Abstract evolution equation

Usually, a differential equation

\begin{equation} \tag{a1} \frac { d u ( t ) } { d t } = A ( t ) u ( t ) + f ( t ) , \quad 0 < t \leq T, \end{equation}

in a Banach space $X$ (cf. also Qualitative theory of differential equations in Banach spaces). Here, $A ( t )$ is the infinitesimal generator of a $C _ { 0 }$-semi-group for each $t \in [ 0 , T]$ (cf. also Semi-group; Strongly-continuous semi-group) and the given (known) function $f ( . )$ is usually a strongly continuous function with values in $X$. The first systematic study of this type of equations was made by T. Kato [a4]. Under the assumptions

i) the domain $D ( A ( t ) )$ of $A ( t )$ is dense in $X$ and is independent of $t$;

ii) $A ( t )$ generates a contraction semi-group for each $t \in [ 0 , T]$;

iii) the bounded operator-valued function $t \mapsto ( I - A ( t ) ) ( I - A ( 0 ) ) ^ { - 1 }$ is continuously differentiable he constructed the fundamental solution (or evolution operator) $U ( t , s )$, $0 \leq s \leq t \leq T$. He required this fundamental solution to be a bounded operator-valued function with the following properties:

a) $U ( t , s )$ is strongly continuous in $0 \leq s \leq t \leq T$;

b) $U ( s , s ) = I$ for $s \in [ 0 , T]$;

c) $U ( t , r ) U ( r , s ) = U ( t , s )$ for $0 \leq s \leq r \leq t \leq T$;

d) a solution of (a1) satisfying the initial condition

\begin{equation} \tag{a2} u ( 0 ) = u _ { 0 }, \end{equation}

if it exists, can be expressed as

\begin{equation} \tag{a3} u ( t ) = U ( t , 0 ) u _ { 0 } + \int _ { 0 } ^ { t } U ( t , s ) f ( s ) d s; \end{equation}

e) if $u _ { 0 } \in D ( A ( 0 ) )$ and $f \in C ^ { 1 } ( [ 0 , T ] ; X )$ or $f \in C ( [ 0 , T ] ; D ( A ( 0 ) )$), then (a3) is the unique solution of (a1), (a2). Since Kato's paper, efforts have been made to relax the restrictions, especially the independence of the domain of $A ( t )$ and the semi-group generated by $A ( t )$ being a contraction. Typical general results are the following.

Parabolic equations.

"Parabolic" means that the semi-group generated by $A ( t )$ is analytic for each $t \in [ 0 , T]$. In this case the domain of $A ( t )$ is not supposed to be dense. Consequently, property b) should be replaced by

\begin{equation*} \operatorname { lim } _ { t \rightarrow s } U ( t , s ) u _ { 0 } = u _ { 0 }\; \text { for } u _ { 0 } \in \overline { D ( A ( s ) ) }. \end{equation*}

P. Acquistapace and B. Terreni [a1], [a2] proved the following result: Suppose that

I) there exist an angle $\theta _ { 0 } \in ( \pi / 2 , \pi )$ and a positive constant $M$ such that:

i) $\rho ( A ( t ) )$ (the resolvent set of $A ( t )$) contains the set $S _ { \theta _ { 0 } } = \{ z \in \mathbf{C} : |\operatorname { arg } z | \leq \theta _ { 0 } \} \cup \{ 0 \}$, $t \in [ 0 , T]$;

ii) $\| ( \lambda - A ( t ) ) ^ { - 1 } \| \leq M / ( 1 + | \lambda | )$, $\lambda \in S _ { \theta _ { 0 } }$, $t \in [ 0 , T]$;

II) there exist a constant $B > 0$ and a set of real numbers $\alpha _ { 1 } , \ldots , \alpha _ { k } , \beta _ { 1 } , \ldots , \beta _ { k }$ with $0 \leq \beta _ { i } < \alpha _ { i } \leq 2$, $i = 1 , \ldots , k$, such that

\begin{equation*} | A ( t ) ( \lambda - A ( t ) ) ^ { - 1 } ( A ( t ) ^ { - 1 } - A ( s ) ^ { - 1 } ) \| \leq \end{equation*}

\begin{equation*} \leq B \sum _ { i = 1 } ^ { k } ( t - s ) ^ { \alpha _ { i } } | \lambda | ^ { \beta _ { i } - 1 } , \lambda \in S _ { \theta _ { 0 } } \backslash \{ 0 \} , \quad 0 \leq s \leq t \leq T. \end{equation*}

