Convergence of difference approximations of quasilinear evolution equations

Nonlinear Analysis. Theory, Methods & Applications, Printed in Great Britain.

Vol.

10. No. 5, pp. 425445.

1986. 0

0X2-546xj86 13.00 + .oO 1986 Pergamon Press Ltd.

CONVERGENCE OF DIFFERENCE APPROXIMATIONS QUASILINEAR EVOLUTION EQUATIONS MICHAEL Mathematics

OF

G. CRANDALL

Research Center, University of Wisconsin-Madison,

Madison, Wisconsin, U.S.A.

and PANAGIO~S Division of Applied Mathematics, (Received Key words and phrases:

E. SOUGANIDIS

Brown University, Providence,

RI 02912, U.S.A.

21 April 1984; received for publication 5 June 1985)

Convergence

difference approximations,

quasilinear evolution equations.

INTRODUCTION WE ARE

interested

in the quasilinear

initial-value problem du -;i; + A(u)u = 0,

(1)

in which A(u) is a linear operator in a Banach space X for each u belonging to a subset W of X. Kato has studied (1) in [8] and [9]. He obtained the existence of a classical solution under assumptions detailed in Section 1 and showed the relevance of these assumptions by applying his theory to a wide variety of problems from mathematical physics. The main goal of this paper is to show that, under these assumptions, the existence theory for (1) can be obtained very directly by showing that the simple difference approximation of (1) given by

UAW- UAQ- A)+A(uA(t-A))u,(t)=O u*(t)=

A Q, for

forO
T,

tG0,

is solvable for uA(t), 0 < t G T (for appropriate $

A and T), that

Uk(C)= u(c)

(3)

exists uniformly on 0 d t =z T and the u so obtained satisfies (1) in the classical sense. Results in this general direction were obtained in [5]. See also [7]. The current work sharpens the results of [5] as applied to (1) in several ways: By restricting attention to (l), the presentation is clearer. We give a simpler proof of the convergence (3) and the proof of the existence of uA solving (2) is given under different assumptions than in [5]. Finally, and this is the point we emphasize, the convergence in (3) is shown to be better than in [5]. This is of numerical interest and the proof allows the current line of attack to obtain the sharpness of some of Kato’s results that was not previously matched by this method. Sponsored by the United States Army under Contract No. DAAG29-80-C-0041 National Science Foundation under grants Nos MC.%8002946 and DMS-8401725. 42s

and partially supported

by the

426

M. G. CRANDALLand P. E. SOUGANIDIS

Kato’s approach to (1) and its generalizations problems of the form g

involves obtaining sharp results for linear

+ B(r)u = 0

(4) u(O) = Q, and then using these with a contraction mapping argument to solve (1). (For a current account of the state of Kato’s theory and more references to other approaches, we refer the reader to [lo] and its bibliography.) Our approach to solving (1) does not require a preliminary linear theory-not even the Hille-Yosida theorem. Indeed, the solvability of (2) under hypotheses of Kato’s type is proved in a straightforward fashion and the convergence (3) follows from standard elementary estimates of “nonlinear semigroup theory”. We will rely on the form given these standard estimates in [3], but other approaches work as well (e.g. [ll, 131). This direct attack on (1) is carried out in Section 3. However, there is ample reason to study (4) by our methods in any case, and this is done in Section 2. It is also a simpler matter to show the optimal convergence of the Us if one has appropriate results for (4) in hand, and the arguments in the case of (4) exhibit clearly several main points which can then be briefly treated in the case of (1). Hence we have organized the presentation by discussing (4) before (l), as is the common practice. The interested reader can take up Section 3 before Section 2, and if he does so he will quickly obtain an existence result for (1) which asserts a little less than both optimal regularity of u and optimal convergence in (3). To obtain these sharper results we have relied on Section 2. The main results concerning (4) are given in Section 2 and state that, under hypotheses of Kato’s type, (4) has a unique solution which may be computed as the limit of solutions of simple difference approximations to (4), and these approximate solutions converge in as strong a sense as is possible. Section 1 collects some preliminaries, notations and precise formulations of the results. Of course, there are many variants and generalizations possible, and we comment on some of these following the proof of theorem 2 in Section 3. In the final Section 4 we briefly sketch how one would prove (known) results on continuous dependence in this setting. 1. PRELIMINARIES

AND STATEMENTS

OF RESULTS

Let Z be a Banach space. We use I] Ilz to denote the norm of Z, as well as the norm of elements of B(Z) (the bounded linear selfmaps of Z). If /i is a real number, we denote by N(Z, /3) the set of densely defined linear operators C in Z such that if A > 0 and A/3< 1, then (I + AC) is one to one with a bounded inverse defined everywhere on Z and ]](I + AC)_‘IJz G (1 - A@)-1. Here and below we use “I” to denote various identity operators depending on the context. The Hille-Yosida theorem-which we will not need in this work-states that C E N(Z, /3) exactly when - C is the infinitesimal generator of a strongly continuous semigroup e-lC, 0 G t, on Z satisfying ]le-fcjjz < e@ for 0 6 t. More generally, if C is a (possibly) nonlinear operator C from its domain D(C) c Z into Z with the property that Z + AC has a well defined inverse (I + AC)-’ on the range of Z + AC with (1 - A@-’ as a Lipschitz constant provided that k > 0 and A0 < 1, then we say that C + 8Z is accretive. We recall a simple lemma about accretive operators that we will have occasion to use. A proof can be found in [3] or [ll].

Convergence

of difference approximations

427

of quasilinear evolution equations

LEMMA 1. Let 8 E R’, C be an operator in a Banach space 2 and C + 61 be accretive. 6 > 0 and 78, 68 < 1, and z, f, w, G E D(C), f, g E 2 satisfy ,. Y+C*=f,

If y,

w-6

-+cw=g 6

then

Throughout call (X):

this paper, X and Y are Banach spaces which have the following properties

X and Y are reflexive and Y is continuously The operator

norm of a bounded

we

and densely imbedded in X.

linear mapping C: Y+ X will be denoted

(X)

by ]/&x.

If

T > 0, the set of continuously differentiable mappings f: [0, T] + X will be written C’[O, T: x] and C[O, T: Y] denotes the continuous maps into Y, etc.

