Abstract hyperbolic differential equation
Consider the Cauchy problem for the symmetric hyperbolic system (cf. also Hyperbolic partial differential equation)
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with the boundary conditions
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Here, is a bounded domain with smooth boundary
(when
, no boundary conditions are necessary), and
,
, and
are smooth functions on
with as values real matrices in
, the
being symmetric. It is assumed that the boundary matrix
,
, is non-singular, where
is the unit outward normal vector to
. Also,
denotes the maximal non-negative subspace of
with respect to
, i.e.
,
, and
is not a proper subset of any other subspace of
with this property. The function
is the unknown function.
One can handle this problem as the Cauchy problem for an evolution equation in a Banach space (cf. also Linear differential equation in a Banach space). Indeed, let be the smallest closed extension in
of the operator
defined by
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with domain
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Then is the negative generator of a
semi-group on
(cf. [a1], [a2]; see also Semi-group of operators). Hence, the Hille–Yoshida theorem proves the existence of a unique solution
to the Cauchy problem
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which is given in the form
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Next to this idea of an abstract formulation for hyperbolic systems, the study of the linear evolution equation
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was originated by T. Kato, and was developed by him and many others (cf. [a3], Chap. 7). Here, denotes a given function with values in the space of closed linear operators acting in a Banach space
;
and
are the initial data, and
is the unknown function with values in
.
Among others, Kato's theorem in [a4] is fundamental: Suppose that
I) is a stable family on
, in the sense that
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for any and any
with some fixed
and
.
II) There is a second Banach space, , such that
, and
is a continuous function of
with values in
.
III) There is an isomorphism from
onto
such that
, with
a strongly continuous function of
with values in
. Then there is a unique solution
, and it is given by
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,
, where
is a unique evolution operator. It is easily seen that III) implies, in particular, the stability of
on
. When
and
are Hilbert spaces, III) can be replaced by the simpler condition [a5]:
III') There exists a positive-definite self-adjoint operator on
with
such that
for any
, with some constants
.
The Cauchy problem for the quasi-linear differential equation
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has been studied by several mathematicians on the basis of results for linear problems, [a7]. Here, depends also on the unknown function
. In [a6], [a7],
, defined for
, where
is a bounded open set, is assumed to satisfy conditions similar to I)–III) and a Lipschitz condition
with respect to
. Under such conditions, the existence and uniqueness of a local solution, continuous dependence on the initial data and applications to quasi-linear hyperbolic systems have been given.
References
[a1] | E. Hille, R.S. Phillips, "Functional analysis and semi-groups" , Amer. Math. Soc. (1957) |
[a2] | K. Yoshida, "Functional analysis" , Springer (1957) |
[a3] | H. Tanabe, "Functional analytic methods for partial differential equations" , M. Dekker (1997) |
[a4] | T. Kato, "Linear evolution equations of "hyperbolic" type" J. Fac. Sci. Univ. Tokyo , 17 (1970) pp. 241–248 |
[a5] | N. Okazawa, "Remarks on linear evolution equations of hyperbolic type in Hilbert space" Adv. Math. Sci. Appl. , 8 (1998) pp. 399–423 |
[a6] | K. Kobayashi, N. Sanekata, "A method of iterations for quasi-linear evolution equations in nonreflexive Banach spaces" Hiroshima Math. J. , 19 (1989) pp. 521–540 |
[a7] | T. Kato, "Abstract evolution equations, linear and quasilinear, revisited" J. Komatsu (ed.) , Funct. Anal. and Rel. Topics. Proc. Conf. in Memory of K. Yoshida (RIMS, 1991) , Lecture Notes Math. , 1540 , Springer (1991) pp. 103–125 |
Abstract hyperbolic differential equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Abstract_hyperbolic_differential_equation&oldid=50266