scholarly journals Possibility of amoebas' aggregation in finite time

2013 ◽  
Vol 21 (1) ◽  
pp. 101-120
Author(s):  
Saïd Hilmi ◽  
Chérif Ziti
Keyword(s):  
Partial Differential Equations ◽  
Nonlinear System ◽  
Differential Equations ◽  
Finite Time ◽  
Blow Up ◽  
Dimensional Space ◽  
Partial Differential ◽  
Self Similar ◽  
Similar Solutions

Abstract The dynamic of amoebas in favorable circumstances is modeled by a nonlinear system of Partial Differential Equations arising in chemotaxis. The competition between different parameters of this system plays a major role in the process of aggregation. Throughout this paper, we prove the existence of self-similar solutions that blow up in finite time in a dimensional space and under specific circumstances depending upon the position of those parameters.

The evolution of turbulent bursts: the b—ε model

1994 ◽  
Vol 5 (4) ◽  
pp. 537-557 ◽  
Author(s):  
M. Bertsch ◽  
R. Dal Passo ◽  
R. Kersner
Keyword(s):  
Partial Differential Equations ◽  
Energy Dissipation ◽  
Energy Density ◽  
Differential Equations ◽  
Dissipation Rate ◽  
Turbulent Energy ◽  
Energy Dissipation Rate ◽  
Self Similar ◽  
Similar Solutions ◽  
Semi Empirical

We study the semi-empirical b—ε model which describes the time evolution of turbulent spots in the case of equal diffusivity of the turbulent energy density b and the energy dissipation rate ε. We prove that the system of two partial differential equations possesses a solution, and that after some time this solution exhibits self-similar behaviour, provided that the system has self-similar solutions. The existence of such self-similar solutions depends upon the value of a parameter of the model.

scholarly journals Variational symmetries and pluri-Lagrangian structures for integrable hierarchies of PDEs

2020 ◽  
Author(s):  
Matteo Petrera ◽  
Mats Vermeeren
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Integrable Hierarchies ◽  
Dimensional Space ◽  
Original System ◽  
Lagrange Equations ◽  
Partial Differential ◽  
Dimensional Space Time ◽  
Variational Symmetries ◽  
Nonlinear Math

Abstract We investigate the relation between pluri-Lagrangian hierarchies of 2-dimensional partial differential equations and their variational symmetries. The aim is to generalize to the case of partial differential equations the recent findings in Petrera and Suris (Nonlinear Math. Phys. 24(suppl. 1):121–145, 2017) for ordinary differential equations. We consider hierarchies of 2-dimensional Lagrangian PDEs (many of which have a natural $$(1\,{+}\,1)$$ ( 1 + 1 ) -dimensional space-time interpretation) and show that if the flow of each PDE is a variational symmetry of all others, then there exists a pluri-Lagrangian 2-form for the hierarchy. The corresponding multi-time Euler–Lagrange equations coincide with the original system supplied with commuting evolutionary flows induced by the variational symmetries.

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