Flow (mathematics)
In mathematics, a flow formalizes the idea of the motion of particles in a fluid.
Flows are ubiquitous in science, including engineering and physics.
The notion of flow is basic to the study of ordinary differential equations.
Informally, a flow may be viewed as a continuous motion of points over time.
More formally, a flow is a group action of the real numbers on a set.
The idea of a vector flow, that is, the flow determined by a vector field, occurs in the areas of differential topology, Riemannian geometry and Lie groups.
More explicitly, a flow is a mapping such that, for all x ∈ X and all real numbers s and t, It is customary to write φt(x) instead of φ(x, t), so that the equations above can be expressed as
Flows are usually required to be compatible with structures furnished on the set X.
In particular, if X is equipped with a topology, then φ is usually required to be continuous.
In these cases the flow forms a one-parameter group of homeomorphisms and diffeomorphisms, respectively.
It is very common in many fields, including engineering, physics and the study of differential equations, to use a notation that makes the flow implicit.
and one might say that the variable x depends on the time t and the initial condition x = x0.
Informally, it may be regarded as the trajectory of a particle that was initially positioned at x.
If the flow is generated by a vector field, then its orbits are the images of its integral curves.
In general it may be hard to show that the flow φ is globally defined, but one simple criterion is that the vector field F is compactly supported.
Namely, the mapping indeed satisfies the group law for the last variable: One can see time-dependent flows of vector fields as special cases of time-independent ones by the following trick.
Furthermore, then the mapping φ is exactly the flow of the "time-independent" vector field G. The flows of time-independent and time-dependent vector fields are defined on smooth manifolds exactly as they are defined on the Euclidean space
However, the global topological structure of a smooth manifold is strongly manifest in what kind of global vector fields it can support, and flows of vector fields on smooth manifolds are indeed an important tool in differential topology.
The bulk of studies in dynamical systems are conducted on smooth manifolds, which are thought of as "parameter spaces" in applications.
maps each point to an element of its own tangent space.
Denote by Γ its boundary (assumed smooth).
The equation u = 0 on Γ × (0, T) corresponds to the Homogeneous Dirichlet boundary condition.
The mathematical setting for this problem can be the semigroup approach.
To use this tool, we introduce the unbounded operator ΔD defined on
and is the closure of the infinitely differentiable functions with compact support in Ω for the
Thus, the flow corresponding to this equation is (see notations above) where exp(tΔD) is the (analytic) semigroup generated by ΔD.
We denote by Γ its boundary (assumed smooth).
(the operator ΔD is defined in the previous example).
That is, if ψ(x, t), is another flow with the same entropy, then ψ(x, t) = φ(x, t), for some constant c. The notion of uniqueness and isomorphism here is that of the isomorphism of dynamical systems.
Many dynamical systems, including Sinai's billiards and Anosov flows are isomorphic to Bernoulli shifts.
