CC BY
7
MSC 34K32, 47D06, 47D07
DOI: 10.14529/mmp 180307
CAUSAL RELATIONS IN SUPPORT OF IMPLICIT EVOLUTION EQUATIONS
N. Sauer1, J. Banasiak1, Wha-Suck Lee1 1 University of Pretoria, South Africa
E-mail: [email protected], [email protected], [email protected]
This is a brief exposition of dynamic systems approaches that form the basis for linear implicit evolution equations with some indication of interesting applications. Examples in infinite-dimensional dissipative systems and stochastic processes illustrate the fundamental notions underlying the use of double families of evolution equations intertwined by the empathy relation. Kisynski's equivalent formulation of the Hille-Yosida theorem highlights the essential differences between semigroup theory and the theory of empathy. The notion of K-bounded semigroups, a more direct approach to implicit equations, and related to empathy in a different way, is included in the survey.
Keywords: implicit equations; empathy theory; semigroups.
Dedicated, to Professor Jan Kisyrlski on the occasion
of his 85th birthday.
Introduction
The classic autonomous evolution equation in a Banach space Y is
d
-u(t) = Au(t) (1)
with A (usually) an unbounded linear operator in Y. The causal relation that underpins such an equation is the semigroup E(t + s) = E(t)E(s). Indeed, the solution curve is of the form u(t) = E(t)y, but it is not the only trajectory defined by the semigroup. The method is effective if A is the generator of the semigroup and the initial state y is in the A
In fluid mechanics and in dynamic boundary condition problems, evolution equations of the form
d
Jt [Bu(t)] = Au(t) (2)
arise. Here two Banach spaces X and Y are involved with A and B unbounded linear operators defined on D С X with values in Y. These are called implicit evolution equations. In early studies of implicit equations, modelled according to equations of
AB
not necessarily of the same order. This led to an assumption that the operators involved were closed or could be extended separately to closed operators. This, in turn, made possible to use the initial condition lim u(t) = u0 so that lim [Bu(t)] = Bu0 = y. For
dynamic boundary condition problems this setting can be untenable. Indeed, an example B
}im\[Bu(t)] = y. (з)
For the classic evolution equation (in a single space) treatment by means of semigroups made it necessary that the operator A be closed or closeable. Thus we are led to the question: what is the analog of the notion of semigroup when we treat implicit equations? It turned out to be important to consider two families of "evolution operators". The "solution operators" S(t) : Y ^ X which will represent the solution in the form u(t) = S(t)y and another family E(t) : Y ^ Y that will describe the curve v(t) = Bu(t) in Y. If we free ourselves temporarily from the operator B, this leads to the empathy relation S (t + s) = S (t)E (s).
If causal relations such as semigroup and empathy are used to investigate the solvability of evolution equations, a major question not always kept in mind, is the nature of the initial states that evolve into solutions. Our approach to the problem is via the Laplace transform which leads in an almost natural way to an answer.
In what follows we give a survey of the development of the empathy relation. Some proofs are given to share some secrets of the trade. Long and technical proofs have been omitted. The flow of text is as follows: Section 1 gives the mathematical setting for empathy theory and the very basic results. The bearing of empathy theory on implicit evolution equations is discussed in Section 2 where we introduce the notion of generator of an empathy. Section 3 gives a very short indication of how the theory is applied. The important case of holomorphic empathies is briefly discussed in Section 4. As in the case of holomorphic semigroups, the "admissible" class of initial states is quite large. In Section 5 we discuss integrated empathy which is the analog of integrated semigroup. An adaptation of Jan Kisyriski's algebraic approach to the Hille-Yosida construction to empathy theory is discussed in Section 6. Section 7 is devoted to the role of empathy-considerations in Markov processes. It leads to implicit Fokker-Planck equations. The topic of K-bounded semigroups, which represents an alternative approach to implicit evolution equations, is explained in Section 8. We conclude with Section 9 where additional background is supplied and references are given.
1. Empathy
XY
linear operators, {S(t) : Y ^ X} and {E(t) : Y ^ Y} defined for t > 0, with the following properties:
S(t + s) = S(t)E(s) for all t,s> 0. (4)
For every A > 0 and y E Y the Laplace transforms
P (A)y = j exp{-At}S (t)ydt, (5)
0
X
R(A)y = j exp{-At}E (t)ydt (6)
0
exist as Lebesgue (Bochner) integrals.
As stated before, the requirement (4) is known as the empathy relation. The conditions (5) and (6) are akin to similar requirements used in semigroup theory when the C 0-property is not imposed.