In most of this paper X and Y will be related via a linear isometric isomorphism We denote this condition by (S):

S: Y+ X.

S : Y + X is a linear isometric isomorphism.

(S)

We next formulate our results in the case of the equation U’ + A(u)u = 0. Concerning operators A(u) we assume: there is a /3 5 0, an open subset W of Y and a mapping A : W-, The next assumption

restricts the domain D(A(w))

N(X, p).

the (Al)

of A(w) and the joint continuity

of

“A(u)u”.

For every w E W, Y L D(A(w)).

Moreover,

there are constants PA, yA such that

for u, ti E Wand u E Y [/(A(u) - A(4)& The next assumption

(A2) c PAIIu - Nell+

and

lBA(u>ull~ s YAIIuII~.

is more subtle:

there is a mapping P : W + B(X) and a constant yp such that (i) SA(w) = A(w)!3 + P(w)S

for

w E W,

(A3)

and (ii) IIP(w)llx C yp

for

w E W.

The assumptions (Al)-(A3) will suffice to guarantee the solvability and convergence of the scheme (2) to the classicai solution of (1). However, we will obtain sharper convergence results under the further restriction: there is a ,up such that I/P(u) - P(Li)ljx s ppJIu- ~21~ for Kato [8] has shown the relevance of these assumptions examples which enjoy these properties. We will prove:

U, 6 E W.

by exhibiting

644)

many important

428

M. G. CRANDALL and P. E. SOLJGANIDIS

1. Let (X), (S), (Al), (A2) and (A3) hold and cp E W. Then there are T, A,, > 0 such that there is a unique finite sequence Xi, i = 0,. . . , N, in W which satisfies

THEOREM

Xi -Xi-l

+ A(xi-l)Xi

A

i= 1,. . .,N

= 0,

provided that 0 < II =Zho and T d NA c T + A. Moreover, ~~(0) = q

and

u*(t) =xi

for

if u,(f) is defined by (1

(i-

l)A
and

\’

i = 1,. . , N

3

then (1.3) exists in X uniformly on [0, T] and the function u so defined is continuously X, continuous into Y, satisfies u([O, T]) s W and u’(f) + A(u(f))u(t) If (A4) also holds, then the convergence

=0

for

differentiable

0 G t < T.

into (1.4)

in (1.3) holds in Y uniformly on [0, T].

Remarks. The description (1.2) of Us coincides with the scheme (2) (which produces piecewise constant functions). The assumptions (Al)-(A4) are an amalgam of conditions used by Kato in [8] and [9]. (A4) was used by Kato to establish strong results concerning the dependence of the solution of (1) on A and q, and its role in our work is related to this. In [8] Kato imposed an extra condition which was also used by us in [5] to obtain the existence of Us. This was dropped in [9] and is now dropped here. (However, one can relax (A3) if this extra condition is imposed-see [2, Section 41 for a simple account.) This work obtains the existence of u E C’[O, T:X] II C[O, T: Y] solving u’ + A(u)u = 0, u(O) = Q,via the scheme (1.1). This sharpens the result of [5] which, under somewhat different assumptians, produced only a Lipschitz continuous function. If one takes the existence as given via Kato, then our main result is the fact that the solutions of (1.1) converge so nicely to Kato’s solution. As was mentioned in the introduction, we will first study the associated linear problem u’ + B(t)u = 0. Th e assumptions on B(t) parallel (Al)-(A3) above. T>Oandthereisap

5 0, such that B(r) E N(X, /3) for 0 G r 6 T.

Y C D( B(t)) for 0 C t G T and the mapping [0, T] 3 t--, B(t)\,

(the

(B1) (B2)

restriction of B(t) to Y) is continuous into B(Y, X). There is a strongly measurable mapping D : [0, T] + B(X) and a constant yD such that

(B3)

SB(r) = B(t)S + D(t).S Before formulating in the sequel.

and

IID(t)llx s y.

for 0 < t s T.

the result in this case, we recall a standard lemma which is often used

Convergence

of difference approximations

of quasilinear evolution equations

429

LEMMA 2. Let (S) hold, C E N(X, /3), Y C D(C), P E B(X), and SC = CS + PS. Set 8 = P + IIJ’llx. Th en f or every y E X and A > 0 such that Af3< 1, the problems x+ACx=y

(1.5)

and f + A(Ci + P2) = y

(l-6)

have unique solutions x and 2 in x. Moreover llxllx s (1 - A@)-’ IIYIIX and llfllx c (1 - ~Wllvll~

(1.7)

andifyEY,thenxEYand II4

(1.8)

c (1 - w-‘IIYllr.

Proof The unique solvability of (1.5) and the estimate (1.9)

llxllx s (1 - w-‘IIYIIX

are due to the definition of N(X, p). We have weakened (1.9) to the first estimate of (1.7) for later convenience in writing. The assertions concerning (1.6) are standard perturbation remarks, and can be deduced easily and directly from the unique solvability of (1.5) with the estimate (1.9). (We leave it as an exercise for the reader who may not be familiar with the perturbation results.) If y E Y in (1 S), write p = Sy, apply S to (1.5) and use the assumptions to arrive at the equivalent problem i + A(C_t?+ Pi) = j for R = Sx. The auxiliary assertions in the case y E Y then follow at once from the case discussed and the assumption (S). We will abbreviate the information contained in lemma 2 when it applies by writing ]I(z + K)-lI/z

s (1 - A8)-’

with appropriate Let

for

2 = X

or

Y

and

ll(Z + iz(C + P))-‘llx s (1 - A@-’

choices of C and P. P = (0 = ro s tl

s . . .StN=T)

be a partition of [0, TJ. The mesh size m(P) of P is the largest step ti - ti-1, i = 1, . . . , N. If (Bl)-(B3) hold, 8 = p + yo, m(P)0 -C 1 and ~1E X, then lemma 2 guarantees that the scheme xi

-

xi-1+ B(ti)Xi = 0,

ti - ti-1

i=l

7.. ., N,

x0 = (P,

(1.10)

is uniquely solvable. Indeed, the solution is given by iterating Xi =

(I + (ti - tj_,)B(ti))-‘Xi-,

to find

where the product (and all others in this paper) is “time-ordered”. partition P as above, with a sufficiently fine mesh, we set