Then the fundamental solution $U ( t , s )$ exists, is differentiable in $t \in ( 0 , T ]$ and there exists a constant $C$ such that

\begin{equation*} \| \frac { \partial } { \partial t } U ( t , s ) \| \leq \frac { C } { t - s } , \quad 0 \leq s < t \leq T. \end{equation*}

If $u _ { 0 } \in \overline { D ( A ( 0 ) ) }$ and $f$ is Hölder continuous (cf. also Hölder condition), i.e. for some constant $\alpha \in ( 0,1 ]$,

\begin{equation*} \| f ( t ) - f ( s ) \| \leq C _ { 1 } | t - s | ^ { \alpha } , \quad s , t \in [ 0 , T ], \end{equation*}

then the function (a3) is the unique solution of the initial-value problem (a1), (a2) in the following sense: $u \in C ( [ 0 , T ] ; X ) \cap C ^ { 1 } ( ( 0 , T ] ; X )$, $u ( t ) \in D ( A ( t ) )$ for $t \in ( 0 , T ]$, $A u \in C ( ( 0 , T ] ; X )$, (a1) holds for $t \in ( 0 , T ]$ and (a2) holds. A solution in this sense is usually called a classical solution. If, moreover, $u _ { 0 } \in D ( A ( 0 ) )$ and $A ( 0 ) u_0 + f ( 0 ) \in \overline { D ( A ( 0 ) ) }$, then $u \in C ^ { 1 } ( [ 0 , T ] ; X )$, $u ( t ) \in D ( A ( t ) )$ for $t \in [ 0 , T]$, $A u \in C ( [ 0 , T ] ; X )$ and (a1) holds in $[0 , T]$. Such a solution is usually called a strict solution.

The above result can be applied to initial-boundary value problems for parabolic partial differential equations (cf. also Parabolic partial differential equation). The study of non-linear equations is also extensive. For details, see [a3], [a10].

Hyperbolic equations.

Here, equations of hyperbolic type are written as

\begin{equation} \tag{a4} \frac { d u ( t ) } { d t } + A ( t ) u ( t ) = f ( t ), \end{equation}

conforming to the notations of the papers quoted below, so that $- A ( t )$ generates a $C _ { 0 }$-semi-group. A general result on this class of equations was first established by Kato [a5], (and extended in [a6]), by K. Kobayashi and N. Sanekata [a8], and by A. Yagi [a11] and others. A typical general result is as follows. Suppose that $D ( A ( t ) )$ is dense in $X$. Let $Y$ be another Banach space embedded continuously and densely in $X$, and let $S$ be an isomorphism of $Y$ onto $X$. Suppose that

A) $\{ A ( t ) \}$ is stable with stability constants $M$, $\beta$, i.e. $\rho ( A ( t ) ) \supset ( \beta , \infty )$, $t \in [ 0 , T]$, and for every finite sequence $0 \leq t _ { 1 } \leq \ldots \leq t _ { k } \leq T$ and $\lambda > \beta$ the following inequality holds:

\begin{equation*} \left| \prod _ { j = 1 } ^ { k } ( \lambda - A ( t _ { j } ) ) ^ { - 1 } \right\| _ { X } \leq M ( \lambda - \beta ) ^ { - k }, \end{equation*}

where the product is time ordered, i.e. a factor with a larger $t_j$ stands to the left of all those with smaller $t_j$;

B) there is a family $\{ B ( t ) \}$ of bounded linear operators in $X$ such that $B ( . )$ is strongly measurable in $[0 , T]$, $\sup_{t \in [0,T]} ||B(t)||_X <\infty$, and

\begin{equation*} S A ( t ) S ^ { - 1 } = A ( t ) + B ( t ) , \quad t \in [ 0 , T ], \end{equation*}

with exact domain relation;

C) $Y \subset D ( A ( t ) )$, $t \in [ 0 , T]$, and $A ( . )$ is strongly continuous from $[0 , T]$ to ${\cal{L}} ( Y , X )$, i.e. to the set of bounded linear operators on $Y$ to $X$. Then there exists a unique evolution operator $U ( t , s )$, $( t , s ) \in \Delta = \{ ( t , s ) : 0 \leq s \leq t \leq T \}$, having the following properties:

$\| U ( t , s ) \| _ { X } \leq M e ^ { \beta ( t - s ) } , \quad ( t , s ) \in \Delta$;

