Flow (mathematics)

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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.

Flow in phase space specified by the differential equation of a pendulum. On the horizontal axis, the pendulum position, and on the vertical one its velocity.

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 Anosov flows. Flows may also be defined for systems of random variables and stochastic processes, and occur in the study of ergodic dynamical systems. The most celebrated of these is perhaps the Bernoulli flow.

Formal definition

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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

 

such that, for all xX 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   (the identity function) and   (group law). Then, for all   the mapping   is a bijection with inverse   This follows from the above definition, and 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 group of homeomorphisms and diffeomorphisms, respectively.

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

 

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

Alternative notations

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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   and one might say that the variable x depends on the time t and the initial condition x = x0. 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 a way that its generator is made explicit. For example,

 

Orbits

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Given x in X, the set   is called the orbit of x under φ. 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

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Algebraic equation

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Let   be a time-dependent trajectory which is a bijective function. Then a flow can be defined by

 

Autonomous systems of ordinary differential equations

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Let   be a (time-independent) vector field and   the solution of the initial value problem

 

Then   is the flow of the vector field F. It is a well-defined local flow provided that the vector field   is Lipschitz-continuous. Then   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

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In the case of time-dependent vector fields  , one denotes   where   is the solution of

 

Then   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

 

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. Define

 

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

 

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

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The flows of time-independent and time-dependent vector fields are defined on smooth manifolds exactly as they are defined on the Euclidean space   and their local behavior is the same. 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.

Formally: Let   be a differentiable manifold. Let   denote the tangent space of a point   Let   be the complete tangent manifold; that is,   Let   be a time-dependent vector field on  ; that is, f is a smooth map such that for each   and  , one has   that is, the map   maps each point to an element of its own tangent space. For a suitable interval   containing 0, the flow of f is a function   that satisfies  

Solutions of heat equation

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Let Ω be a subdomain (bounded or not) of   (with n an integer). Denote by Γ its boundary (assumed smooth). Consider the following heat equation on Ω × (0, T), for T > 0,

 

with the following initial value condition u(0) = u0 in Ω .

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   by its domain

 

(see the classical Sobolev spaces with   and

 

is the closure of the infinitely differentiable functions with compact support in Ω for the  norm).

For any  , we have

 

With this operator, the heat equation becomes   and u(0) = u0. Thus, the flow corresponding to this equation is (see notations above)

 

where exp(tΔD) is the (analytic) semigroup generated by ΔD.

Solutions of wave equation

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Again, let Ω be a subdomain (bounded or not) of   (with n an integer). We denote by Γ its boundary (assumed smooth). Consider the following wave equation on   (for T > 0),

 

with the following initial condition u(0) = u1,0 in Ω and  

Using the same semigroup approach as in the case of the Heat Equation above. We write the wave equation as a first order in time partial differential equation by introducing the following unbounded operator,

 

with domain   on   (the operator ΔD is defined in the previous example).

We introduce the column vectors

 

(where   and  ) and

 

With these notions, the Wave Equation becomes   and U(0) = U0.

Thus, the flow corresponding to this equation is

 

where   is the (unitary) semigroup generated by  

Bernoulli flow

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Ergodic dynamical systems, that is, systems exhibiting randomness, exhibit flows as well. The most celebrated of these is perhaps the Bernoulli flow. The Ornstein isomorphism theorem states that, for any given entropy H, there exists a flow φ(x, t), called the Bernoulli flow, such that the flow at time t = 1, i.e. φ(x, 1), is a Bernoulli shift.

Furthermore, this flow is unique, up to a constant rescaling of time. 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.

See also

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References

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  • 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
  • This article incorporates material from Flow on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.