Abstract differential equation
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 privileged 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 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 Einar Hille in several papers and in his book Functional Analysis and Semi-Groups. Other main contributors were Kōsaku Yosida, Ralph Phillips, Isao Miyadera, and Selim Grigorievich Krein. Abstract Cauchy problem Definition Let 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 and its derivative is . A solution of the equation is a function such that: the strong derivative exists and for any such , and the previous equality holds . 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: for any it has a unique solution, and this solution depends continuously on the initial data in the sense that if (), then for the corresponding solution at every 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 .