May  2017, 16(3): 1013-1036. doi: 10.3934/cpaa.2017049

Boundary layer analysis from the Keller-Segel system to the aggregation system in one space dimension

1. 

Department of Mathematics, University of Mannheim, 68131 Mannheim, Germany

2. 

Weierstraß-Institut, 10117 Berlin, Germany

3. 

Department of Mathematics, Shanghai Normal University, 200234

Received  March 2016 Revised  December 2016 Published  February 2017

Fund Project: Li Chen is partially supported by the National Natural Science Foundation of China (NSFC), No. 11271218. Jing Wang is supported by NSFC (No.11101286), and the Doctoral Discipline Foundation for Young Teachers in the Higher Education Institutions of Ministry of Education, No.20113127120007

This article is devoted to the discussion of the boundary layer which arises from the one-dimensional parabolic elliptic type Keller-Segel system to the corresponding aggregation system on the half space case. The characteristic boundary layer is shown to be stable for small diffusion coefficients by using asymptotic analysis and detailed error estimates between the Keller-Segel solution and the approximate solution. In the end, numerical simulations for this boundary layer problem are provided.

Citation: Jiahang Che, Li Chen, Simone GÖttlich, Anamika Pandey, Jing Wang. Boundary layer analysis from the Keller-Segel system to the aggregation system in one space dimension. Communications on Pure & Applied Analysis, 2017, 16 (3) : 1013-1036. doi: 10.3934/cpaa.2017049
References:
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J. BedrossianN. Rodr$\acute{i}$guez and A. Bertozzi, Local and global well-posedness for aggregation equations and Patlak-Keller-Segel models with degenerate diffusion, Nonlinearity, 24 (2011), 1683-1714. doi: 10.1088/0951-7715/24/6/001.

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D. BenedettoE. Caglioti and M. Pulvirenti, A kinetic equation for granular media, RAIRO Mod l. Math. Anal. Numr., 31 (1997), 615-642.

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A. L. Bertozzi and J. Brandman, Finite-time blow-up of L-weak solutions of an aggregation equation, Comm. Math. Sci., 8 (2010), 45-65.

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A. L. BertozziJ. B. Garnett and T. Laurent, Characterization of radially symmetric finite time blowup in multidimensional aggregation equations, SIAM J. Math. Anal., 44 (2012), 651-681. doi: 10.1137/11081986X.

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A. L. Bertozzi and T. Laurent, Finite-time blow-up of solutions of an aggregation equation in Rn, Comm. Math. Phys., 274 (2007), 717-735. doi: 10.1007/s00220-007-0288-1.

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A. L. Bertozzi and T. Laurent, The behavior of solutions of multidimensional aggregation equations with mildly singular interaction kernels, Chin. Ann. Math. Ser. B, 30 (2009), 463-482. doi: 10.1007/s11401-009-0191-5.

[7]

S. Bian and J.-G. Liu, Dynamic and steady states for multi-dimensional Keller-Segel model with diffusion exponent m ≥ 0, Comm. Math. Phys., 323 (2013), 1017-1070. doi: 10.1007/s00220-013-1777-z.

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P. Biler, Local and global solvability of some parabolic systems modeling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743.

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P. BilerM. CannoneI. A. Guerra and G. Karch, Global regular and singular solutions for a model of gravitating particles, Math. Ann., 330 (2004), 693-708. doi: 10.1007/s00208-004-0565-7.

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A. BlanchetJ. A. Carrillo and P. Laurençot, Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions, Calc. Var., 35 (2009), 133-168. doi: 10.1007/s00526-008-0200-7.

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M. Bodnar and J. J. L. Velazquez, An integro-differential equation arising as a limit of individual cell-based models, J. Differential Equations, 222 (2006), 341-380. doi: 10.1016/j.jde.2005.07.025.

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M. BurgerV. Capasso and D. Morale, On an aggregation model with long and short range interactions, Nonlinear Anal.: Real World Appl., 8 (2007), 939-958. doi: 10.1016/j.nonrwa.2006.04.002.