More generally,

given a

M.G. CRANDALLandP.E. SOUGANIDIS

430

UP(t, S) = fi

j=m

for with the understanding THEOREM

t,_,

(I +

=ss
(fj

-

(1.11) and

that &(t, t) = I for 0 G f d

2. Let (X), (S) and (Bl)-(B3)

lim

tj-1)B(lj))-'

t,_l


T.

hold and x E X. Then the limit (1.12)

U,(t, s)x = U(t, s)x

m(P)-+0

exists uniformly in X on A = (0 G s G t s T) and defines a strongly continuous mapping U(t, s) from A to B(X) with the property that if q.~E Y and u(t) = U(t, s)q on s G t G T, then u E C’[s, T:x] IIC[s, T:Y],u(s) = tpand u’(r) + B(f)u(t) = 0 for s G r < T.If, moreover, D(t) in (B3) is strongly continuous into B(X) and x E Y, then the limit (1.12) is uniform in Y. The proof of theorem corollary of theorem 2.

2 is given in Section 2. Here we will be interested

in the following

COROLLARY 1. Let P(n)= (0 = t; < ty < . . . < f;;cn, = 7’j be a partition of [0, q for n = 1, 2 . . . Letx;,fl EXfori = 1,. . . , N(n) andfE L’[O, T :x](the strongly integrable functions

f;dm [0, Tj to X). Assume that lim m(P(n)) n-5

= 0

and

lim x

llfi -f(t)11 dt = 0. t-1

Let xl - xf-‘-1 ty - ty-,

+B(tl)x;=fl

fori=l....,N(n),

x0 = q,

(1.13)

and u”(t) = xl for ty_ 1 < t s tl. Then f

(1.14)

lim u”(t) = U(t, O)q + U(t, s)f(s) d.r. n-t= I0 in X uniformly on [0, T]. Proof of corollary

1. If (1.13) is solved explicitly, one finds I” u”(t) = U,(,,(t;,O)cp + ’ U,,,,,(t7,s)f”(~)ds fort:_, I0


(1.15)

where p(s) = fi on (tj’_i, tl]. It is an elementary matter to use the convergence asserted for Up in theorem 2 and the assumed convergence off” to f in L’ to pass to the limit in (1.15) to find (1.14), and we leave it to the reader to supply details as desired. 2.THE LINEARCASE

We begin the proof of theorem 2. The proof is broken into four steps. Step 1 establishes the convergence of Up to a limit U. Step 2 establishes properties of U. Step 3 proves that

Convergence u(t) = U(t, O)q

of difference approximations

of quasilinear evolution equations

431

is continuous into Y for Q, E Y and the final step 4 proves that the convergence

holds in Y. Step 1. The convergence

of Us, Let us begin by remarking that this step involves only routine arguments and could be deduced from various references, but we give it here for completeness and later convenience. We assume that (Bl)-(B3) are satisfied and let e=p+

yo.

When A > 0 and A8 < 1, lemma 2 and the assumptions

(2.1) imply that the operator

Ji(f) = (I + M(t)) -1

(2.2)

satisfies llJA(t)((z S (1 - M)-’

for 2 E {X, r).

(2.3)

Hereafter we will always assume that whenever we use an operator JA(t) then A is positive and satisfies A8 < l/2, in which case the elementary inequality (1 - At?-’ 6 ezAeholds and (2.3) implies ~/JA(f) llz S e22e

In particular,

with this implicit restriction 11 ~p(t, s)/IZ = 11fi j=m

for 2 E {X, Y}.

(2.4)

on m(P), it follows that

J,,_,,_, (t,)li

s fi e2(‘i-r~-1)es e2(r-s+m(P))e Z

j=m

(2.5)

where the notation is that of (1.11) and 2 is either X or Y. We will also assume the mesh of every partition we deal with is at most 1. Let P={O=to
Fix s E [0, T) and choose iO,ja according to SjO-1 s S < Sio and tjo-l ss < tjo.

(2.6)

Next choose Q; E Y and put Y; = Up(rl,s)Q),xj Then the proof will proceed

= Up(tj,s)q>

Yi =~isi-l

and6j

=rj -tj-i.

(2.7)

by estimating the numbers

ai,j = IlYi - xjllX = ll”BCsi, 4V - uP(fj2 s)q?llX

(2-g)

for i0 s i and j0 s j. Indeed, the Ui,,satisfy certain inequalities which allow us to estimate them in a standard way. First, observe that by (2.5) IIxillV3IlYjIlY d KIIIVll~

(2.9)

where we introduce the practice of denoting by Kj a constant which may be estimated in terms of the “data”

M. G. CRANDALLand P. E. SOUGANIDIS

432

T,P,YD, andy,,

(2.10)

YE = ,~~xrIP(~)llYX7

(2.11)

which includes the constant

which is well defined by (B2). For example, in this case we may take K, = e2(r+lje. We begin by estimating ai,jo for i,, 6 i. To this end, first observe that for y E Y llJn(OY -YllX=llJA(o(Y

-(Y

+w)Y))llX”IIJ~(ollXw(oYllX~~2~llYll,.

(2.12)

Now, by definition,

so, using the triangle inequality,

(2.5) and (2.12) we first find

and then ai,jo c K3(6j,

+ k2, Y/c)ll(PllU = K3(6j,

+ $i - sio-l)ll~lIY*

(2.13)

Similarly, if jO d j (2.14)

uio,j s K3(Yio + ‘j - fj~-l)llVllV*

Next observe that, by definition, Yi + YiBtsi)Yi

and, writing it in a complicated

= Yi-1

way,

Xi + 8jB(Si)Xj

= Xi-1 + 6j(B(Si)

- B(fj))Xj.

