“Abstract differential equation”的意思、由来-开放百科全书

In mathematics, an abstract differential equation is a differential equation in which the unknown function and its derivatives take values in some generic abstract space (a Hilbert space, a Banach space, etc.). Equations of this kind arise e.g. in the study of partial differential equations: if to one of the variables is given a privilegiate position (e.g. time, in heat or wave equations) and all the others are put together, an ordinary "differential" equation with respect to the variable which was put in evidence is obtained. Adding boundary conditions can often be translated in terms of considering solutions in some convenient function spaces.

The classical abstract differential equation which is most frequently encountered is the equation^[1]

where the unknown function

belongs to some function space

and

is an operator (usually a linear operator) acting on this space. An exhaustive treatment of the homogeneous (

) case with a constant operator is given by the theory of C₀-semigroups. Very often, the study of other abstract differential equations amounts (by e.g. reduction to a set of equations of the first order) to the study of this equation.

The theory of abstract differential equations has been founded by professor Einar Hille in several papers and in his book Functional Analysis and Semi-Groups.^[2] Other main contributors were^[3] Kōsaku Yosida, Ralph Phillips, Isao Miyadera and Selim Grigorievich Krein.

Abstract Cauchy problem

Definition

Let^[4]^[5]^[6]

and

be two linear operators, with domains

and

, acting in a Banach space

. A function

is said to have strong derivative (or to be Frechet differentiable or simply differentiable) at the point

if there exists an element

such that

The Cauchy problem consists in finding a solution of the equation, satisfying the initial condition

Well posedness

According to the definition of well-posed problem by Hadamard, the Cauchy problem is said to be well posed (or correct) on

if:

A well posed Cauchy problem is said to be uniformly well posed if

implies

uniformly in

on each finite interval

Semigroup of operators associated to a Cauchy problem

To an abstract Cauchy problem one can associate a semigroup of operators

, i.e. a family of bounded linear operators depending on a parameter

(

) such that

Consider the operator

which assigns to the element

the value of the solution

of the Cauchy problem (

) at the moment of time

. If the Cauchy problem is well posed, then the operator

is defined on

and forms a semigroup.

Additionally, if

is dense in

, the operator

can be extended to a bounded linear operator defined on the entire space

. In this case one can associate to any

the function

, for any

. Such a function is called generalized solution of the Cauchy problem.

is dense in

and the Cauchy problem is uniformly well posed, then the associated semigroup

is a C₀-semigroup in

Conversely, if

is the infinitesimal generator of a C₀-semigroup

, then the Cauchy problem

Nonhomogeneous problem

with

, is called nonhomogeneous when

. The following theorem gives some sufficient conditions for the existence of the solution:

Theorem. If

is an infinitesimal generator of a C₀-semigroup

and

is continuously differentiable, then the function

Time-dependent problem

where the unknown is a function

is given and, for each

is a given, closed, linear operator in

with domain

, independent of

and dense in

, is called time-dependent Cauchy problem.

An operator valued function

with values in

(the space of all bounded linear operators from

), defined and strongly continuous jointly in

for

, is called a fundamental solution of the time-dependent problem if:

is also called evolution operator, propagator, solution operator or Green's function.

A function

is called a mild solution of the time-dependent problem if it admits the integral representation

There are various known sufficient conditions for the existence of the evolution operator

. In practically all cases considered in the literature

is assumed to be the infinitesimal generator of a C₀-semigroup on

. Roughly speaking, if

is the infinitesimal generator of a contraction semigroup the equation is said to be of hyperbolic type; if

is the infinitesimal generator of an analytic semigroup the equation is said to be of parabolic type.

Non linear problem

where

is a nonlinear operator with domain

, is called nonlinear Cauchy problem.

See also

References

1. ^{{cite web |last1=Dezin |first1=A.A. |title=Differential equation, abstract |url=http://www.encyclopediaofmath.org/index.php?title=Differential_equation,_abstract&oldid=14482 |website=Encyclopedia of Mathematics}}
2. ^{{cite book |last1=Hille |first1=Einar |title=Functional Analysis And Semi Groups |date=1948 |publisher=American mathematical Society|url=https://archive.org/details/functionalanalys017173mbp}}
3. ^{{cite book |last1=Zaidman |first1=Samuel |title=Abstract differential equations |date=1979 |publisher=Pitman Advanced Publishing Program}}
4. ^{{cite book |last1=Krein |first1=Selim Grigorievich |title=Linear differential equations in Banach spaces |date=1972 |publisher=American Mathematical Society}}
5. ^{{cite book |last1=Zaidman |first1=Samuel |title=Topics in abstract differential equations |date=1994 |publisher=Longman Scientific & Technical}}
6. ^{{cite book |last1=Zaidman |first1=Samuel |title=Functional analysis and differential equations in abstract spaces |date=1999 |publisher=Chapman & Hall/CRC}}
7. ^¹{{cite book |last1=Lakshmikantham |first1=V. |last2=Ladas |first2=G. E. |title=Differential Equations in Abstract Spaces |date=1972}}