$U ( t , s )$ is strongly continuous from $\Delta$ to $\mathcal{L} ( Y ) = \mathcal{L} ( Y , Y )$ with

\begin{equation*} \| U ( t , s ) \| _ { Y } \leq \overline { M } e ^ { \overline { \beta } ( t - s ) } , \quad ( t , s ) \in \Delta, \end{equation*}

for certain constants $\overline{M}$ and $\beta$;

for each $v \in Y$, $U ( ., . ) v \in C ^ { 1 } ( \Delta ; X )$ and

\begin{equation*} \frac { \partial } { \partial t } U ( t , s ) v = - A ( t ) U ( t , s ) v, \end{equation*}

\begin{equation*} \frac { \partial } { \partial s } U ( t , s ) v = U ( t , s ) A ( s ) v. \end{equation*}

For $u _ { 0 } \in Y$ and $f \in C ( [ 0 , T ] ; X ) \cap L ^ { 1 } ( 0 , T ; Y )$, the function $u$ defined by (a3) belongs to $C ^ { 1 } ( [ 0 , T ] ; X ) \cap C ( [ 0 , T ] ; Y )$ and is the unique solution of (a4), (a2).

The notion of stability was introduced by Kato [a5] and generalized to quasi-stability in [a6]. In [a5], [a6] it was assumed that $t \mapsto A ( t )$ is norm continuous in ${\cal{L}} ( Y , X )$.

For equations in Hilbert spaces, N. Okazawa [a9] obtained a related result which is convenient in applications to concrete problems.

Hyperbolic quasi-linear equations

\begin{equation*} \frac { d u ( t ) } { d t } + A ( t , u ( t ) ) u ( t ) = f ( t , u ( t ) ) \end{equation*}

have also been extensively studied. Especially deep research was carried out by Kato (see [a7] and the bibliography there). The assumption with the most distinctive feature in [a7] is the intertwining condition

\begin{equation*} Se ^ { - s A ( t , u ) } \supset e ^ { - s \hat{A} ( t , u ) } S, \end{equation*}

where $\hat{A} ( t , u )$ is considered to be a perturbation of $A ( t , u )$ by a bounded operator in some sense and $S$ is a closed linear operator from $X$ to a third Banach space such that $D ( S ) = Y$ (see [a7] for the details). The result can be applied to a system of quasi-linear partial differential equations

\begin{equation*} \frac { \partial u } { \partial t } + \sum _ { j = 1 } ^ { m }a _ { j } ( t , u ) \frac { \partial u } { \partial x _ { j } } = f ( t , u ), \end{equation*}

where the unknown $u$ is a function from $\mathbf{R} \times \mathbf{R} ^ { m }$ into $\mathbf{R} ^ { N }$, and $a_j ( .,. )$ are simultaneously diagonalizable $( N \times N )$-matrix valued functions.

The theory and methods for abstract evolution equations have been applied to many physical problems, such as the wave equation, the Navier–Stokes equations and the Schrödinger equation.

References

[a1]	P. Acquistapace, B. Terreni, "On fundamental solutions for abstract parabolic equations" A. Favini (ed.) E. Obrecht (ed.) , Differential equations in Banach spaces, Bologna, 1985 , Lecture Notes Math. , 1223 , Springer (1986) pp. 1–11
[a2]	P. Acquistapace, B. Terreni, "A unified approach to abstract linear non-autonomous parabolic equations" Rend. Sem. Univ. Padova , 78 (1987) pp. 47–107
[a3]	H. Amann, "Linear and quasilinear parabolic problems I: Abstract linear theory" , Monogr. Math. , 89 , Birkhäuser (1995)
[a4]	T. Kato, "Integration of the equation of evolution in a Banach space" J. Math. Soc. Japan , 5 (1953) pp. 208–234
[a5]	T. Kato, "Linear evolution equations of `hyperbolic' type" J. Fac. Sci. Univ. Tokyo , 17 (1970) pp. 241–258
[a6]	T. Kato, "Linear evolution equations of `hyperbolic' type II" J. Math. Soc. Japan , 25 (1973) pp. 648–666
[a7]	T. Kato, "Abstract evolution equations, linear and quasilinear, revisited" H. Komatsu (ed.) , Functional Analysis and Related Topics, 1991 , Lecture Notes Math. , 1540 , Springer (1993) pp. 103–125
[a8]	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
[a9]	N. Okazawa, "Remarks on linear evolution equations of hyperbolic type in Hilbert space" Adv. Math. Sci. Appl. , 8 (1998) pp. 399–423
[a10]	A. Lunardi, "Analytic semigroups and optimal regularity in parabolic problems" , Progress in Nonlinear Diff. Eqns. Appl. , 16 , Birkhäuser (1995)
[a11]	A. Yagi, "Remarks on proof of a theorem of Kato and Kobayashi on linear evolution equations" Osaka J. Math. , 17 (1980) pp. 233–243