Since B(sJ + 81 is accretive, the above relations and lemma 1 imply that

(

’ - ‘$)

yi

+ Yi + 6j

Moreover,

using (2.9),

I

I

IlYi -xjllX

s&llYi-l

IlYi - Xj_ 1 [IX + Lyl;“~j

L

I

II(Wsi)

-xjIIX

- B(fj))xjllF

Convergence

of difference approximations

433

of quasilinear evolution equations

so we have

(

1 _

0%) +

6

(Ji,j

$

AUi-l,j

Yi

I

I

+

K411VllYllB(si)

I

SW(E,

q)

-

J

+

E

J

I

B(rj)ll*X*

The results of [3] imply that for E > 0 we can guarantee Ui,j

&ui,j-l

6j

+

that

Si_] d gGSi,fj_l

for

(2.15)

c q dtj

and i,, s i, i. s i, as soon as m(P) and m(P) are small, where w is the solution of the simple

boundary-value

problem w5 + wD - 6~ =

KJldlyIIWf) - Nq)ll~~ fors s 5, rl s T

and ~(5, rl) = &((5 given by integration

- s) + (t7 -

~>)lbll~ if f = s or7 = s,

along characteristics. While we could write the formula for w, it is enough and w(g, 5) = 0 for s d 5 s T. In particular

to know that-w is continuous

as soon as m(P) and m(P) are sufficiently small. We conclude that $m * o ~I4

(2.16)

S)V = U(t, s)Q)

exists in X uniformly on A. Since I&(& s) is bounded in B(X) and Y is a dense in X, the limit in (2.16) exists uniformly if q E X as well, and U : A+ B(X). Step 2. Properties of U

We now establish simple properties of U. If 0 c r < s < t s T, we may choose any partition of [0, T] with each of r, s and t as partition points and see that UP(t, r) = U&, s)U&,

r)

and, in the limit, U(t, r) = U(t, s)U(s, r).

(2.17)

We next establish continuity of U in (t, s). Let P be a partition and I = fj be a point of P. As in the proof of (2.13) one sees that IIu& so,

O)g, -

dlx = 11kfi.J,,Op,

- V/i s

KS(61

+

. * *

X

+dj>ll(?4Y= K5rl1911Y

in the limit,

llm w - dlx s ~s4ldlY.

(2.18)

434

The relation leads to IIVt+h,4g,

M. G. CRANDALL

and P. E. SOUGANIDIS

U(t +.h, s) = U(t + h, t)U(t, s) for 0 G s 6 t s t + h d T and the above estimate - u(t,s)~,ll,=IIv(t+h,r)U(t,s)q,

-

~(~,~)~llx~~shll~(~~~)g7ll~~~shll~~llu

since the restriction of U to Y is bounded in B(Y) by (2.5). In a similar way we see that U(t, 3)~) is Lipschitz continuous into X as a function of s for cp E Y. Since Y is dense in X, we obtain that U(t, s)x is continuous in (t, s) into X for arbitrary x E X. Let Q, E Y and consider u(t) = U(t, 0)~. We want to argue that u([O, T]) C Y and u is weakly continuous as a Y-valued function. But this is obvious, since u is the uniform (in X) limit of the functions Ur(t, O)q which remain bounded in the reflexive space Y. It is also clear that any function which is bounded in Y and continuous into X is weakly continuous into Y, and Ur(t, 0)~ converges weakly to u(t) in Y as m(P) * 0. It now follows from (B2) that the function B(t)u(t) is weakly continuous into X and hence strongly integrable. If the points of P are tj, the relations xj = xi-1 -

(fj -

tj_l)B(fj)Xj

satisfied by Xj = Up(rj, O)q imply, upon summing,

where gP(s) = tk on t&i < t s fk. Choosing partitions P which have t = fj as a partition point, one easily verifies-using (B2) and the above remarks-that the right-hand side of the above relation converges weakly (in X) to the right-hand side of U(t, O)q = Q, -

f mw,

I0

(2.19)

WP d.T

and it follows that (2.19) holds and u(t) satisfies the equation u’(t) + B(t)u(t) = 0 almost everywhere. The weak continuity of B(t)u(t) then implies that the equation holds weakly everywhere. Once we know that u(t) is continuous into Y so that B(t)u(r) is continuous into X, it will follow that u E C’[O, T: x] and the equation holds classically. Step 3. Continuity into Y We wish to establish the strong continuity of u(t) = U(t, 0)q into Y. It is equivalent, by (S), to show that G(r) is continuous into X. The above remarks show that Su(t) is weakly continuous into X and thus it is strongly measurable. By (B3) we then have that D(t)&(t) is bounded and strongly measurable and therefore strongly integrable (in X), and then so is s+ U(t, s)D(s)Su(s). The proof will proceed by showing that * Su(t) = U(t, 0)&p -

I0

U(t, s)D(s)Su(s)

ds

from which it is obvious that Su is continuous into X. Since D(t)Su(t) is strongly integrable in X and u(t) is strongly integrable sequence of partitions P(n) = (0 = t; < ty

c . . . < f”Ncn,= z-j

(2.20)

in Y, there is a

Convergence

of difference approximations

435

of quasilinear evolution equations

such that m(P(n)) --, 0 and Jw)

’ I(D(t)Su(t)

- D(t~)Su(~)ljx

p

dt = lim x tl*x

j=l

’

I/u(t) - u(t))Iju

dt = 0.

(2.21)

I 1"

j-l

/-I

The scheme x7 - xy-‘-1

ty - ty-1

+ B(t;)xy

= 0,

i = 1,2, . . . ) iv(n),

(2.22)

has the solution u”(t) = &(,,(t, O)q where u”(t) = xl on (t;_l, t;]. Consider the auxiliary scheme

(2.23) z”, =

sg,

which defines the values of the piecewise constant function z”(t). By corollary lim z”(t) = u(t, O)S9, n-+x

’ U(t,

s)D(s)Su(s)

1 and (2.21)

ds

I0

holds in X uniformly in t. Define z(t) = lim z”(t). Next we show that z(t) = su(t). To this end, n-)x set u”(t) = S-‘z”(t). The values z$ of on(t) satisfy-using (2.23) and (B3)-

(2.24)

Since z” converges as above, u” converges in Y uniformly in f to a continuous function u(t). We are done upon showing that u = u. Using (2.24), (2.22), (Bl) and lemma 2 we find 11~~- x;llr

S

e2(tf-tf-l)e())ui_l - x~-~JI~

+ (t? - t~_l)~~s-‘(~(t~)Su(t~)- D(t~)su;)ll,) S

e2(r:-r~-l)e(IIu~_1 -

x~_~I)~+ (tl - t~_,)yDllu(t~) - ul\)y).