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J. A. CarrilloL. ChenJ.-G. Liu and J. Wang, A note on the subcritical two dimensional Keller-Segel system, Acta Appl. Math., 119 (2012), 43-55. doi: 10.1007/s10440-011-9660-4.

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S. Childress, Chemotactic collapse in two dimensions, Lecture Notes in Biomath., 55 (1984), 61-68. doi: 10.1007/978-3-642-45589-6_6.

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T. Cieślak and P. Laurencot, Finite time blow-up for radially symmetric solutions to a critical quasilinear Smoluchwski-Poisson system, C. R. Acad. Sci. Paris Ser. Ⅰ, 347 (2009), 237-242. doi: 10.1016/j.crma.2009.01.016.

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J. Dolbeault and B. Perthame, Optimal critical mass in the two dimensional Keller-Segel model in ℝ2, C. R. Acad. Sci. Paris, Ser. Ⅰ, 339 (2004), 611-616. doi: 10.1016/j.crma.2004.08.011.

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H. Dong, The aggregation equation with power-law kernels: Ill-posedness, mass concentration and similarity solutions, Comm. Math. Phys., 304 (2011), 649-664. doi: 10.1007/s00220-011-1237-6.

[23]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3.

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D. Horstmann, From 1970 until now: The Keller-Segel model in chemotaxis and its consequences Ⅰ, Jahresberichte der DMV, 105 (2003), 103-165.

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D. Horstmann, From 1970 until now: The Keller-Segel model in chemotaxis and its consequences Ⅱ, Jahresberichte der DMV, 106 (2004), 51-69.

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D. Horstmann, On the existence of radially symmetric blow-up solutions for the Keller-Segel model, J. Math. Biol., 44 (2002), 463-478. doi: 10.1007/s002850100134.

[27]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107. doi: 10.1016/j.jde.2004.10.022.

[28]

D. Iftimie and F. Sueur, Viscous boundary layers for the Navier-Stokes equations with the Navier slip conditions, Arch. Ration. Mech. Anal., 199 (2011), 145-175. doi: 10.1007/s00205-010-0320-z.

[29]

S. Ishida and T. Yokota, Global existence of weak solutions to quasilinear degenerate KellerSegel systems of parabolic-parabolic type, J. Differential Equations, 252 (2012), 1421-1440. doi: 10.1016/j.jde.2011.02.012.

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F. James and N. Vauchelet, Chemotaxis: from kinetic equations to aggregate dynamics, Nonlinear Diff. Eq. Appl. NoDEA, 20 (2013), 101-127. doi: 10.1007/s00030-012-0155-4.

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E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.

[33]

E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234.

[34]

R. Kowalczyk and Z. Szymańska, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398. doi: 10.1016/j.jmaa.2008.01.005.

[35]

T. Laurent, Local and global existence for an aggregation equation, Comm. Partial Differential Equations, 32 (2007), 1941-1964. doi: 10.1080/03605300701318955.

[36]

S. Luckhaus and Y. Sugiyama, Large time behavior of solutions in super-critical cases to degenerate Keller-Segel systems, M2AN Math. Model. Numer. Anal., 40 (2006), 597-621. doi: 10.1051/m2an:2006025.

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A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Vol. 27, Cambridge University Press, 2002.

[38]

C. Marchioro and M. Pulvirenti, Hydrodynamics in two dimensions and vortex theory, Comm. Math. Phys., 84 (1982), 483-503.

[39]

T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.

[40]

T. Nagai, Global existence and blowup of solutions to a chemotaxis system, Nonlinear Anal., 47 (2001), 777-787. doi: 10.1016/S0362-546X(01)00222-X.

[41]

T. Nagai and T. Senba, Behavior of radially symmetric solutions of a system related to chemotaxis, Nonlinear Anal. Theory Methods Appl., 30 (1997), 3837-3842. doi: 10.1016/S0362-546X(96)00256-8.

[42]

T. Nagai and T. Senba, Global existence and blow-up of radial solutioins to a parabolic-elliptic system of chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 145-156.

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NASA NPARC Alliance Verification and Validation, Examining Spatial (Grid) Convergence, http://www.grc.nasa.gov/WWW/wind/valid/tutorial/spatconv.html.

[44]

K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-470.