Direct calculations lead to Theorem 1. For arbitrary positive X, p and t the following is true:
P (X)E (t) = S (t)R(X), (7)
P (X)R(p) = P (p)R(X),
P(X) — P(p) = (p — X)P(X)R(p). (8)
Additional to the assumptions (5) and (6) we make one more with far-reaching consequences:
The invertibility assumption. There exists £ > 0 for which the linear op era tor P (£) : Y ^ X is invertible.
Theorem 2. The family {E(t)} is a semigroup, strongly continuous in (0, to). Moreover, the family {S(t)} is strongly continuous in (0, to).
Proof. From (4) we see that S(t+r+s)y = S(t)E(r+s)y = S(t+r)E(s)y = S(t)E(r)E(s)y. Taking the Laplace transform at £ with respect to t gives P(£)[E(r + s)y — E(r)E(s)y] = 0 and the semigroup property follows. Assumption (6) means that the function t ^ E(t)y is measurable and therefore continuous in (0, to) [1, Theorem 10.2.3]. From (4) it is seen that if h E (0, t/2), S(t — h)y = S(t/2)E(t/2 — h)y ^ S(t/2)E(t/2)y = S(t)y when h ^ 0. Continuity from the right is easier.
□
We note that continuity at t = 0 makes no sense and {E(t)} does not have to be of class C0. As another important consequence we have the identities
R(X)E (t) = E (t)R(X), (9)
R(X) — R(p) = (p — X)R(X)R(p). (10)
They are derived in the same way as the identities in Theorem 1. Thus we have arrived at two pseudo-resolvent equations namely (8) and (10). These will turn out to be crucial.
Theorem 3. For every X> 0 the operator P (X) is invertible.
Proof. We consider the kernels of P(X) and R(X). First we note from (10) that ker R(X) = ker R(p) =: NE. Then, from (8), if y E NE nker P(p) then P(X)y = (p — X)P(X)R(^)y = 0. Hence P(X) n NE = P(p) n NE for arbitrary X and p. Suppose that P(X)y = 0. From (8) it follows that P(£)y = (£ — X)P(£)R(X)y and from the invertibility assumption therefore, y =(£ — X)R(X)y. Hence R(£)y = (£ — X)R(£)R(X)y. It follows from (10) that R(X)y = 0. Thus y E ker P(X) n NE = ker P(£) n NE = {0}.
□
As can be seen from the proof above, the pseudo-resolvent equations (8) and (10) lead to invariances. There are more. Let us define the domains DE := R(X)[Y] and D := P(X)[Y]. From (8) and (10) we readily see that the definitions do not depend on the choice X
Theorem 4. For all t > 0 E(t)[DE] C DE and S(t)[DE] C D.
Some calculations are needed to obtain the following representations: Let y = R(X)yx e DE. Then
t
E (t)y = ext[y — J e-XsE (s)yx ds], (11)
0
t
S (t)y = ext [P (X)y — J e-XsS (s)yx ds]. (12)
0
From these representations it is possible to prove
Theorem 5. The following statements hold:
(a) For every y e D^ lim^o+ E(t)y = y.
(b) There exists a linear operator B0 : DE ^ D defined by By = limt^0+ S(t)y.
(c) The operators R(X) are invertible for all X > 0.
(d) The operator B0 is invertible, B = P(X)R-1 (X) and B[DE] = D.
The notation B0 indicates that this operator is the inverse of an operator yet to be introduced. To end this section we state a theorem, the proof of which is based on the dominated convergence theorem, to show that Theorem 5 is in accordance with asymptotic behaviour of the Laplace transform.
Theorem 6. For y e DE, lim XR(X)y = y, and lim XP(X)y = By. 2. The Generator of an Empathy
We shall refer to the pair (S(t),E(t)) as an empathy and proceed to define the notion of its generator. To begin with we define the operator B : D ^ DE as the inverse of the operator B. By Theorem 5 this is possible. In fact, we have the representation B = [B]-1 = R(X)P-1 (X) which is free of the choice of X.
Next, we define the operators Ax := [XR(X) — IY]P-1 (X). It takes some effort to prove that Ax = AM =: A and that P(X) = (XB — A)-1. We call the operator pair (A,B) the generator of the empathy. The following result explains the word.
Theorem 7. Let u(t) = S(t)y. For y e DE, u(t) is a solution of the Cauchy problem
d [Bu(t)] = Au(t),
dt Kh) (13)
imBu(t)]=y.
