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Dynamics of incompressible vector fields. (English) Zbl 0989.37012

The paper deals with the topological and geometrical structure of the Lagrange dynamics of incompressible fluid flows. More precisely the dynamics of the ordinary differential equations in the physical space \[ \frac{dx}{dt}= v(x,t), \qquad x\bigl|_{t=0}= x_0 \] are considered where \(v\) is the velocity field of the fluid, satisfying the Navier-Stokes equation, \(x\) is the physical location of the fluid parcel, \(t\) the time. Then the authors study the structural stability and topological classification of the vector field \(v(x,t)\) at each time instant, treating \(t\) as a parameter. Moreover treating time as a parameter, the authors study the bifurcation of the topological structure of the field \(v\) at different time.

MSC:

37C10 Dynamics induced by flows and semiflows
37C75 Stability theory for smooth dynamical systems
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