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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.

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.

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 *x* ∈ *X* and all real numbers s and t,

It is customary to write *φ ^{t}*(

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.

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* = *x*_{0}. 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,

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.

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

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.

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*(

if and only if * x*(

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

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) = *u*^{0} 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 Δ_{D} 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) = *u*^{0}. Thus, the flow corresponding to this equation is (see notations above)

where exp(*t*Δ_{D}) is the (analytic) semigroup generated by Δ_{D}.

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) = *u*^{1,0} 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 Δ_{D} is defined in the previous example).

We introduce the column vectors

(where and ) and

With these notions, the Wave Equation becomes and *U*(0) = *U*^{0}.

Thus, the flow corresponding to this equation is

where is the (unitary) semigroup generated by

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.