Proof. From Theorem 4 we see that u(t) e D so that Au(t) and Bu(t) are well-defined. Let us obtain an expression for Bu(t). If y = R(X)yx then, from the representation of B, (7) and (9) we obtain Bu(t) = BS(t)R(X)yx = R(X)P-1 (X)S(t)R(X)yx = R(X)E(t)yx = E (t)R(X)yx = E (t)y.
From Theorem 5 we immediately see that the initial condition is satisfied. To evaluate the derivative of Bu(t) we use the representation (11) and note that, by Theorem 2, Bu(t) = E(t)y
manipulations, turns out to be Au(t). □
gg Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming
& Computer Software (Bulletin SUSU MMCS), 2018, vol. 11, no. 3, pp. 85-102
Thus the notion of empathy, which involves two families of "evolution operators", turns out to be the replacement of semigroups when implicit evolution equations are concerned. We need to take the analogy further. In semigroup theory the generator needs to be a closed (or closeable) operator (see e.g. [1, Chapter XI]). What should it be in empathy
P(X)
are bounded [1, Theorem 3.8.2]. This means that the operators P-1 (X) = XB — A are closed. What does that say about A and B? For this we introduce the notion of jointly closed. Let A,B : D C X — Y be two linear operators. Then A and B are jointly closed if the operator (A, B) : x E D — {Ax, Bx) E Y x Y is closed. This is the same as saying that if {xn } C D, xn — x in X, Axn — yA and Bxn — yB, then x E D and Ax = yA, Bx = yB. It takes some refection to conclude that if XB — A is closed for two distinct values of X, then A and B is jointly closed. Thus we have
Theorem 8. The generator of an empathy is closed.
We note that if the operators A and B are both closed, the operator (A, B) is closed. The converse is not true.
3. Applications
To apply empathy theory to concrete problems, sufficient conditions for an operator
(S( t) , E( t) )
is uniformly bounded if for all t > 0 there are constants M and N such that \\S(t)|| < M and \\E(t)\\ < N.
Let A,B : D C X — Y be given linear operators, and suppose that P(X) := (XB—A)-1 exists for every X > 0. In accordance with Theorem 5 we let R(X) := BP(X). The Hille-Yosida-type theorem is
Y
P(X) R(X) (A, B)
MN
for all X > 0 and k = 1, 2....
\\XP(X)\\< M and \\XkRk (X)\\< N.
YE DE Y E(t)
This theorem is still far removed from applications. The Radon Xikodyni property Y
concern though, is the requirement imposed by the condition that P(X) = (XB — A)-is bounded for all X which implies that A and B should be jointly closed. For situations based on dissipative phenomena such as heat and diffusion, the evolution equations are often framed in a Hilbert space setting {e.g. L2 and embedded Sobolev spaces) and the
(A, B)
Friedrichs extension had to be found. This was done for the first time to deal with dynamic boundary conditions for the Xavicr Siokcs equations, and later extended to cover a large class of problems including the so-called Sobolev equations (misnamed after Sobolev) and pseudo-parabolic equations.
A telling example that shows the strength of the approach, is the heat equation in a domain Q C R3 with boundary r. The equation is
ut (x, t) — Au(x, t) = 0; x e Q; t > 0,
with A is the Laplacian.
If the boundary r is considered as heat-transferring medium with it own thermal properties, the boundary condition is, in a very simple model
dt [You(y,t)} + khu(y,t) y e r.
Here y0 denotes the boundary trace operator and y1 the normal derivative. If we take X = L2 (Q), Y = L2 (Q) x L2 (r^d D = W2,2 (Q), the system of equations can be expressed in the form
dt (u,Y0u) = (Au, —kj1 u) = 0 e Y.
We use the notation (., .) to denote elements of Y. Thus with Bu = (u,%u) and Au = (Au, —kj1 u), this becomes a respectable implicit equation. But the operator B is not closeable. Fortunately the joint extension exists and the problem can be handled.
4. Holomorphic Empathies
If the semigroup E(t) is holomorphic, in the sense that R(X) can be extended to a sector = { — ($ + n/2) < Arg (X) < n + n/2 : 0 < ^ < n/2} of the complex plane such that
\\R(X)\\< M, for X e ^,
P(X)
\\P(X)\\< N,for X e .
S(t) E(t) associated Cauchy-problem
d
- [Bu(t)] = Au(t),
imBu(t)]=y,
can be solved for y e YE. The holomorphic case therefore allows a larger class of initial states.
