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

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.
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  • 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.

    Mathematics Subject Classification: 35Q92, 35C20, 92C17.


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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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    Table 3.  Grid convergence

    h Eh e
    0.002 0.165698 1.9280
    0.001 0.165216
    0.0005 0.0164966
     | Show Table
    DownLoad: CSV
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