Boundary-value problem for density-velocity model of collective motion of organisms. (English) Zbl 1146.35047
Summary: The collective motion of organisms is observed at almost all levels of biological systems. In this paper the density-velocity model of the collective motion of organisms is analyzed. This model consists of a system of nonlinear parabolic equations, a forced Burgers equation for velocity and a mass conservation equation for density. These equations are supplemented with the Neumann boundary conditions for the density and the Dirichlet boundary conditions for the velocity. The existence, uniqueness and regularity of solution for the density-velocity problem is proved in a bounded 1D domain. Moreover, a priori estimates for the solutions are established, and existence of an attractor is proved. Finally, some numerical approximations for asymptotical behavior of the density-velocity model are presented.
MSC:
| 35K50 | Systems of parabolic equations, boundary value problems (MSC2000) |
| 35K55 | Nonlinear parabolic equations |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 92C15 | Developmental biology, pattern formation |
Keywords:
collective organization; Burgers equation; conservation equation; existence; uniqueness; regularity; positivity; attractors; Neumann boundary conditions; Dirichlet boundary conditionsReferences:
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