The extension procedure mentioned in Section 3 allows for the extension of an operator (A, B)
holomorhic empathies associated with dynamic boundary conditions for heat transfer and diffusion such as the one discussed above where the initial state can be in L2 (Q) x L2 (r). For the Sobolev-type of Cauchy-problem
dt [Mu(x,t)]= Lu(x,t); x E Q, tli0+[Mu(y,t)} = y; y E г,
}
with L and M strongly elliptic partial differential operators of order 21 and 2m respectively, the underlying empathy is holomorphic. The joint (Friedrichs) extension of (L, M) leads
LM
Within the framework of holomorhic empathy there is a happy idea of studying perturbations. This is based on comparison within a fixed solution space X and using the complex plane as another steady entity while allowing the operators A and B to have different range spaces. The framework is as follows: For n = 1, 2,... let An ,Bn : Dn C X ^ Yn and consider the sequence of Cauchy-problems
d
— [Bn u(t)] = An u(t); dt
lim u(t) = yn e Yn.
Let (S(t),E(t)) be holomorphic empathy in the spaces X and Y with sectorial domain £,. If there is a sectorial domain £ . such that
90 V
X
n=0
and for A G £
lim \\Pn (X)yn - P(A)y|| =0,
then Sn (t)yn ^ S(t)y in X. This result can be used to study singular perturbations of dynamic boundary conditions and Sobolev-type equations.
5. Integrated Empathy
Y
excludes cases where spaces of continuous functions under the supremum norm are concerned. For semigroups this has been overcome by introducing the notion of integrated
(S( t) , E( t) )
of evolution operators, this concept is adapted by replacing the causal relation (4) by
t
S(t)E(s)y = j[S(s + a) - S(a)]yda. (14)
0
(S( t) , E( t) )
(6) we introduce the Laplace transforms p(A) and r(A) by
(
p(A)y = J exp{-At}S (t)ydt,
0
(
r(A)y = exp{-At}E (t)ydt
and let P(A) := Ap(A), R(A) := Ar(A). The analysis now follows the same pattern as in Section 1 with some deviations. The representations (11) and (12) are different and one needs to replace lim E(t)y and lim S(t)y in Theorem 5 by lim t-1 E(t)y and
lim t-1 S(t)y.
The implicit evolution equation in Theorem 7 satisfied by u(t) = S(t)y; y e DE is replaced by the implicit integral equation
t
Bu(t) = y + A J u(s) ds. 0
For a domain smaller than DE the original implicit differential equation is still satisfied.
Within this setting it is possible to study wave motion described by the system of equations
u1, t(x,t) — u2,x (x,t) = 0 u2, t(x,t) — u1,x (x,t) = 0,
for 0 < x < 1, under the dynamic boundary condition
d
— [lim u1 (x, t)] + lim u2 (x, t) = 0, dt x ^ i x ^ i
in the space C([0,1]) of continuous functions. The boundary condition at x = 0 can be one of many.
6. The Kisyriski Construction
The Hille-Yosida-like generation Theorem 9 is based on considering, for given
(unbounded) linear operators A and B from D C X to Y that define, for X > 0 the operators
P (X) = (XB — A)-1; (15)
R(X) = BP (X), (16)
and assuming that they exist and are bounded. Moreover, the result, (Theorem 9), leans on
Y
property. The question is, can these assumptions be weakened in some sense?
An answer to this question can be found in Kisynski's algebraic approach to the Hille— Yosida theorem for semigroups [4, Theorem 12.5]. But there is a price to be paid.
The point of departure is to consider two families Px : Y ^ X and Rx : Y ^ Y
X>0
equations
Rx — R, = — X)Rx r, A
Px — P, = (p — X)Px R,,
P(X) R(X)
associated with Laplace transforms. In this way we free ourselves from the assumption that Y
theorem to hold if the Hille-Yosida inequalities in Theorem 9 are invoked. Instead, we assume that the pseudo-resolvent Rx : Y ^ Y; X > 0 satisfies the strong Widder growth condition [5]
sup{\\ [XRx]k \\ : X > 0; k e N} < to. (18)
In addition it is assumed that the operators Rx are invertible, in contrast to the (invertibility) assumption that Px be invertible.
We define the class of "initial states" DE := Rx [Y] analogous to DE. The growth condition (18) ensures the existence of a unique bounded Banach algebra representation T : rx M Rx on the convolution algebra Z := (L1 (0, to), ®) of integrable scalar functions. The family of exponentials {rx : A > 0} with rx (x) = exp{ — Ax} for x > 0 is a canonical Z
T
{E(t) : t > 0} of class C0 on the subspace AK := {y E Y : \\ARxy — y\\ = 0}. This
analytic object has the algebraic reconstruction
A, = De = U T ШL
Ф
which is called the T-regularity space. Now a unique C0-semigroup E(t) on AK is constructed from the canonical C0-semigroup of right-shifts Rt on Z % letting E(t)y := [T (Rt (4>))}y9 for y = T ($>)y9 e AK. The (vector) sub space DE is precisely the domain of the infinitesimal generator of E(t).
