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:
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:
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 [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.
http://www.livingreviews.org/lrr-2012-9 |
Living Rev. Relativity 15, (2012), 9
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