Initial value problems for the Carleman equation

0362.546X!80/0201-0341

Nonlinear Analysis, Theor.y, Mefhods & Applications, Vol. 4, No. 2, pp 343-362 Q Pergamon Press Ltd 1980. Printed in Great Bntam

TNITTAL VALUE

PROBLEMS

FOR THE

HANS G. KAPER

CARLEMAN

SO2.00/0

EQUATION*

and GARY K. LEAF

Applied Mathematics Division, Argonne i%tional Laboratory, Argonne, IL 60439, U.S.A. (Received 26 February 1979) Key words: Nonlinear evolution equation, Carleman equation, discrete velocity model Boltzmann equation

1. INTRODUCTION THE CARLEMAPU’ equation

is a system of coupled nonlinear differential equations,

(1.1) at4, au, --_-_+u dt

2 2’

ax

for the vector-valued function u = (u,, u2) defined on the domain Q = ((x, t):x E R, t 3 0). It was proposed by Carleman in the 1930’s and appeared in print for the first time in 1957 in [l], Note II. The Carleman equation is supposed to model the spatio-temporal behavior of the velocity distribution function of a gas whose molecules move parallel to the x-axis with constant and equal speed, either in the direction of increasing x (density ui) or in the direction of decreasing x (density u2). Interaction between two molecules of the former type results in two molecules of the latter type, and vice versa; each of these interactions is equally probable. The normalization is such that the molecular speed and the mean free path (i.e. the average distance travelled by a molecule between two successive collisions) are unity. Thus, (1.1) represents a discrete velocity model Boltzmann equation. The Carleman equation has been studied by several authors using various analytical techniques. Basically, two different types of problems can be identified. The first one is a pure initial value problem: for which initial distributions U(X,0) = g(x), x E R, does the Carleman equation define a distribution u(x, t) at all later times t > 0 and, if so, is it unique and what are its properties‘? The second one is a problem of an asymptotic nature. Suppose the mean free path had not been normalized to unity, but had been left in the equation as a parameter, E.Instead of (1. l), we would have the following system of equations,

au, au, ---= dt

8X

Qu:- u;,. E

* Work performed under the auspices of the U.S. Department of Energy. 343

(1.2)

344

H. G. KAPERand G. K. LEAF

The question then arises: what is the limiting form of this system as E 1 0, and how do the initial data of the limiting equation match the initial data associated with (1.2)‘! Since the limit (e JO) corresponds to the transition from the description of a gas as a collection of individual molecules to that of a gas as a continuum, we refer to the asymptotic problem as the problem of the hydrodynamic limit associated with the Carleman equation. The initial value problem for the Carleman equation is the object under discussion in the present paper. A discussion of the problem of the hydrodynamic limit is given in a companion article [2]. The objective of both these articles is to review the existing literature on the subject, to correct and improve some results of previous authors, and to give a unified presentation of the entire subject matter. The initial value problem for the Carleman equation was first discussed by Kolodner in 1963, [3]. Using lixed point arguments and special properties of the Riccati equation, Kolodner showed that for nonnegative initial data gl, g2 E C’(R), there exist nonnegative functions ul, u2 E C’(Q), such that u = (ul, UJ is a solution of (1.1). In 1969, Temam [4] applied the method of fractional steps, cf. [IS], to an equation similar to the Carleman equation, viz., &A, z8% + >y = uf - u;,

au, au, at + _ = u; - u;, ay

(1.3)