As in Section 1 the domain D := Px [Y] is free of the choice of A, and assume that that there is, in analogy to (16), a linear operator B : D C X ^ Y such that Rx = BPx. Thus
we replace the invertibility of Px by the weaker condition that BPx is invertible.
y
(S(t),E(t))t>0 intertwined by the empathy relation such that u(t) = S(t)y satisfies (13).
Towards this goal we define the class A2K := Rx [AK ] and call it the T2 -regularity space. Then A2K is a dense subspace (like DE) of AK. We use the operator C := P(A)R-1 (A) : A2K ^ D to construct a map T2 := CT on the algebra Z. Note that T2 is not an algebra representation and need not even be closed. Then we construct, (a) the semigroup E(t) : AK ^ AK by the bounded Banach algebra representation T and (b) the (family of) operators S(t) : A2 ^ D by the map T2 as follows:
S(t)[RxT(Ф)] = T2(rx ® Rф) = P(X)[E(t)T(Ф)].
The bounded operator S(t) constructed on the dense subspace AK can be extended by continuity to the Banach space AK. Moreover, the map T2 that generates the operators S(t)
semigroup theory. Then from the relation Rx = BPX and the commutativity of E(t) and Rx we have the following result:
Theorem 10. Let (Px, Rx) be an entwined pseudo-resolvent (17) and B : D С X ^ Y be a linear operator such that Rx = BPX. If Rx satisfies the growth condition (18), then u(t) = S(t)y; y E A2K solves the implicit Cauchy problem (13) with A = AEB. Here AE
E(t)
P(A) B
invertible and C = B-1. This is at the core of the theory of empathy.
7. Implicit Fokker-Planck Equations
Causal relations explain how a state u(t) at time t evolves to a state u(t + s) at a later time t + s where s > 0. The semigroup relation E(t + s) = E(s)E(t) is a causal relation that is based on the assumption that the effect of the state u(t) on any later state u(t + s) is determined solely % the the time difference s. In many real world situations,
s
This is captured in the notion of a stochastic kernel Qs (x, B) of a Markov process X. The Chapman-Kolmogorov equation
Q+ (x, b )
Qt (x, [dy])Qs (y, B ) for alls, t> 0,
(19)
yeR
is a causal relation that follows from the assumption of memory-less transitions. For a homogeneous Markov process X with a transition function Q = {Qs (x, B)}s>0 intertwined by the Chapman-Kolmogorov equation, Q has an operator representation {Es : Y ^ Y}s>0 on a suitable space Y of measurable functions and Et+s = Es Et. The question is, what causal relation similar to (19) is described by the empathy relation S(t + s) = S (s)E (t)l
An answer to this question can be found in a dynamic boundary condition approach to diffusion with an absorbing barrier when the boundary is a object in its own right, interacting meaningfully with the system as another body. Two distinct intensities arise: the absorbing barrier is seen as a distinct collection of states with zero intensity, while the system is a typical pseudo-Poisson process with uniform intensity a > 0. Also, two distinct state spaces arise: SX := {1,... ,m}, the set of m safe (life) states that all have the same intensity a, and SY := {1,..., n}, the set of n death states that all have the same intensity 0
which is prohibited in classical diffusion equations. The bar notation distinguishes between a life state and a death state.
The pair of state spaces (SX , SY) gives rise to the pair (qj , r(n) )n>1 of n-step transition functions,
(n)
q,
(n-lik
where pi
Sx |
Ei
k=l
Sx |
ij ^ v ^ik
k=l
denotes the one step transition P(X,
pkj wit h q
(i)
pi
(n)
(n-1)
■sk. wit h r
kj
(1)
(20)
n + 1
j\Xn = i) and s.-. denotes the one step
transition P(Yn+1 = j\Xn = i). For the continuous-time version of the discrete Markov chains (20), we partition the event of a transition in the time interval (0, t] into mutually exclusive events determined by the number of transitions in the time interval (0, t] :
Qt (i, r = {j}) = P(Xt = j\X0 = i) = e-at £ qin (at)
n=0
n!
rc ( t)r
Rt (i, r = {j}) = P(Y = j\X„ = i) = — Y r'"'(at)
£
n=0
ij n!
Now, the pair (Q, R) := (Qt (i, Г), Rt (i, r))t>0 is intertwined by a countable version of the backward extended Chapman-Kolmogorov equation/.
