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
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:
Example 24. As a simple example consider the Hilbert space with the linear operator defined by
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
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  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  and the linearized Einstein equations  with elliptic gauge conditions.
Living Rev. Relativity 15, (2012), 9
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