for the vector-valued function u = (ur, u2) defined on the cylinder Qr = R x [0, T], where R is a bounded rectangular domain in R2, Sz = {(x, y) E R2:a d x d b, c d y d d], assuming zero incoming densities at the left and lower boundaries (~~(a, y, t) = u2(x, c, t) = 0). Temam showed that, for nonnegative initial data gr, g2 E H’(n) x L”(Q), there exist unique nonnegative functions u,, u2E L"(0, T ;H'(R)) n Lm(QT) such that u = (u,, u2) is a solution of (1.3). Temam’s results were reproduced by Lions in [6], Section 4.2. In 1971, Crandall [7] studied the same equation (1.3) on R$ x [0, co), again assuming zero incoming densities, to illustrate some concepts from the theory of nonlinear semigroups in Banach spaces. He showed that, with the appropriate definitions, (1.3) corresponds to an evolution equation in the positive cone of L,(R2) x L,(R2), viz. u’(t) + Au(t) = 0, and that the operator - A generates a contraction semigroup S(t). It follows that, if the initial value problem u’(r) + Au(t) = 0 for t > 0, u(O) = g, has a strong solution for g E D(A), then it is given by u(t) = S(t) g for all t 3 0. The initial value problem for the Carleman equation (1.2) was taken up by Kurtz in 1973 [S], primarily to study the problem of the hydrodynamic limit. In the course of his investigation Kurtz states that the evolution equation associated with (1.2) in the positive cone of L,(R) x L,(R) is solved by a contraction semigroup S,(t). However, his proof is somewhat sketchy and incomplete. In a survey article on abstract evolution equations published in 1976, Tartar [9] mentions the initial value problem for the Carleman equation (1.1) and states, without proof, the result that for bounded initial data gl, g2 E L”(R) with 0 < g,(x), g2(x) < M, (1.1) has a unique solution u = (u,, uJ such that u,(t), u,(t) E L”(R) and 0 < u,(x, t), u2(x,t) d M for all x E R, t 3 0. Moreover, u increases as g increases, and l(u,(t)// ,,- iR, + I(u,(t)ll,,, ,R,d ecreases as a function of r E [0, cc). Finally, in an article on Broadwell’s discrete velocity model Boltzmann equation presented in

345

Initial value problems for the Carleman equation

1975, Nishida and Mimura [lo] comment on the initial value problem for the Carleman equation. They consider solutions of (1.1) which are close to a spatially uniform steady state solution. If the latter is given by (a, a), where a is some positive constant, then the perturbation v = (u, - a, u2 - a) must satisfy the equation * at

+ ?L + 2U(Vl - Y/J = q; - II:, ax

(1.4) ayl2 _---

arl2

at

ax

2&i

- V2) = V: - 6

and the initial condition q(. , 0) = 4, where 4 = (4i, 4,) = (gi - a, g2 - a). Nishida and Mimura that for bounded continuous initial data 4 which are strictly positive and such that 41, $2 E H’(R), there exist differentiable functions yli, q2 such that v] = (vi, q2) is a solution of (1.4). Moreover, for i = 1,2, indicate

L*(R)

tend to zero as t -+ co. No proofs are given, however.

In Section 2 we discuss the abstract initial value problem associated with the Carleman equation (1.1). The classical results of Kolodner [3] can be recovered from the abstract results of Section 2 by a suitable restriction of the class of initial data. This is shown in Section 3. In Section 4 we study the decay of solutions generated by perturbations of spatially uniform steady states. Section 5 contains a summary of our results.

2. THE CARLEMAN

EQUATION

AS AN ABSTRACT

EVOLUTION

EQUATION

In this section we discuss the initial value problem for the Carleman equation (1.1) 2

au + $

= u; - u;,

au, au, ----_+u at ax for the vector-valued

function

u = (u,, UJ defined

(2.1) 2 2,

on the domain

Q = {(x, t): x E R, t > 01.

The initial conditions are UI(? 0) = g,(x), where gi and g2 are given functions.

a&, 0) = 92(x),

x E R,

(2.2)

H. G. KAPERand G. K. LEAF

346

We interpret (2.1) as an evolution equation for the vector u = (ui, uz): [0, co) + L+, where

L" is the positive cone in the real Banach space L = L,(R) x L,(R). The topology in L is the prod= IuljI,, + Iu21LI for u = (u,, UJ EL. uct topology; 1.iI. d enotes the norm in L,sojulL Let the expression t be defined by z:u = (u,, u2)

b

T(U)= (u; +

u:- u;,-u; - u;-

Id:,.

