nonlinear Equations, Second edition. Applied Mathematical Sciences, 117. Springer, New York, 2011. doi: 10.1007/978-1-4419-7049-7.
[18]	Z.-A. Wang and T. Hillen, Shock formation in a chemotaxis model, Math. Methods Appl. Sci., 31 (2008), 45-70. doi: 10.1002/mma.898.
[19]	Z.-A. Wang, Z. Y. Xiang and P. Yu, Asymptotic dynamics on a singular chemotaxis system modeling onset of tumor angiogenesis, J. Differential Equations, 260 (2016), 2225-2258. doi: 10.1016/j.jde.2015.09.063.
[20]	Y. Yang, H. Chen and W. A. Liu, On existence of global solutions and blow-up to a system of reaction diffusion equations modeling chemotaxis, SIAM J. Math. Anal., 33 (2001), 763-785. doi: 10.1137/S0036141000337796.
[21]	Y. Yang, H. Chen, W. A. Liu and B. D. Sleeman, The solvability of some chemotaxis systems, J. Differential Equations, 212 (2005), 432-451. doi: 10.1016/j.jde.2005.01.002.
[22]	M. Zhang and C. J. Zhu, Global existence of solutions to a hyperbolic-parabolic system, Proc. Amer. Math. Soc., 135 (2007), 1017-1027. doi: 10.1090/S0002-9939-06-08773-9.

show all references

References:

[1]	M. Bramanti, L. Brandolini, E. Lanconelli and F. Uguzzoni, Non-divergence equations structured on Hörmander vector fields: Heat kernels and Harnack inequalities, Mem. Amer. Math. Soc., 204 (2010). doi: 10.1090/S0065-9266-09-00605-X.
[2]	L. Corrias, B. Perthame and H. Zaag, A chemotaxis model motivated by angiogenesis, C. R. Math. Acad. Sci. Paris, 336 (2003), 141-146. doi: 10.1016/S1631-073X(02)00008-0.
[3]	L. Corrias, B. Perthame and H. Zaag, Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math., 72 (2004), 1-28. doi: 10.1007/s00032-003-0026-x.
[4]	C. Deng and T. Li, Well-posedness of a 3D parabolic-hyperbolic Keller-Segel system in the Sobolev space framework, J. Differential Equations, 257 (2014), 1311-1332. doi: 10.1016/j.jde.2014.05.014.
[5]	E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.
[6]	L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998.
[7]	E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.
[8]	O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R.I. 1968.
[9]	D. Li, T. Li and K. Zhao, On a hyperbolic-parabolic system modeling chemotaxis, Math. Models Methods Appl. Sci., 21 (2011), 1631-1650. doi: 10.1142/S0218202511005519.
[10]	P. Li, Geometric Analysis, Cambridge Studies in Advanced Mathematics, 134. Cambridge University Press, Cambridge, 2012. doi: 10.1017/CBO9781139105798.
[11]	T. Li, R. H. Pan and K. Zhao, Global dynamics of a hyperbolic-parabolic model arising from chemotaxis, SIAM J. Appl. Math., 72 (2012), 417-443. doi: 10.1137/110829453.
[12]	G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302.
[13]	H. G. Othmer and A. Stevens, Aggregation, blowup, and collapse: The ABCs of taxis in reinforced random walks, SIAM J. Appl. Math., 57 (1997), 1044-1081. doi: 10.1137/S0036139995288976.
[14]	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.
[15]	H. Y. Peng, H. Y. Wen and C. J. Zhu, Global well-posedness and zero diffusion limit of classical solutions to 3D conservation laws arising in chemotaxis, Z. Angew. Math. Phys., 65 (2014), 1167-1188. doi: 10.1007/s00033-013-0378-1.
[16]	T. Suzuki, Free Energy and Self-Interacting Particles, Progress in Nonlinear Differential Equations and their Applications, 62. Birkhäuser Boston, Inc., Boston, MA, 2005. doi: 10.1007/0-8176-4436-9.
[17]	M. E. Taylor, Partial Differential Equations III: Nonlinear Equations, Second edition. Applied Mathematical Sciences, 117. Springer, New York, 2011. doi: 10.1007/978-1-4419-7049-7.
[18]	Z.-A. Wang and T. Hillen, Shock formation in a chemotaxis model, Math. Methods Appl. Sci., 31 (2008), 45-70. doi: 10.1002/mma.898.
[19]	Z.-A. Wang, Z. Y. Xiang and P. Yu, Asymptotic dynamics on a singular chemotaxis system modeling onset of tumor angiogenesis, J. Differential Equations, 260 (2016), 2225-2258. doi: 10.1016/j.jde.2015.09.063.
[20]	Y. Yang, H. Chen and W. A. Liu, On existence of global solutions and blow-up to a system of reaction diffusion equations modeling chemotaxis, SIAM J. Math. Anal., 33 (2001), 763-785. doi: 10.1137/S0036141000337796.
[21]	Y. Yang, H. Chen, W. A. Liu and B. D. Sleeman, The solvability of some chemotaxis systems, J. Differential Equations, 212 (2005), 432-451. doi: 10.1016/j.jde.2005.01.002.
[22]	M. Zhang and C. J. Zhu, Global existence of solutions to a hyperbolic-parabolic system, Proc. Amer. Math. Soc., 135 (2007), 1017-1027. doi: 10.1090/S0002-9939-06-08773-9.

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On the vanishing viscosity limit of a chemotaxis model

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