[45]

C. S. Patlak, Random walk with persistenc and external bias, Bull. Math. Biol. Biophys., 15 (1953), 311-338.

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P. J. Roache, Verification and validation in computational science and engineering, Computing in Science Eng., 1 (1998), 8-9.

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F. Rousset, Characteristic boundary layers in real vanishing viscosity limits, J. Differential Equations, 210 (2005), 25-64. doi: 10.1016/j.jde.2004.10.004.

[48]

F. Rousset, Stability of small amplitude boundary layers for mixed hyperbolic-parabolic systems, Trans. Amer. Math. Soc., 355 (2003), 2991-3008. doi: 10.1090/S0002-9947-03-03279-3.

[49]

M. Sammartino and R. E. Caflisch, Zero viscosity limit for analytic solutions of the NavierStokes equation on a half-space. Ⅰ.Existence for Euler and Prandtl equations, Comm. Math. Phys., 192 (1998), 433-461. doi: 10.1007/s002200050304.

[50]

T. Senba and T. Suzuki, A quasi-linear parabolic system of chemotaxis, Abstr. Appl. Anal. , V (2006), Article ID 23061, 1-21. doi: 10.1155/AAA/2006/23061.

[51]

D. Serre and K. Zumbrun, Boundary layer stability in real vanishing viscosity limit, Comm. Math. Phys., 221 (2001), 267-292. doi: 10.1007/s002200100486.

[52]

Y. Sugiyama, Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate Keller-Segel systems, Diff. Int. Equa., 19 (2006), 841-876.

[53]

Y. Sugiyama, Application of the best constant of the Sobolev inequality to degenerate KellerSegel models, Adv. Diff. Eqns., 12 (2007), 121-144.

[54]

Y. Sugiyama and H. Kunii, Global existence and decay properties for a degenerate Keller-Segel model with a power factor in drift term, J. Differential Equations, 227 (2006), 333-364. doi: 10.1016/j.jde.2006.03.003.

[55]

C. M. TopazA. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), 1601-1623. doi: 10.1007/s11538-006-9088-6.

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G. Toscani, One-dimensional kinetic models of granular flows, ESAIM Math. Model. Numer. Anal., 34 (2000), 1277-1291. doi: 10.1051/m2an:2000127.

[57]

J. Wang, L. Chen and L. Hong, Parabolic elliptic type Keller-Segel system on the whole space case, Discrete Contin. Dyn. Syst. , accepted. doi: 10.3934/dcds.2016.36.1061.

[58]

J. Wang, Boundary layers for compressible Navier-Stokes equations with outflow boundary condition, J. Differential Equations, 248 (2010), 1143-1174. doi: 10.1016/j.jde.2009.12.001.

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J. Wang and L. Tong, Stability of boundary layers for the inflow compressible Navier-Stokes equations, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2595-2613. doi: 10.3934/dcdsb.2012.17.2595.

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J. Wang and F. Xie, Characteristic boundary layers for parabolic perturbations of quasilinear hyperbolic problems, Nonlinear Anal., 73 (2010), 2504-2523. doi: 10.1016/j.na.2010.06.022.

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J. Wang and F. Xie, Zero dissipation limit and stability of boundary layers for the heat conductive boussinesq equations in a bounded domain, Proc. Roy. Soc. Edinburgh Sect. A, accepted. doi: 10.1017/S0308210513000875.

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Z. P. Xin and T. Yanagisawa, Zero-viscosity limit of the linearized Navier-Stokes equations for a compressible viscous fluid in the half-plane, Comm. Pure Appl. Math., 52 (1999), 479-541. doi: 10.1002/(SICI)1097-0312(199904)52:4<479::AID-CPA4>3.3.CO;2-T.

show all references

References:
[1]

J. BedrossianN. Rodr$\acute{i}$guez and A. Bertozzi, Local and global well-posedness for aggregation equations and Patlak-Keller-Segel models with degenerate diffusion, Nonlinearity, 24 (2011), 1683-1714. doi: 10.1088/0951-7715/24/6/001.

[2]

D. BenedettoE. Caglioti and M. Pulvirenti, A kinetic equation for granular media, RAIRO Mod l. Math. Anal. Numr., 31 (1997), 615-642.