Here, u1 and :!2 are functions of x alone, and ’ denotes differentiation. The expression r will define an operator 4 in L+ which, when properly defined, is accretive and satisfies the range condition, i.e., J, = (, + AA)-’ is a contraction and R(Z + AA) I D(A) for all i > 0. Let the operator A, be defined by the expression ‘I on the domain &A,), ‘4,u

= z(u),

u E W,),

where D(A,) = (u E L’ :ul, u2 locally absolutely continuous, u’ = (u;, u;) E L] n {u~L:Iu~l~, luzlco d Ml, M being some positivenumber.

Thus, A,:D(A,) LEMMA 2.1.

Note that ifu E D(A,), then lim (ul(x), U,(X)) = 0 and (UT,uz) E L. x+fcc

+ L.

The operator A, is closed and accretive in L.

Proof: Suppose that u” = (a;, u:) -+ u = (u,, u2) in L and that AMu” converges in L. There exists a subsequence of {u”) that converges pointwise a.e. and, therefore, Ju, lli, Iu21r d M a.e. It follows that ((u;)‘, (u”,)“)-+ (uf, u’) 2 in L. Thus, the sequence {(ul)‘} converges in L,(R) to y,, say. Since x

IS

(u;)‘(t) dt -

-cc

sx

y,(t)

-cc

I

dt G I(u;)l- Y&

+ 0,

we have U,(X) = r_, y,(t) dt, u;(x) = yl(x) and, hence, (u”,)’-+ u; in L,. Similarly, (u”,)’--f u; in L,, so (u”)’ -+ u’ in L. It follows that A, is closed. To prove that A, is accretive in L, take u, o E D(A,) and I 3 0. Then,

341

Initial value problems for the Carleman equation

where CI= u1 - vl, j3 = u2 - v2. IfI=(xER:a(x)>OjandA=R\T,then Jx + J.a’ + /z(Uf - v: - u; + v& 2

(D.+ Au’ + A(u; - v; - u; + 11;))dx sr +

(-l)(cr+M+~(+v;-u;+v;))dx sA

=

$a/

+

n/u;

-

v;/)dx

- 2

(u; - v;)dx

+

(1 r

s

(-l)(u; sA

- v;)dx

, >

where we have used the fact that, if c1= u1 - a2 is locally absolutely continuous, so is la] and

The sum of the integrals over I and A is at most equal to {,Iui - vi ( dx, so Ia + id

+ qu:

- v: - u:+v:)JL*

Similarly,

R(Ipj + A+; - $1 -

3

s

/u;-

c;I))dx.

Adding these two results we obtain the inequality )(I + ;iA,)u - (I + M&j, which implies that (I + AA,)-’

is a contraction

3 IU - t&,

in L.

We now define A =

u A,. ‘%4=-O

Note that D(A) = {UE L+ : a,, u2 locally absolutely continuous, U’ = (u;, U;)E L}, because if u E L’ is sucn mat u, and u2 are absolutely continuous and u’ E L, then ul, t’r E L,(R). Clearly, A is accretive in L and D(A) = L+. In addition, A is closed and satisfies a range condition, as expressed in the following lemma LEMMA2.2. The operator A is closed and R(Z + ,?A) = L+ for all 1 > 0.

Proof. To prove that A is closed, we assume that u” -+ u in L and that {Au”) converges in L, Au” + z, say. It suffices to prove that u” E D(A,) for some M.

H. G. KAPER and G. K. LEAF

348

We have in L.,(R).

Adding

and subtracting

we obtain (u; -

u”,)’ -+ z, +

(u’l + u;)’ + 2((u32 From

the relation

(*)

Z2)

(u;,‘) + z1 -

(**I

-2.

(*) it follows that \u? -

u;(,

d I(u; -

a;)$,

< M,

and, thus, (u;)~]~, < 2M,lu;+ u&

l(u;)‘Then it follows from the relation

(**) that ((u; + u;)(L,

This result, together

d M,.