Iterating this yields liu”(t;) - u”(t;)llY G e2re YLJ($l

1’ Il@l) - u”(& I” k-l

d+

(2.25)

Since u” converges in Y uniformly and u” converges weakly in Y and (2.21) holds, we may take limits in (2.25) upon appealing to the lower semicontinuity of the Y-norm with respect

M. G. CRANDALL

436

and P. E. SOUGANIDIS

to the weak topology to conclude that

II44 - WIIY s ezTeyD II44 - 4.9llud.7 Since IIu(f) - u(t)llr is integrable,

forOsts

T.

(2.26)

this implies that u(t) = u(t) and we are done.

Step 4. Convergence in Y We now impose the condition that D(t) is strongly continuous. Since we established above that u(t) = U(t, 0)~ E C[O, T: Y], the relation (2.21) holds for an arbitrary sequence of partitions P(n) satisfying m(P(n)) + 0. By the analysis of step 3 we conclude that if P = (0 = to < . . . < tN = ZJ and zp is the piecewise constant function on P whose values zi are given by zi

?i

1 :i_l + B(ti)Zi

= 0,

+ D(ti)SU(ti)

rl

(2.27)

then zp+ Su uniformly in X as m(P) + 0. To show that Up(t, 0)cp converges in Y we need to show that wp(f) = SU,(t, 0)~ converges in X. The values Wi of wp are given by

z,“r-’+B(ti)Wi

wi

4

11

wo =

+ D(ti)Wi =0

(2.28)

scp.

Rewriting the relations (2.27) as zi fi

I i::l 11

+ B(tj)Zi

and using (2.28) and the accreditivity l/w;

Iteration

+ D(ti)Zi

- SU(ti))

of B(ti) + D(tJ + 131,we find

) (IIwi-1-Li-l

-zillX~e2~‘i-~i-l

= D(ti)(Zi

IX +

Cti

-ti-I)llD(fi>(zi-Su(ti>)llX)*

of this inequality yields

llwi- IillX s e2eTYD&II*Ok- hdlh

- wx).

(2.29)

Since zp-, Su in X as m(P) + 0 and Su E C[O, T:X], the right-hand side tends to zero as m(P) + 0. We conclude that wP - zp+ 0 in X uniformly as m(P) + 0 and so wp-* Su as claimed. 3. PROOF OF THEOREM

1

The proof is again broken into four pieces, and (A4) is invoked only in the fourth part. In step 1 we show that the scheme Xi

-

Xi-1

A x0 = Q1

+ A(Xi_,)Xi

= 0,

i= 1,. . .,N (3.1)

Convergence of difference approximations of quasilinear evolution equations

is solvable and obtain some appropriate uA(t) = xi

437

estimates along the way. In step 2 we show that if

on

((i - l)& iA],

UA(O)= V,

(3.2)

then the limit FE UA@)= u(t)

(3.3)

exists in X uniformly. Moreover, in this step it will be proved that the limit is a solution of the evolution problem in a strong-but not quite classical- sense. Up to this point, the results of Section 2 will not be used. In step 3 it is shown that the limit u in (3.3) lies in C[O, T: Y] and for this we will rely on the results of Section 2. In step 4 we demonstrate that the limit in (3.3) exists in the topology of Y. Step 1. Existence of ui We will now discuss the solvability of (3.1). To this end let Q, E W and d, = inf{Ilq - u(ly : u E Y\m. be the distance in Y of Q, to the boundary LEMMA 3.

Let (Al)-(A3)

of W. We have:

hold and (3.4)

e=P+Yp Let T > 0 satisfy

)gf,((1+ e2eT)IISv - 4x

+ T(~~llzll~

+ YPII~Ix>) < 4.

(3.5)

Then there is a & > 0 such that for 0 < A G AZ0 there is a unique sequence xi, i = 0,. . . , N, in W satisfying (3.1) and T s NJ s T + A. Proof. The desired relation between Xi and Xi-1 given by (3.1), can be, if A > 0 and )L8 < 1, rewritten as Xi = (I + AA(xi_ I))-‘xi- 1e (3.6) By lemma 2, xi E Y is uniquely determined by (3.6) so long as Xi-1 E W. We thus seek to estimate llxi - qllu and keep this below d, since the open Y ball centered at Q, of radius d, lies in W. Assuming Xi- 1 E W, we put wi = Sx,, wi_ i = Sx,_ i and operate on (3.1) with S to obtainvia (A3)Wi + A(A(xi_,)Wi+ P(xi_l)Wi) = Wi-1. (3.7) Choose z E Y. We have Wi -Z =

Using lemma 2 in conjunction Iiwi

-

4x

+

A(A(Xi_,)(Wi- Z)+ P(Xi_l)(Wi-Z))

Wi-1 -

Z -

A(A(xi_l)Z

+ P(Xi_l)Z).

(3.8)

with (3.8) we obtain

s (1 -

W-‘(Ilwi-1 -

Zllx + h(y,4llzllv

+ YPII~Ix))

(3.9)

Again we assume that AB < l/2 so that (1 - A@-’ 4 e2*’ everywhere below. Then we can iterate (3.9) to obtain 11 Wi - Zllx L e2iA8 (IIS~ - 41~ + i~(yMl~ + YPII~Ix)).

M. G. CRANDALLand P.E. SOUGANIDIS

438

(recall w. = 5x0 = Sq). This further implies 11~ - WI,

s (1 + e2”‘%IPg, -

4~ + Nx41/41~ + r~llzll~)).

(3.1oj

By (3.5) and considerations of continuity we can choose (Y> 0 and z E Y such that the righthand side of (3.10) is less than d, if iA s T + a. Set A0 = min(cu, 40). By what we have shown (3.5) implies the existence of an r < d, so that for 0 < ,J 6 A0 and T d NA G T + A, one can solve (3.1) and Xi E

By(Ty

q)

= {U E

Y 1IIU- qllu s I}

fori=l,.

. .,N.

(3.11)

Remark.