Qt+s (x, Г)= J Qt (x, {dy})Qs (y, Г); s,t> 0;
yeR
Rt+S (x, Г)= J Qt (x, {dy})Rs (y, Г); s,t> 0. (21)
yeR
Their operator representation (St : X ^ Y,Et : Y ^ Y)t>0 produces a pair of function spaces X := BM(SY ^d Y := BM(SX) of bounded measurable functions on SX and SY, respectively. The two-state space uni-directional transition (21) forces the two state spaces X and Y to be disjoint. Indeed, the pair (St ,Et )t>0 has exponential (pencil-operator) representations St = e-ateatKR and Et = e-ateatKQ where (KR ,KQ) are the operator representations of the pair of defective transition matrices ([s. ].ex , [pij ]i jeX), respectively. Moreover,
ea(t+s)KR = eatKQ easKR ^22)
By (22), the pair (St ,Et )t>0 is intertwined % the reverse empathy relation St+s = Et Ss ; s, t > 0
an empathy (St : Y ^ X,Et : Y ^ Y). The machinery of empathy theory then yields an implicit Fokker-Planck equation.
8. K-Bounded Semigroups and Implicit Evolution Equations
Very often the existence of the semigroup {E(t)}t>0 solving (1) is established in a non-constructive way and then very little quantitative information on the evolution is available. On the other hand, there may exist an operator K such that t ^ KE(t) can be calculated constructively yielding some information about the evolution. In other words, it may be possible to look at the evolution through the "lens" of another operator and filter out the information we are not able to quantify. This idea led to the following definition:
Definition 1. Let X and Y be Вanach spaces and let L : DL С Y ^ Y and K : DK С Y ^ X be linear operators. Supp ose that DL С D K and for so me ш E R the resolvent set L
p(L) D (ш, ж). (23)
A family of operators {Z(t)}t>0 from DK to X, which satisfies
1) for every t > 0 md f E DK
\\Z(t)f ||x < Mexp{ut}\\Kf ||x , (24)
2) for every f E DK the function t ^ Z(t)f E C([0, m),X),
3) for f E Dkl := {f E Dl n Dk : Lf E Dk }
t
Z(t)f = Kf + J Z(s)Lfds; t > 0, 0
KL
For an operator A we write A E G(M,u,Y) if A generates a C0-semigroup in Y, that satisfies estimate such as (24). If there is a need to indicate that a semigroup is generated by A we write {EA (t)}t>0. Similarly, we write L E K — G(M,u,Y, X) if L generates a K-bounded semigroup as in Definition 1.
Let YK be the completion of the quotient space DK / ker K with respect to the seminorm \\f || K = \\Kf 11X. The n DK / ker K is isometrically isomorphic to a dense subspace of YK1 say y. The canonical injection of Y into YK (and onto y) will be denoted by p. In a standard way, K can be extended by density to an isometry K: YK ^ X.
L K L
preserves cosets of DK / ker K and therefore it can be defined to act from pDK(L) C y into y. We denote by LK the part of L in QK, i.e. LK = L\D . It can be also proved that if
L E K — G(M,u,Y, X), then the shift of L to y is closeable in YK; we denote its closure by L.
LK
yp
precisely projection) and for any operator C defined in YK and f E DK, the symbol Cf is to be understood as Cpf, if the latter is defined.
K
Theorem 11. If L E K — G(M,u,Y,X) and K[DK(L)] = XK, then L E G(M,u,YK). Conversely, if there is L D L such that L E G(M,u,YK), then L = L and L E K — G(M, u, Y, X).
The K-bounded semigroup {Z(t)} for f E DK is given by
Z(t)f = EKLK-1 (t)Kf = KEl (t)f. (25)
The assumption that K [DK(L) ] is dense in XK can be discarded if X (and consequently
XK
YK
often used in the following version.
L K L
K
Z(0)f = Kf for all f E Dk , (26)
if and only if the following conditions hold:
1) K[DK(L)] is dense in XK,
2) there exist M > 0 mid u E R such that for every f E DK, X > u and n E N:
\\K (XI — L)-n f \\x < x-un\\Kf \\x •
{ Z( t) }
We observe that it is not necessary for L to generate a semigroup in Y and Y need
LY
(2.1') The space Y is a linear space and the operator LK is closeable in YK. Denoting
-Yk
L = LK , we assume further that there exist subspaces: Y satisfying DK С Y С YK, and D such that DK(L) С D С X П DL such that (X — L|D) : D — Y is bijective for all X > ш.
