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.
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 (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
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 fieldV on a
smooth manifoldX, the flow is often denoted in such a way that its generator is made explicit. For example,
Orbits
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
Algebraic equation
Let be a time-dependent trajectory which is a bijective function. Then a flow can be defined by
Autonomous systems of ordinary differential equations
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
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
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.
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
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
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
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.
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.