In contrast with [5] we have used the full force of (A3) here. This is because we do not assume any bounds on expressions like IIA(w)ylly in this case. Step 2. Convergence

of ui

For the rest of the discussion, r is fixed at the value above. LEMMA

4. If x, f E Bv(r, q) and ,uA, r3 be as in (A3) and (3.5) respectively.

w= e +

~A(bb

Let (3.12)

+ +

If x + M(x)x=

2,

% + nA(f)R = 2

(3.13)

then [Ix - qx 6 (1 - Alj+-‘llz - fllx.

(3.14)

Before we give the simple proof, let us explain why lemma 4 and standard results establish step 2. The conclusion of lemma 4 is that the mapping By(r, 9) 3 x ---, A(x)x + yx is accretive. That is, if C(x) = A(x)x for x E D(C) = B~(T, q), then C + I/JZis accretive (in X) according to lemma 3. It is known that if C + ~/JZis accretive in X for some w and for each small A > 0, Xi’s are given so that x0 E D(C) and xi -Axi-i + C(Xi) = Ei

I$llEillX+o

fori=O,l,...,N

withTcNAcT+A,

(3.15)

asA i 0,

then the uA(t) given as xi on ((i - l)A, d] converge uniformly on [0, T] in X to a Lipschitz continuous u E C[O, T:X]. This is a basic result of [2] when Ei = 0, i = 1, . . . , N. In our case we have, with the xifs of lemma 3, and C(x) = A(x)x,

xi-Axi-+ C(Xi) ~0, by (A%

= (A(Xi)

- A(Xi_ 1))Xi

Convergence

of difference approximations

of quasilinear evolution equations

439

By Xi - X;_ 1 = AA(x, _ 1)~; and (A3) Ilx, -

xi-lIIX

s

AYAIlxlllY c AYi4(IIQ?IIY + r>

and thus llqllx G CA. It follows that for some constant Ci. asA .J 0.

i~/,E,,,~CI(N~)h~CI(TfA)I--*O 1

The convergence of Us when I(.sij/+ 0 uniformly in i as A J 0 is a simple extension of [2]. More generally, the results of [l] or Kobayashi [ll] or Takahashi (131, or Crandall and Evans [3] can (as we have already done in Section 1) be applied. Indeed, from these works one has an error estimate of the form IlG)

- @)]ix c C(C,(T+

A)A + A”*]IAW&)

where C depends on T and q. Proof of lemma 4. Forming

the difference

of the relations (3.13) and rearranging

suitably

yields x - 2 + clA(x)(x - a) = z - f + L(A(i)

- A(x))f.

Since x, f E Br(r, Q?) and A(x) E N(X, /I) this implies

M>llx- 4x c lb - fllx + ~l&W - A(x))& c lb - 41x+ AP,4 IIX- fllxllfllY c lb - 4x + b,4 IIX- 4lxMlr + 4

(1 -

and rearranging this proves lemma 4. By the above, the convergence (3.3) takes place in X uniformly in t and the limit u is Lipschitz continuous. Since the values of uI are bounded in Y (they lie in &(r, q)). and Y is reflexive, the limit u therefore takes its values in Y (in fact in BY(rr q)). Since u is continuous into X it is weakly continuous into Y. Similarly, the convergence Us to u takes place weakly in Y. Iterating the relations (3.1) we find UA(iA) = q -

” A(u& I0

- A))u,(s)

d.s.

It is a simple matter, using the above remarks, (Al) and (A2), to see that as ih + t and h --, 0 (e.g. let A = r/i and i-+ x) the right-hand side above tends to the right-hand side of u(t) = q -

’A(u(s))u(s)

I0

ds

weakly in X, thus establishing the validity of the integral relation. Observe that A(u(t))u(t) is weakly continuous into X-and thus integrable-by the assumptions and the properties of U. Thus U’ + A(u)u = 0 holds strongly a.e. and weakly everywhere. In the next step we will prove that u is continuous into Y. This will make A(u(t))u(t) continuous into Xand so u E C’[O, T:X’j. Step 3. Continuity into Y

Set . B(f) = +40)

forOGtGT.

(3.16)

M. G. CRANDALL

440

and P. E.

SOUGANIDIS

where A and u are as above. It follows from (Al)-(A3) that B(t) satisfies (Bl)-(B3). We may take p of (Al) for /3 of (Bl), (A2) and the continuity of u into X and u([O, T]) E W imply (B2) while D(t) = P(u(t)) works in (B3). We briefly recall Kato’s reasoning concerning this latter point. From the assumptions on A and the properties of u it follows that = S-‘A(u(t))y

S-‘P(u(t))y

- A(u(t))S-‘y

fory E Y,

and the right-hand side of this expression is continuous into X. Thus S-‘P(U(~))~ is continuous (in t) into X and bounded into Y, hence it is weakly continuous into Y and then P(u(t))y is weakly continuous (and therefore strongly measurable) into X. Since Y is dense in X, P(u(t)) is strongly measurable. We want to show next that the schemes (3.1) and yi

are equivalent.

-Ayi-l+A(u(iA))yi = 0,

i=l,...,N, (3.17)

More generally, let us argue that if (t, s) E A = {(t, s) : 0 c s G t 6 2-J and v~(r, 4 = kfim (I+ AA(+ u,(r, 4 = $m (I+ A+@@))

-’

when (m - 1)L G s < mA, and (n - 1)A < I < nL, then IIV*(r, s)llz, II&(6

4llz s

Cl

forZE{X,Y)

(3.18)

forxEX

(3.19)

and lim V,(t, s)x = U(f, s)x

A+0

in X uniformly on A where U(t, s)x = ;“, U,(t, s)x

(3.20)

exists by Section 2 and the fact that B(t) satisfies (Bl)-(B3). We now adopt the convention that the C’s are constants estimable in terms of the data. The first estimate of (3.18) is proved just like the second, and this is part of the proof of theorem 2. It suffices to consider the case s = 0, as the general case is entirely similar. Assume that (3.1) and (3.17) hold, but allow x0 = y. = x to be an arbitrary element of Y (and not necessarily q). Writing (3.17) as yi

-~yi-l+ A(Xi_l)yi

= (A(x~_,) - A(~(t;)))y,

and using (3.1) and the accretivity of A(Xi_ J + /3Zyields

IIXi-Y~IIx s (1 - A/J)-‘(IIXi-I- Yi-lIIx + AII(A(xi-1)-A(u(t;)))YiIIx) s e2*s(llxi-l -Yi-1IlX + APAIIXi-l- u(ti)IIXllYill~) se2’B(llxi-1 -Yi-lllx

+ ~Gll4lvllxt-1 - ~Wllx).

