### 3.4 Abstract evolution operators

A general framework for treating evolution problems is based on methods from functional analysis. Here, one considers a linear operator with dense domain, , in a Banach space and asks under which conditions the Cauchy problem
possesses a unique solution curve, i.e., a continuously differentiable map satisfying Eqs. (3.125, 3.126) for each . Under a mild assumption on this turns out to be the case if and only if the operator is the infinitesimal generator of a strongly continuous semigroup , that is, a map , with denoting the space of bounded, linear operators on , with the properties that
1. ,
2. for all ,
3. for all ,
4. and .

In this case, the solution curve of the Cauchy problem (3.125, 3.126) is given by , , . One can show [327, 51] that always possesses constants and such that

which implies that for all and all . Therefore, the semigroup gives existence, uniqueness and continuous dependence on the initial data.

There are several results giving necessary and sufficient conditions for the linear operator to generate a strongly continuous semigroup; see, for instance, [327, 51]. One useful result, which we formulate for Hilbert spaces, is the following:

Theorem 4 (Lumer–Phillips). Let be a complex Hilbert space with scalar product , and let be a linear operator. Let . Then, the following statements are equivalent:

1. is the infinitesimal generator of a strongly continuous semigroup such that for all .
2. is dissipative, that is, for all , and the range of is equal for some .

Example 24. As a simple example consider the Hilbert space with the linear operator defined by

where denotes the Fourier–Plancharel operator; see Section 2. Using Parseval’s identity, we find
hence is dissipative. Furthermore, let , then
defines an element in satisfying . Therefore, the range of is equal to , and Theorem 4 implies that generates a strongly continuous semigroup on such that for all . The curves , , are the weak solutions to the heat equation on ; see Section 3.1.2.

In general, the requirement for to be dissipative is equivalent to finding an energy estimate for the squared norm of . Indeed, setting and using we find

for all and , which yields the estimate
for all . Given the dissipativity of , the second requirement, that the range of is for some , is equivalent to demanding that the linear operator be invertible. Therefore, proving this condition requires solving the linear equation
for given . This condition is important for the existence of solutions, and shows that for general evolution problems, requiring an energy estimate is not sufficient. This statement is rather obvious, because given that is dissipative on , one could just make smaller, and still have an energy estimate. However, if is too small, the Cauchy problem is over-determined and a solution might not exist. We will encounter explicit examples of this phenomenon in Section 5, when discussing boundary conditions.

Finding the correct domain for the infinitesimal generator is not always a trivial task, especially for equations involving singular coefficients. Fortunately, there are weaker versions of the Lumer–Phillips theorem, which only require checking conditions on a subspace , which is dense in . It is also possible to formulate the Lumer–Phillips theorem on Banach spaces. See [327, 152, 51] for more details.

The semigroup theory can be generalized to time-dependent operators , and to quasilinear equations where depends on the solution itself. We refer the reader to [51] for these generalizations and for applications to examples from mathematical physics including general relativity. The theory of strongly continuous semigroups has also been used for formulating well-posed initial-boundary value formulations for the Maxwell equations [354] and the linearized Einstein equations [309] with elliptic gauge conditions.