LK
assumption (23) replaced by 2.1'. Then L E K — G(M^,Y, X) and (26) holds if and only if the following conditions are satisfied:
1) K [D] is dense in XK,
2) there exist M > 0 an d ш E R such that for e very y EY, X > ш and n E N:
\\K(XI — lIdгy\\x < \\Ky\x.
If (26) is not assumed, the assumption 1. is sufficient but not necessary. In both cases the K
Let us consider again the original Cauchy problem (2), (3). It is often the case that XY
point of view. We are usually interested to keep the values of the solution in the original space which may be related to some physical properties like the finite total energy space, finite mass, etc., but for (2) to hold in the strict sense may be too restrictive and often it is enough that it holds in some other Banach (or even linear topological) space Y with B and A replaced by appropriate extensions B and A acting from X to Y.
To be able to link B and A with B and A we restrict these extensions to the closures of respective operators. In other words, D A and DB are required to be со res for A and B, respectively.
As we mentioned above, in general, thanks to Theorem 13, we don't need any Y
BA
assumptions imposed on either AB-1 or B-1 A. Let us introduce the following definition.
Definition 2. Let Y С Y an d Л = A , B = B .A n X-valued function t — u(t) is called
YY
d
— [B u(t)] = Au
dt e } (27)
lim [Bu(t)] = U,
that is, t — Bu(t) is continuously differentiable in Y, the differential equation holds for t > 0 YY YY
This definition suggests an alternative way of approaching implicit problems (2) with BB
K
theorem shows, the space YB-1 turns out to be a good choice as on this space a suitable extension of B is an isomorphism (and even an isometry).
Theorem 14. Let us suppose that we are given operators B : DB — Y and A : D A — Y with Da , DB C X, where X is a Banaeh and Y is a linear space. Assume that B is a densely defined, one-to-one operator. Define L = AB-1 with the natural domain Dl = B[Da n DB] and K = B-1. If L e K - G(M,u,Y,X), then for u e DK(L) = {u e B[Da n DB] : AB-1 u e B[DB}}, the function t — Z(t)u, where {Z(t)}t>0 is the K L YK
Assumption (23), specified to the present conditions, means that the operator (AB-1, DAB-1) satisfies p(AB-1) D [u, <x>) for some u e R. If this assumption is satisfied, we can combine Theorems 14 and 12 to obtain the following result specified to the holomorphic case.
Theorem 15. Assume that
1) the set Db-1a = {y e DA n DB : Ay e ImB} is dense in >
2) for f e Db
||(AI - B-1 A)-1 f II* f llx, (28)
for A in some sector with the opening greater than n, then for any u e DK(L), the t - Z(t)u YK
In reflexive spaces Assumption 1 is superfluous.
We illustrate the approach by considering the same example as in Section 3 where, for simplicity, we consider ^ = (0,k = 1. Let us begin with the setting of Section 3; that is, with X = L2(0,1). The boundary operators are y0u = (u(0),u(1)^d y1 u = {—ux (0),ux (1)), for sufficiently regular u. Let DA = W2,2 (0,1) and, on DA,
Au = (uxx,-y1 u) = (uxx ,ux (0), -ux (1))
and DB = W1,2 (0,1) with
Bu = (u, y0u) = (u, u(0),u(1)).
However, we see that with such a domain we cannot achieve A[DA] C ImB as uxx does not have traces at x = Let us now change the setting and take X = W1,2 (0,1). Then B is well-defined and bounded on X with ImB = (u,u(0),u(1)) C Y := X x C2.
YB
14 we should define A on such a domain that Au = (uxx, ux (0), -ux (1)) e ImB; that is,
Da = {u e W3,2 (0,1) : uxx (0) - ux (0) = 0,uxx (1) + ux (1) = 0}. Now, B-1A = uxx and (28) with u = 0 can be proved by standard Hilbert space methods.
DB-1A X
therefore the problem is solvable. Here XK = X, the semigroup t — EB-1 (t) acts in
X and the solution operator is defined as t ^ Eb-1a (t)B \ see (25). Thus the solution
operator acts on the subspace of W1 ' 2 (0,1)xC2 of initial conditions satisfying compatibility conditions, (u,u(0),u(1)), in contrast to the solution in [3] where a weaker regularity of the solution does not require such a restriction.
Notes and Remarks
For the heuristic underpinning of evolution equations by semigroups, consult S.G. Krein [6].