G M,.

with the inequality

which is implied by the relation (*), leads to the inequalities ((ul)‘IL,, I(u;)‘(~, 6 M, and, hence, ~u~~~,~u~~~ d M,. Weconcludethat u”ED(A,)forM = M,. To prove the second part of the lemma it suffices to solve the equation (I + AA,)u = f for some f = (f,, f,) E L+ with If, (m, If, I m 6 M. For k > 1 + 2JVM2 we define the mappings R, and Q by the expressions R,/ u = (u,, u2) H ((k -

l)u,

- L(u; - u”,) + f,, (k -

Q: u = (u,, uJ H (Au; + ku,, -Au;

l)u,

- i(u;

- u:) + fi),

+ ku,),

D(A,). We are looking for a point u E @A,) such that Qu = R,,u. If we introduce an ordering in L+ in the obvious manner: u 3 u if u I(x) 3 u 1(x) and u2(x) 3 uz(x) for all XE R, then R, is increasing on D(A,), R/(O) = f 2 0 and, hence, R, maps D(A,) into L‘ . Furthermore, the map p,: R’ -+ K’ defined by the expression on

~a(xi,x2)

= ((k -

1)x, - 2(x: - x;) + a,, (k - 1)x, - L(x; - x:) + a,),

where a = (a,, a2) E R,~ = [0, M] x [0, M], is increasing on R,, so its maximum can occur only of p&x 1, x2) are at most on the boundary CC&. It is easily verified that on dC&, both components equal to kM, whence we conclude that R, maps D(A,w) into (g E L’: lgl Im, 1g21I < kMj. For g E L’, jglj,,

lg21m Q kM, let

(+7 :( co ‘01)I ?;h +(co ‘01 JO slasqns IceduxoD uo ama%amo3- ‘7 JO 6%oIodol aq$ IJI~M paddlnba ‘(co ‘01JO slasqns IceduroD uo alq&?a]uy L@IOIIS a.m ivy] +7 + (GE‘01: a suo!punJ JO aDads aql s! (+7 ‘(cc ‘0])““\7 j co ‘01JO slasqns IwduroD uo acma%amo3 moJ!un JO d%olodoi aql r.p~ +7 OIU~ sdetu snonu!juoD JO aceds aql sr (+7 ‘(co ‘01)~ : sands uopq SI!MO[[OJ ayl ampoll -ur aM (~‘2) ‘(vz) uralqold anlm leggy aqlp uognlos (8uoas) ego &mm~ ay$ augap 01 laplo UI (co $1 UIOJJ

(S’Z)

‘B = (0)n

(t7.z) ‘+7 us uoymba

[t?guamjyp f! se (1’~) suoymba JO mawis aql laldIaw!

MOU

aA

.pawy se ‘(Wv)~ 3 n ‘pasop s! “y acme; ‘{,@‘y} amanbas aql saop OS put? sa%anuoD {(,(tn) ‘,(j,n))} amanbas aq1 ‘,,nJx = I +.t@ amls ‘.raAoaJoly .(:n ‘in) c (&n) ‘,@)) ‘jy 3 (x)% ‘(~)~n > 0 ams .(“n ‘In) = n dq if alouaa ‘“n = onu~ JO ~rurg aql s! lu!od paxg aqy leyl alou ‘s!y$ aas 0~ ‘J[aslr (mv)~ ur ‘KJIZJ u! ‘s! yu!od paxg srq~ ]vq~ uqep aA fWt/)a JO amsop aql UI lured paxg anb!un a seq ‘s ]eql SMO[[OJ 11

xp #x)(‘n

- +~)I(x)(~n + ‘n)y+~(~)(~~ - “n)i((x) (Zn + Zn)y - 1 - y) +

j(x) (% - Zn)l(x)(Zn + Zn)y+ j(x) (Ia - In)/@) (Ia + In)r - T - 7))

B Y i 3 s

(“I@ - :n)y + (z.T- Zn) ((7~ + %)y - 1 - $1 + “I@ - zn)y + (I:, - ‘n)((‘o + ‘n)y - 1 - y)ljf =

y3Eq (“~)a sdatu /‘xr _a = /‘s leql smaw srql ‘6 = na amis jWpi)a 3 (Zn ‘In) = n ‘snqJ .zn uo!wng aql JOJ sagladold sum aq$ uyqo aM ‘L~JIZI~!S .snonupuo3 6[a$nIosqE il[[e~ol s! Jlasi! 51 ‘asuaq ‘pm suoyuty snonuguos Alaltqosqe 61p3301ohi1 ~0 WipOJd aql s! 52 layI sag!JaA au0

