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:
[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. Google Scholar

[2]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[24]

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

[25]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[32]

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

[33]

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

[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. Google Scholar

[35]

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

[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. Google Scholar

[37]

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

[38]

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

[39]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[43]

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

[44]

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

[45]

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

[46]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[53]

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

[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. Google Scholar

[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. Google Scholar

[56]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[61]

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. Google Scholar

[62]

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. Google Scholar

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. Google Scholar

[2]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[24]

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

[25]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[32]

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

[33]

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

[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. Google Scholar

[35]

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

[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. Google Scholar

[37]

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

[38]

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

[39]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[43]

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

[44]

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

[45]

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

[46]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[53]

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

[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. Google Scholar

[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. Google Scholar

[56]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[61]

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. Google Scholar

[62]

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. Google Scholar

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
[1]

Masahiro Suzuki. Asymptotic stability of a boundary layer to the Euler--Poisson equations for a multicomponent plasma. Kinetic & Related Models, 2016, 9 (3) : 587-603. doi: 10.3934/krm.2016008

[2]

Fujun Zhou, Junde Wu, Shangbin Cui. Existence and asymptotic behavior of solutions to a moving boundary problem modeling the growth of multi-layer tumors. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1669-1688. doi: 10.3934/cpaa.2009.8.1669

[3]

Güher Çamliyurt, Igor Kukavica. A local asymptotic expansion for a solution of the Stokes system. Evolution Equations & Control Theory, 2016, 5 (4) : 647-659. doi: 10.3934/eect.2016023

[4]

Thierry Paul, Mario Pulvirenti. Asymptotic expansion of the mean-field approximation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1891-1921. doi: 10.3934/dcds.2019080

[5]

Gung-Min Gie, Chang-Yeol Jung, Roger Temam. Recent progresses in boundary layer theory. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2521-2583. doi: 10.3934/dcds.2016.36.2521

[6]

X. Liang, Roderick S. C. Wong. On a Nested Boundary-Layer Problem. Communications on Pure & Applied Analysis, 2009, 8 (1) : 419-433. doi: 10.3934/cpaa.2009.8.419

[7]

D. Sanchez. Boundary layer on a high-conductivity domain. Communications on Pure & Applied Analysis, 2002, 1 (4) : 547-564. doi: 10.3934/cpaa.2002.1.547

[8]

Lizhi Ruan, Changjiang Zhu. Boundary layer for nonlinear evolution equations with damping and diffusion. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 331-352. doi: 10.3934/dcds.2012.32.331

[9]

Liping Wang, Chunyi Zhao. Solutions with clustered bubbles and a boundary layer of an elliptic problem. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2333-2357. doi: 10.3934/dcds.2014.34.2333

[10]

Liping Wang, Juncheng Wei. Solutions with interior bubble and boundary layer for an elliptic problem. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 333-351. doi: 10.3934/dcds.2008.21.333

[11]

Marissa Condon, Jing Gao, Arieh Iserles. On asymptotic expansion solvers for highly oscillatory semi-explicit DAEs. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4813-4837. doi: 10.3934/dcds.2016008

[12]

Walter Allegretto, Liqun Cao, Yanping Lin. Multiscale asymptotic expansion for second order parabolic equations with rapidly oscillating coefficients. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 543-576. doi: 10.3934/dcds.2008.20.543

[13]

Hua Chen, Shaohua Wu. The moving boundary problem in a chemotaxis model. Communications on Pure & Applied Analysis, 2012, 11 (2) : 735-746. doi: 10.3934/cpaa.2012.11.735

[14]

Micol Amar. A note on boundary layer effects in periodic homogenization with Dirichlet boundary conditions. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 537-556. doi: 10.3934/dcds.2000.6.537

[15]

Tómas Chacón-Rebollo, Macarena Gómez-Mármol, Samuele Rubino. On the existence and asymptotic stability of solutions for unsteady mixing-layer models. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 421-436. doi: 10.3934/dcds.2014.34.421

[16]

Rafał Celiński, Andrzej Raczyński. Asymptotic profile of solutions to a certain chemotaxis system. Communications on Pure & Applied Analysis, 2020, 19 (2) : 911-922. doi: 10.3934/cpaa.2020041

[17]

Christos Sourdis. Analysis of an irregular boundary layer behavior for the steady state flow of a Boussinesq fluid. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 1039-1059. doi: 10.3934/dcds.2017043

[18]

Shu Wang, Chundi Liu. Boundary Layer Problem and Quasineutral Limit of Compressible Euler-Poisson System. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2177-2199. doi: 10.3934/cpaa.2017108

[19]

O. Guès, G. Métivier, M. Williams, K. Zumbrun. Boundary layer and long time stability for multi-D viscous shocks. Discrete & Continuous Dynamical Systems - A, 2004, 11 (1) : 131-160. doi: 10.3934/dcds.2004.11.131

[20]

Eunice Mureithi. Effects of buoyancy on the lower branch modes on a Blasius boundary layer. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 613-622. doi: 10.3934/dcdsb.2007.8.613

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (59)
  • HTML views (81)
  • Cited by (0)

[Back to Top]