[3]

A. L. Bertozzi and J. Brandman, Finite-time blow-up of L-weak solutions of an aggregation equation, Comm. Math. Sci., 8 (2010), 45-65.

[4]

A. L. BertozziJ. B. Garnett and T. Laurent, Characterization of radially symmetric finite time blowup in multidimensional aggregation equations, SIAM J. Math. Anal., 44 (2012), 651-681. doi: 10.1137/11081986X.

[5]

A. L. Bertozzi and T. Laurent, Finite-time blow-up of solutions of an aggregation equation in Rn, Comm. Math. Phys., 274 (2007), 717-735. doi: 10.1007/s00220-007-0288-1.

[6]

A. L. Bertozzi and T. Laurent, The behavior of solutions of multidimensional aggregation equations with mildly singular interaction kernels, Chin. Ann. Math. Ser. B, 30 (2009), 463-482. doi: 10.1007/s11401-009-0191-5.

[7]

S. Bian and J.-G. Liu, Dynamic and steady states for multi-dimensional Keller-Segel model with diffusion exponent m ≥ 0, Comm. Math. Phys., 323 (2013), 1017-1070. doi: 10.1007/s00220-013-1777-z.

[8]

P. Biler, Local and global solvability of some parabolic systems modeling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743.

[9]

P. BilerM. CannoneI. A. Guerra and G. Karch, Global regular and singular solutions for a model of gravitating particles, Math. Ann., 330 (2004), 693-708. doi: 10.1007/s00208-004-0565-7.

[10]

A. BlanchetJ. A. Carrillo and P. Laurençot, Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions, Calc. Var., 35 (2009), 133-168. doi: 10.1007/s00526-008-0200-7.

[11]

M. Bodnar and J. J. L. Velazquez, An integro-differential equation arising as a limit of individual cell-based models, J. Differential Equations, 222 (2006), 341-380. doi: 10.1016/j.jde.2005.07.025.

[12]

M. BurgerV. Capasso and D. Morale, On an aggregation model with long and short range interactions, Nonlinear Anal.: Real World Appl., 8 (2007), 939-958. doi: 10.1016/j.nonrwa.2006.04.002.

[13]

J. A. CarrilloL. ChenJ.-G. Liu and J. Wang, A note on the subcritical two dimensional Keller-Segel system, Acta Appl. Math., 119 (2012), 43-55. doi: 10.1007/s10440-011-9660-4.

[14]

J. A. CarrilloM. DifrancescoA. FigalliT. Laurent and D. Slepcev, Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations, Duke Math. J., 156 (2011), 229-271. doi: 10.1215/00127094-2010-211.

[15]

J. A. CarrilloR. J. McCann and C. Villani, Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates, Rev. Mat. Iberoam., 19 (2003), 971-1018. doi: 10.4171/RMI/376.

[16]

L. ChenJ.-G. Liu and J. H. Wang, Multidimensional degenerate Keller-Segel system with critical diffusion exponent 2n=(n +2), SIAM J. Math. Anal., 44 (2012), 1077-1102. doi: 10.1137/110839102.

[17]

L. Chen and J. H. Wang, Exact criterion for global existence and blow up to a degenerate Keller-Segel system, Doc. Math., 19 (2014), 103-120.

[18]

S. Childress, Chemotactic collapse in two dimensions, Lecture Notes in Biomath., 55 (1984), 61-68. doi: 10.1007/978-3-642-45589-6_6.

[19]

T. Cieślak and P. Laurencot, Finite time blow-up for radially symmetric solutions to a critical quasilinear Smoluchwski-Poisson system, C. R. Acad. Sci. Paris Ser. Ⅰ, 347 (2009), 237-242. doi: 10.1016/j.crma.2009.01.016.

[20]

R. DiPerna and A. Majda, Concentrations in regularizations for 2-D incompressible flow, Comm. Pure Appl. Math., 40 (1987), 301-345. doi: 10.1002/cpa.3160400304.