Convergence

Iteration

of difference approximations

441

of quasilinear evolution equations

yields

and then, since ~4~convergcj uniformly to u, we conclude that Ui(t, 0)x - VA(t, 0)x-* 0 in X uniformly as desired. It is now established that u(t) = U(t, Ojq is the solution of U’ + B(t)u = 0 produced in Section 2, and so u E C[O, T: Y] II C’[O, T:x] and u’(t) + A(u(t))u(t) = 0 by the results of Section 2. Step 4. Convergence in Y

We now assume (A4). We will be considering four families of functions: The functions K~ whose values x, are given by (3.1), the functions We= SUMwhose values w, satisfy w; -

wj-1

+ A(X;_ 1)W; + P(x;_ I)w, = 0,

A

wo =

i=l

,. . ., N,

(3.21)

sq,

the functions z1 whose values zi satisfy ” -Lzi-l

+ A(xi_,)zi

+ P(u(iA))Su(iA)

= 0,

i=l

, . . . , N,

(3.22) and the functions Vi -Lvi-l

V~= S-‘.Z~ whose values vi satisfy

+ A(x;_~)v,

+

S-‘P(u(iL))Su(ik)

- S-‘P(x;_l)Sv;

= 0

fori=l,...,N, (3.23)

vg = cp.

Concerning these we claim several things. First, it is obvious that Us converges in Y exactly when wA:converges in X. Since we cannot show the convergence of the We directly, we begin by observing that .zi. satisfies lim z*(t) = U(t, OjSq - J’ U(t, s)P(u(s))Su(s) AJO 0

d.7,

(3.24)

The in X where U(t,s), given by (3.20), is the evolution generated by -B(t) = -A(u(t)). relation (3.24) holds in X uniformly in f, because of arguments like that sketched in the proof of corollary 1 in Section 1 together with (3.19) and (3.22), the convergence of the function whose value on ((i - l)A, iA] is P(u(iA))Su(ih) to P(u(t))u(t) -which follows in turn from (A4) and the continuity of u into Y and of G(t) into X from step 3. Second, lim vi(t) = u(t)

(3.25)

I,*0

holds in X. The reasoning here parallels the corresponding Indeed, from (3.1) and (3.23) we deduce that (1 -A8)/1v, -xillyGIIvi_r ~J/Vj-l

arguments which led to (2.24).

-xi_l(~y+A~~P(u(iA))Su(i~)-P(xi_l)Svi~~x -xi-l

IIy

+AIl(P(U(iL))

-P(X;-l))SU(iL)+P(x;-l)S(U(iA)-

Vjjllx

442

M. G. CRANDALL

and P. E. SOUGANIDIS

and then, using (A4) and letting C denote a bound on jlu(t)llr and (IUiJ(y,we find

The rest of the proof of (3.25) is essentially the same as that of (2.26) and is left to the reader. At this point we have identified pg zi with su(t). One can then show, using (3.21) and (3.22). that (l -

Ae)llzi- willXd lIzi-* - wi7111X + AYPllzi - S”(iA)llX + APPc(IIwi-I - zi-lIIX + lIzi- - s”(iA)llX)*

Iterating this inequality and using the uniform convergence fg in Xuniformly

of zL to Su establishes

(WA- z*)(t) = 0

in c in the same way as established (2.27) in Section 2, and the proof is complete.

Remarks on generalizations. The problem % + A(u)u = f(u),

(3.26)

generalizes

(1) and is in turn generalized

by

du x + A(t, u)u = f(f, u),

(3.27)

u(0) = f.p. The methods of this paper succeed, under appropriate assumptions, in the generality of (3.27). However, these methods do not entirely subsume the t-dependence in either A or f assumed in Kato’s original works. Roughly, the conditions on the t-dependence are required to be more uniform in u than Kato needs (but are otherwise quite general). We will not discuss this point further here-see e.g. [5] and [6]. Instead, let us indicate the situation with respect to (3.26). Kato used the following two conditions on f: f maps W into a bounded subset of Y and there is a constant ,uf such that for every u, li E W we have IIf - f(li)llxs pfJIu - tillx.

(fl)

there is a constant pf such that for every u, ri E W we have Ilf(4 - f(fi)llYs pfllu- fib

(W

and

Convergence

of difference approximations

of quasilinear evolution equations

443

The following modification of Theorem 1 is true and has essentially the same proof. If (X), (S), (Al), (A2), (A3) and (fl) hold, the difference scheme in (1.1) is replaced by

Xi-*xi-1+ A(xi_,)xi

=f(Xi-1)

and (1.4) by the equation of (3.26), then the assertions preceding (1.4) remain true. If also (A4) and (f2) hold, then the convergence holds in Y uniformly on [0, T]. Remark.

Results completely approximation

analogous

UiO>- UA(f- A) A

UA(O= in place of the semi-implicit

to the above can be proved for the fully implicit

+

Atu~(4hO) = 0

fort > 0,

for t d 0.

Q,

scheme (2).

Remark. One can play a bit with the assumptions on P(U). For example, it is enough to require u+ P(U) to be continuous into the strong operator topology from the X topology on W in

order to assert the convergence point of view of applications.)

in Y. (However,

4. CONTINUITY

this does not seem a good assumption from

WITH RESPECT

TO DATA

In this section we state and prove a result of [8] concerning the continuous dependence of the solution of (1) as an element of C[O, T: Y] on the data. Nothing new is proved in the process and we include this section primarily to indicate how one might prove such results in the current setting. To formulate the result, we consider a sequence of equations %

+ A”(P)&

=f”(u”),

n=l

u”(0) = cp”,

, . . . , ~0,

(1)”

n=l,....m,

where n = m is explicitly allowed. We assume A” and f” satisfy (Al)-(A4)

and constants independent

and (f l)-(f2) with the same X, Y, S, W of n = 1, 2, . . . , 00.

