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. Specific examples of vector flows include the geodesic flow, the Hamiltonian flow, the Ricci flow, the mean curvature flow, and the Anosov flow.

Formal definition

A flow on a set X is a group action of the additive group of real numbers on X. More explicitly, a flow is a mapping

$\varphi :X\times \mathbb {R} \rightarrow X,\qquad \varphi (x,0)=x$

such that if s and t are real numbers and x belongs to X, then

$\varphi (\varphi (x,t),s)=\varphi (x,t+s).$

It is customary to denote φ^t(x):=φ(x, t). Then φ^t:X→X is a bijection with the inverse φ^t:X→X for each t∈R. This is justified by the above definition, since the real parameter t may be taken as a generalized functional power, as in function iteration.

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. If X is equipped with a differentiable structure, then φ is usually required to be differentiable. In these cases the flow forms a one parameter subgroup of homeomorphisms and diffeomorphisms, respectively.

In certain situations one might also consider local flows, which are defined only in some subset

$\operatorname {dom} (\varphi )=\{(x,t)\ |\ t\in ]a_{x},b_{x}[,\ a_{x}<0<b_{x},\ x\in X\}\subset X\times \mathbb {R}$

called the flow domain of φ. This is often the case with the flows of vector fields.

Alternative notations

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. Thus, x(t) is written for φ^t(x), and one might say that the "variable x depends on the time t". Examples are given below.

In the case of a flow of a vector field V on a smooth manifold X, the flow is often denoted in such way that its generator is made explicit. For example,

$\Phi _{V}:X\times \mathbb {R} \to X;\qquad (x,t)\mapsto \Phi _{V}^{t}(x).$

Orbits

Given x in X, the set $\{\varphi (x,t):t\in \mathbb {R} \}$ is called the orbit of x under $\varphi .$ 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.

Examples

Autonomous systems of ordinary differential equations

Let F:Rⁿ→Rⁿ be a (time-independent) vector field and x:R→Rⁿ the solution of the initial value problem

${\dot {\boldsymbol {x}}}(t)={\boldsymbol {F}}({\boldsymbol {x}}(t)),\qquad {\boldsymbol {x}}(0)={\boldsymbol {x}}_{0}.$

Then φ(x₀,t)=x(t) is the flow of the vector field X. It is a well-defined local flow provided that the vector field F:Rⁿ→Rⁿ is Lipschitz-continuous. Then φ:Rⁿ×R→Rⁿ is also Lipschitz-continuous wherever defined. 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.

Time-dependent ordinary differential equations

In the case of time-dependent vector fields F:Rⁿ×R→Rⁿ one denotes φ^t,t₀(x₀)=x(t), where x:R→Rⁿ is the solution of

${\dot {\boldsymbol {x}}}(t)={\boldsymbol {F}}({\boldsymbol {x}}(t),t),\qquad {\boldsymbol {x}}(t_{0})={\boldsymbol {x}}_{0}.$

Then φ^t,t₀(x_0,t,t₀) is the time-dependent flow of F. It is not a "flow" by the definition above, but it can easily be seen as one by rearranging its arguments. Namely, the mapping

$\varphi :(\mathbb {R} ^{n}\times \mathbb {R} )\times \mathbb {R} \to \mathbb {R} ^{n}\times \mathbb {R} ;\qquad \varphi ({\boldsymbol {x}}_{0},t_{0},t)=(\varphi ^{t,t_{0}}({\boldsymbol {x}}_{0}),t+t_{0})$

indeed satisfies the group law for the last variable:

$\varphi (\varphi ({\boldsymbol {x}}_{0},t_{0},t),s)=\varphi (\varphi ^{t,t_{0}}({\boldsymbol {x}}_{0}),t+t_{0},s)=(\varphi ^{s,t+t_{0}}({\boldsymbol {x}}_{0}),s+t+t_{0})=\varphi ({\boldsymbol {x}}_{0},t_{0},s+t).$

One can see time-dependent flows of vector fields as special cases of time-independent ones by the following trick. Define

${\boldsymbol {G}}({\boldsymbol {x}},t):=({\boldsymbol {F}}({\boldsymbol {x}},t),1),\qquad {\boldsymbol {y}}(t):=({\boldsymbol {x}}(t),t+t_{0}).$

Then y(t) is the solution of the "time-independent" initial value problem

${\dot {\boldsymbol {y}}}(s)={\boldsymbol {G}}({\boldsymbol {y}}(s)),\qquad {\boldsymbol {y}}(0)=({\boldsymbol {x}}_{0},t_{0})$

if and only if x(t) is the solution of the original time-dependent initial value problem. Furthermore, then the mapping φ is exactly the flow of the "time-independent" vector field G.

Flows of vector fields on manifolds

The flows of time-independent and time-dependent vector fields are defined on smooth manifolds exactly as they are defined on the Euclidean space Rⁿ and their local behavior is the same. However, the global topological structure of a smooth manifold is strongly manifested 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. Also a bulk of study in dynamical systems is conducted on smooth manifolds, which are thought of as "parameter spaces" is applications.

References

• D.V. Anosov (2001) [1994], "Continuous flow", Encyclopedia of Mathematics, EMS Press
• D.V. Anosov (2001) [1994], "Measureable flow", Encyclopedia of Mathematics, EMS Press
• D.V. Anosov (2001) [1994], "Special flow", Encyclopedia of Mathematics, EMS Press
• Flow at PlanetMath.