[21]

J. Dolbeault and B. Perthame, Optimal critical mass in the two dimensional Keller-Segel model in ℝ2, C. R. Acad. Sci. Paris, Ser. Ⅰ, 339 (2004), 611-616. doi: 10.1016/j.crma.2004.08.011.

[22]

H. Dong, The aggregation equation with power-law kernels: Ill-posedness, mass concentration and similarity solutions, Comm. Math. Phys., 304 (2011), 649-664. doi: 10.1007/s00220-011-1237-6.

[23]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3.

[24]

D. Horstmann, From 1970 until now: The Keller-Segel model in chemotaxis and its consequences Ⅰ, Jahresberichte der DMV, 105 (2003), 103-165.

[25]

D. Horstmann, From 1970 until now: The Keller-Segel model in chemotaxis and its consequences Ⅱ, Jahresberichte der DMV, 106 (2004), 51-69.

[26]

D. Horstmann, On the existence of radially symmetric blow-up solutions for the Keller-Segel model, J. Math. Biol., 44 (2002), 463-478. doi: 10.1007/s002850100134.

[27]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107. doi: 10.1016/j.jde.2004.10.022.

[28]

D. Iftimie and F. Sueur, Viscous boundary layers for the Navier-Stokes equations with the Navier slip conditions, Arch. Ration. Mech. Anal., 199 (2011), 145-175. doi: 10.1007/s00205-010-0320-z.

[29]

S. Ishida and T. Yokota, Global existence of weak solutions to quasilinear degenerate KellerSegel systems of parabolic-parabolic type, J. Differential Equations, 252 (2012), 1421-1440. doi: 10.1016/j.jde.2011.02.012.

[30]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824. doi: 10.2307/2153966.

[31]

F. James and N. Vauchelet, Chemotaxis: from kinetic equations to aggregate dynamics, Nonlinear Diff. Eq. Appl. NoDEA, 20 (2013), 101-127. doi: 10.1007/s00030-012-0155-4.

[32]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.

[33]

E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234.

[34]

R. Kowalczyk and Z. Szymańska, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398. doi: 10.1016/j.jmaa.2008.01.005.

[35]

T. Laurent, Local and global existence for an aggregation equation, Comm. Partial Differential Equations, 32 (2007), 1941-1964. doi: 10.1080/03605300701318955.

[36]

S. Luckhaus and Y. Sugiyama, Large time behavior of solutions in super-critical cases to degenerate Keller-Segel systems, M2AN Math. Model. Numer. Anal., 40 (2006), 597-621. doi: 10.1051/m2an:2006025.

[37]

A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Vol. 27, Cambridge University Press, 2002.

[38]

C. Marchioro and M. Pulvirenti, Hydrodynamics in two dimensions and vortex theory, Comm. Math. Phys., 84 (1982), 483-503.

[39]

T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.

[40]

T. Nagai, Global existence and blowup of solutions to a chemotaxis system, Nonlinear Anal., 47 (2001), 777-787. doi: 10.1016/S0362-546X(01)00222-X.

[41]

T. Nagai and T. Senba, Behavior of radially symmetric solutions of a system related to chemotaxis, Nonlinear Anal. Theory Methods Appl., 30 (1997), 3837-3842. doi: 10.1016/S0362-546X(96)00256-8.

[42]

T. Nagai and T. Senba, Global existence and blow-up of radial solutioins to a parabolic-elliptic system of chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 145-156.

[43]

NASA NPARC Alliance Verification and Validation, Examining Spatial (Grid) Convergence, http://www.grc.nasa.gov/WWW/wind/valid/tutorial/spatconv.html.

[44]

K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-470.

[45]

C. S. Patlak, Random walk with persistenc and external bias, Bull. Math. Biol. Biophys., 15 (1953), 311-338.

[46]

P. J. Roache, Verification and validation in computational science and engineering, Computing in Science Eng., 1 (1998), 8-9.

[47]

F. Rousset, Characteristic boundary layers in real vanishing viscosity limits, J. Differential Equations, 210 (2005), 25-64. doi: 10.1016/j.jde.2004.10.004.

[48]

F. Rousset, Stability of small amplitude boundary layers for mixed hyperbolic-parabolic systems, Trans. Amer. Math. Soc., 355 (2003), 2991-3008. doi: 10.1090/S0002-9947-03-03279-3.