The results of Sections 1, 2 and 3 are based on the paper [7] where complete proofs and discussions can be found. Of historical interest is that the notion of empathy, in a primitive form was introduced in an earlier paper [8]. That was preceded by the notion of B-evolution introduced in [3] which involves only the solution operators S(t), pre-supposes the operator B and uses the causal relation S(t + s) = S(t)BS(s) on the space Y. If one sets E(t) = BS(t) this is in line with the same expression derived in Section 2 (valid only on the domain DB). The example of a non-closeable operator B can be found in [9].
Equations of Sobolev-type were studied by Showalter and Ting in [10,11]. Within the framework of B-evolutions this was studied by Van der Merwe [12]. Also of note, are the studies of Favini-Yagi where the vantage point is that the operators A and B are assumed to be closed [13].
The Friedrichs extension of an operator-pair was first introduced in [24] and effectively used in a study of dynamic boundary conditions for the Xavicr Siokcs equations [25]. This was extended in [26] where applications to dynamic boundary conditions and equations of Sobolev-type were treated as examples. An example to illustrate that the empathy-approach gives better results for Sobolev equations can be found in [27].
The idea of using B-evolution theory to study perturbations of evolution problems came from Alna van der Merwe [28] who formulated it in a Hilbert space context. Extension to empathy theory in Banach spaces is without difficulty.
Integrated semigroups were introduced by Arendt in [29] although it was anticipated in earlier work of Favini [30]. Adaptation to double families can be found in [31]. The system of equations in Section 5 is closer to the physics of the problem than the wave equation would be, and the dynamic boundary condition is physically and mathematically "realistic".
The Kisyiiski-result has been proved to be equivalent to the classical Hille-Yosida theorem (Chojnacki, [32]). A complete account of the discussion of Section 6 can be found in [33].
Very recently an algebraic-analytic approach to causal relations was pioneered in [34]. In this work a generalized convolution-type algebra, which extends the notion of convolution algebra in abstract harmonic analysis, is developed. The basic idea is the consider linear operators from so-called test spaces of uniformly bounded continuous functions to a Banach space as homomorphisms and define a convolution product of such homomorphisms. The convolution product, in turn, induces a dualism map back to the test space which can be implemented to describe, in a different way, a number of known causal relations such as semigroups, integrated semigroups, empathy and integrated empathy. It also provides a new understanding of linear operators associated with probability measures and the semigroups associated with stochastic processes that satisfy the Chapman-Kolmogorov relation.
In probability theory the occurrence of implicit Fokker-Planck equations is unknown. It was recently introduced in [35] where complete details of the discussion in Section 7 can be found.
KB to change the name to avoid the conflict of notation) was introduced by Belleni-Morante, [14,15]. The generation theorems can be found in [16,17,19,20]. The latter work also contains a comparison of K-bounded semigroups and C-semigroups. Definition 2 is based
K
K
K
the lumpability theory, [23], where one seeks an operator that can aggregate the states of the system, decreasing its complexity without changing salient aspects of its dynamics — the idea possibly being the closest to what originally Belleni-Morante had in mind.
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Received July 2, 2018
УДК 517.9+519.21+517.958 Б01: 10.14529/ттр180307
ПРИЧИННО-СЛЕДСТВЕННЫЕ ОТНОШЕНИЯ В НЕЯВНЫХ ЭВОЛЮЦИОННЫХ УРАВНЕНИЯХ
Н. Сауэр1, Я. Банасяк1, Ва-Сак Ли1
Университет Претории, г. Претория, Южно-Африканская Республика
Данная статья является кратким изложением подходов в динамических системах, которые составляют основу для линейных неявных эволюционных уравнений, ведущих к интересным приложениям. Примеры в бесконечномерных диссипативных системах и стохастических процессах иллюстрируют фундаментальные понятия, лежащие в основе использования двойных семейств эволюционных уравнений, связанных отношением эмпатии. Эквивалентная формулировка Кисиньского теоремы Хилле - Иосиды подчеркивает существенные различия между теорией полугрупп и теорией эмпатии. В обзоре представлено понятие ^-ограниченных полугрупп, являющихся более естественным подходом к неявным уравнением с одной стороны и отношением эмпатии с другой.
Ключевые слова: неявные уравнения; теория эмпатии; полугруппы.
Нико Сауэр, Центр содействия стипендии, Университет Претории (г. Претория, Южно-Африканская Республика), [email protected].
Яцек Банасяк, доктор физико-математических наук, профессор, кафедра математики и прикладной математики, Университет Претории (г. Претория, ЮжноАфриканская Республика), [email protected].
Ва-Сак Ли, кафедра математики и прикладной математики, Университет Претории (г. Претория, Южно-Африканская Республика), [email protected].
Поступила в редакцию 2 июля 2018 г.
