TY - GEN
T1 - Logarithmic-type scaling of the collapse of Keller-Segel equation
AU - Dyachenko, Sergey A.
AU - Lushnikov, Pavel M.
AU - Vladimirova, Natalia
PY - 2011
Y1 - 2011
N2 - Keller-Segel equation (KS) is a parabolic-elliptic system of partial differential equations with applications to bacterial aggregation and collapse of self-gravitating gas of brownian particles. KS has striking qualitative similarities with nonlinear Schrodinger equation (NLS) including critical collapse (finite time point-wise singularity) in two dimensions. The self-similar solutions near blow up point are studied for KS in two dimensions together with time dependence of these solutions. We found logarithmic-type modifications to (t0-t)1/2 scaling law of self-similar solution in qualitative analogy with log-log modification for NLS. We found very good agreement between the direct numerical simulations of KS and the analytical results obtained by developing a perturbation theory for logarithmic-type modifications. It suggests that log-log modification in NLS also could be verified in a similar way.
AB - Keller-Segel equation (KS) is a parabolic-elliptic system of partial differential equations with applications to bacterial aggregation and collapse of self-gravitating gas of brownian particles. KS has striking qualitative similarities with nonlinear Schrodinger equation (NLS) including critical collapse (finite time point-wise singularity) in two dimensions. The self-similar solutions near blow up point are studied for KS in two dimensions together with time dependence of these solutions. We found logarithmic-type modifications to (t0-t)1/2 scaling law of self-similar solution in qualitative analogy with log-log modification for NLS. We found very good agreement between the direct numerical simulations of KS and the analytical results obtained by developing a perturbation theory for logarithmic-type modifications. It suggests that log-log modification in NLS also could be verified in a similar way.
KW - Cellular aggregation
KW - Chemotaxis
KW - Collapse and formation of singularities
KW - Logarithmic modification of self-similar solution
KW - Self-gravitating Brownian particles
UR - https://www.scopus.com/pages/publications/81855176332
U2 - 10.1063/1.3636829
DO - 10.1063/1.3636829
M3 - Conference contribution
AN - SCOPUS:81855176332
SN - 9780735409569
T3 - AIP Conference Proceedings
SP - 709
EP - 712
BT - Numerical Analysis and Applied Mathematics, ICNAAM 2011 - International Conference on Numerical Analysis and Applied Mathematics
T2 - International Conference on Numerical Analysis and Applied Mathematics: Numerical Analysis and Applied Mathematics, ICNAAM 2011
Y2 - 19 September 2011 through 25 September 2011
ER -