[49]

M. Sammartino and R. E. Caflisch, Zero viscosity limit for analytic solutions of the NavierStokes equation on a half-space. Ⅰ.Existence for Euler and Prandtl equations, Comm. Math. Phys., 192 (1998), 433-461. doi: 10.1007/s002200050304.

[50]

T. Senba and T. Suzuki, A quasi-linear parabolic system of chemotaxis, Abstr. Appl. Anal. , V (2006), Article ID 23061, 1-21. doi: 10.1155/AAA/2006/23061.

[51]

D. Serre and K. Zumbrun, Boundary layer stability in real vanishing viscosity limit, Comm. Math. Phys., 221 (2001), 267-292. doi: 10.1007/s002200100486.

[52]

Y. Sugiyama, Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate Keller-Segel systems, Diff. Int. Equa., 19 (2006), 841-876.

[53]

Y. Sugiyama, Application of the best constant of the Sobolev inequality to degenerate KellerSegel models, Adv. Diff. Eqns., 12 (2007), 121-144.

[54]

Y. Sugiyama and H. Kunii, Global existence and decay properties for a degenerate Keller-Segel model with a power factor in drift term, J. Differential Equations, 227 (2006), 333-364. doi: 10.1016/j.jde.2006.03.003.

[55]

C. M. TopazA. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), 1601-1623. doi: 10.1007/s11538-006-9088-6.

[56]

G. Toscani, One-dimensional kinetic models of granular flows, ESAIM Math. Model. Numer. Anal., 34 (2000), 1277-1291. doi: 10.1051/m2an:2000127.

[57]

J. Wang, L. Chen and L. Hong, Parabolic elliptic type Keller-Segel system on the whole space case, Discrete Contin. Dyn. Syst. , accepted. doi: 10.3934/dcds.2016.36.1061.

[58]

J. Wang, Boundary layers for compressible Navier-Stokes equations with outflow boundary condition, J. Differential Equations, 248 (2010), 1143-1174. doi: 10.1016/j.jde.2009.12.001.

[59]

J. Wang and L. Tong, Stability of boundary layers for the inflow compressible Navier-Stokes equations, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2595-2613. doi: 10.3934/dcdsb.2012.17.2595.

[60]

J. Wang and F. Xie, Characteristic boundary layers for parabolic perturbations of quasilinear hyperbolic problems, Nonlinear Anal., 73 (2010), 2504-2523. doi: 10.1016/j.na.2010.06.022.

[61]

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Figure 1.  Time evolution of the Keller-Segel, the aggregation and the approximate solution for $\varepsilon = 0.01$
Table 1.  Error bound comparison for different values of $\varepsilon$
$\varepsilon$ $ \sup\limits_{0\leq t\leq T}\|\rho^{\varepsilon}-\rho_a\|_{\infty} $
0.01 0.165216
0.005 0.112758
0.001 0.048119
0.0005 0.033776
0.0001 0.015221
$\varepsilon$ $ \sup\limits_{0\leq t\leq T}\|\rho^{\varepsilon}-\rho_a\|_{\infty} $
0.01 0.165216
0.005 0.112758
0.001 0.048119
0.0005 0.033776
0.0001 0.015221
Table 2.  Elapsed time in seconds for the Keller-Segel model and the approximate solution
$\varepsilon$ $ k_{fac} $ Keller-Segel approximate solution
0.01 9 13.25 8.94
0.005 18 13.17 4.35
0.001 50 13.26 1.58
0.0005 100 13.18 0.78
0.0001 120 13.05 0.65
$\varepsilon$ $ k_{fac} $ Keller-Segel approximate solution
0.01 9 13.25 8.94
0.005 18 13.17 4.35
0.001 50 13.26 1.58
0.0005 100 13.18 0.78
0.0001 120 13.05 0.65
Table 3.  Grid convergence
h Eh e
0.002 0.165698 1.9280
0.001 0.165216
0.0005 0.0164966
h Eh e
0.002 0.165698 1.9280
0.001 0.165216
0.0005 0.0164966
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