(4.1)

The result is: THEOREM

3. Let (4.1) hold. Moreover, (i)

A”(w) + A”(w)

for each w E W let: strongly in B(X, Y)

(ii) P”(w) + P”(w) strongly in B(X) (iii) f”(w)+f”(w)

in Y

asn-*

asn+m; asn-,m;

”

(4.2)

m.

If@EWforn=l,..., x and q”-* cp” in Y as n --, CQ,then there is a T > 0 such that the solution of (1)” constructed in Section 3 satisfies U” E CIO, T: Y] fl C’[O, T: x] (i.e. the interval

M. G. CRANDALL and P. E.

444

SOUGANIDIS

of definition of u” includes [0, T]) for n = 1,2, . . . , CQ.Moreover, u” + urnin Y uniformly on [0, T]. Remark.

Theorem

3 shows, in particular,

that u depends continuously

on Q, in the Y norm.

Proof of theorem 3. For simplicity (and of necessity, since we did so before) we assume that f” = 0. The existence of T and u” as in the statement of the theorem is an immediate consequence of theorem 1, (3.4) and the assumption that q”+ @ in Y. We need the following lemma: LEMMA 5. Let V(f, S) correspond

to A” and cp” for n = 1,. . . , x as U corresponded

to A and

~1 in (3.20). Let x E X. Then lim U” (t, s)x = V(t. S)Xin X uniformly on A. n-05

(4.3)

We continue with the proof of theorem 3 and then we prove the lemma. In step 4 of the demonstration of theorem 1 we established that &P(t) = U”(t, O)Sq” - 1’ U”(f, s)P”(u”(s))Su”(s)

ds

forn=l,...,x.

0

If we subtract the nth and 33th equations and use the triangle inequality several times we can obtain IPV)

c e *%Pn

- uV)llr

-

vllu + ll(W7 0) - TfY w-wx

+ r llU”(t,s)P”(u”(s))Su”(s)

- U”(t, S)P”(U”(S))SU”(S)~~~ d.s

I0

t IIP”(u”(s))Su”(s)

+ e2er i

- P”(u”(s))Su”(s)ll,y

ds

0

+ e2e*(y

+

C) J’ Ilun(s) - uz(s)lly ds 0

where 8 is given by (3.9, y is the bound on IIP”(w)llx and C = y (bound on ]/.Su”(t)llx. Using lemma 5, (4.2) (ii) and @‘-* Q,x in Y we see that all the terms on the right-hand side above except the last one tend to zero as n + =. Elementary estimates complete the proof. Sketch of proof of Zemma 5. This result is followed from those in [5], but we sketch the proof here in this context for completeness. Since Y is dense in X and (Al)-(A4) hold with constants

uniform in n, U” is bounded in B(X) and it suffices to check (4.3) for x E Y. To this end, we note (with the obvious notations) that (4.4) This is evident from step 2 in the proof of theorem

1. In addition, one can easily check that

lim u!(t) = u;(t) in X uniformly for 0 G t G T. IZ-==

(4.5)

Convergence

of difference approximations

for small A > 0. Using (4.4) and x E Y, the proof of theorem

1 adapts to show that

lim V’j (t, s)x = U”(t, s)x in X uniformly in (t, S) E A and n. Al0 Finally, a straightforward

445

of quasilinear evolution equations

(4.6)

estimate shows that

IIvz(c Sk - K 07MIX s + ,z;J,

tI(A”(G(r))

Coot$s$uP (Ik(r)

- G(r)llx

- A”(u”(t)))V~(t,s)xllx)

(4.7)

and the right-hand side can be made small for fixed A > 0 by choosing IZ large. (Recall that every function subscripted by L has finitely many values, (4.5) and (4.2)(i).) But then (4.5), (4.6) and (4.7) together yield (4.3).

Acknowledgements-The

authors are grateful to R. Pego for useful discussions about this work.

REFERENCES M. G., An introduction to evolution governed by accretive operators, in Dynamical Systems-An International Symposium (Edited by L. CESARI, J. K. HALE, J. LA SALLE), pp. 131-165, Academic Press, New York (1976). CRANDALLM. G., Nonlinear semigroups and evolution governed by accretive operators, Proceedings of Symposium on Nonlinear Functional Analysis and Applications, Berkeley, 1983 (to appear). CRANDALLM. G. & EVANS L. C., On the relation of the operator a/s + a/t to evolution governed by accretive operators, Israel J. Math. 21, 261-278 (1975). CRANDALLM. G. & LIGGEI-~T. M., Generation of semi-groups of nonlinear transformations on general Banach spaces, Am. J. Math. 93. 265-298 (1971). CRANDALLM. G. & SOUGANIDISP. E., Quasinonlinear evolution equations, Mathematics Research Center TSR No. 2352, University of Wisconsin-Madison (1982). CRANDALLM. G. & SOUGANIDISP. (in preparation). HAZAN M., Nonlinear and quasilinear evolution equations: existence, uniqueness, and comparison of solutions: rate of convergence of the difference method, Zap. Nauchn., Sem. Leningrad. Otdel. mat. Inst. Steklov. 127,

1. CRANDALL

2. 3. 4. 5.

6. 7.

181-199 (1983). 8. KATO T., Quasi-linear equations of evolution, with applications to partial differential equations, Spectral Theory and Differential Equations, Lecture Notes in Mathematics 448, 25-70 Springer, Berlin (1975). 9. KATOT., Linear and quasi-linear equations of evolution of hyperbolic type, C.I. M. E. II Ciclo 1976, Hyperbolicity,

pp. 125-191. 10. KATO T.. Nonlinear

of evolution in Banach spaces, in Proceedings of Symposium on Nonlinear Berkeley 1983 (to appear). (Also PAM 202, Center for Pure and Applied Mathematics. University of California-Berkeley (1984)). 11. KOBAYASHIY., Difference approximation of Cauchy problems for quasi-dissipative operators and generation of nonlinear semigroups, J. Mach. Sot. Jauan 27. 64&665 (1975). 12. KOBAYASHIK.:Onhifference approximation of time dependent nonlinear evolution equations in Banach spaces, equations

Functional Analysis and Applications,

Mem. Sagami Inst. Technol. 17, 59-69 (1983). 13. TAKAHASHIT.. Convergence of difference approximation semigroups. J. Math. Sot. Japan 28. 96-113 (1976).

of nonlinear evolution equations and generation

of