.(@I7 3 ‘,n ‘IB = :ny + ny aaurs ‘pw

Uo~ssaJdxaaql ur0.1~‘os[y ‘7)16/I_y = 1711nl‘a 3 x 11” .mJ MI 3 (x)‘n 3 0 uaqJ

H. G. KAPERand G. K. LEAF

350

is the space of all functions u:[O, 00) -+ L+ for which there exists a u E L:,,([O, m); L’) such that u(t) - u(s) =

L V(T)ds i9

t, s E [O, ml),

in which case (du/dt) (t) = v(t) for almost all t E [0, co). We say that u is a (strong) solution of the initial value problem (2.4), (2.5) if u E C([O, co);L+) n i+$ ‘([O, co); L+), the initial condition (2.5) is satisfied, u(t) E D(A) and (2.4) holds for almost all t > 0. We have the following theorem. THEOREM2.3. The operator -A generates a semigroup {s(t): t 3 0) of contractions in L+. If, for some g E D(A), the initial value problem (2.4), (2.5) has a strong solution u, then u(t) = 8(t) g for all t > 0. Proof: Cf. [7], Theorems 1.4 and 1.8.

3. CLASSICAL

SOLUTIONS

In this section we indicate how the results of Kolodner [3] on the existence of classical solutions of the Carleman equation can be related to the abstract theory of the previous section. To this end, we interpret the Carleman equation as an evolution equation for the vector u = (ai, uz): [O, 00) + C’, where C+ is the positive cone in the real Banach space C = C(R)x C(R). The topology in C is the product topology: 1.Ic d enotes the norm in C, \ulc = sup{)u,(x)(:x E R, i= 1,2}foru=(u,,u,)EC. We introduce the following notation. For any triplet (5, cr, T) with 4 E R, CT> 0,T 2 0,let QT(g, o) be the trapezoidal region QT(& a) = ((x, t) E Q:O < t d T, x - t 3 5. - (T,x + t d < + a), and let Qr be the strip QT = R x [O, T] = u Qr(5,~). 0,let T = (4\g,\,)-‘. If gl, g2 E C'(R), then the Carleman equation (2.1) has exactly one solution u E C’(QT(5, cr)) x Cl(Qr(& a)). Th e solution satisfies the pointwise estimates.

for all (x, t) E QT(<, o);i = 1,2.

Initial value problems

for the Carleman

351

equation

Proof: Cf. [3], Theorems 1 and 2. -OHM 3.2. If gr, g2 E C’(R) and gl(x), gJx) >, 0 for all x E R, then the Carleman equation (2.1) has a unique nonnegative global solution u = (II1, I+) with ur, u2 E C(Q). Furthermore, ifg,(x) d c, then ui(x, t) 6 c for all (x, t) E Q, i = 1,2, and sup{lu( ., t)l,: t 2 01 = /glc. Finally, if u and U are two solutions which correspond to the initial data g and g respectively, where g,(x) < ii(x) for all x E R, then ui(x, t) < i&t, t) for all (x, t) E Q, i = 1,2. Proof.

Cf. [3], Theorem 5.

THEOREM 3.3. If g E L” and g,, g2 E C’(R), then the solution u = (u,, u,) of the Carleman equation (2.1) satisfies u E L’ and ui, u2 E C’(Q).

ProoJ The existence of a global solution which is continuously differentiable follows immediately from Theorems 3.1 and 3.2. It remains to show that u E L+. The following proof is implicitly given in Kolodner’s article [3]. Let A and B be two arbitrary positive numbers. We consider the integral B Cn,(x, t) + nz(xI t) - g,(x) - gz(x)l dx. 1.4,s(t) = s -A From (2.1) we obtain, upon integration, ur(x,

t)

=

g&x

-

t)

+

* [u;(x

-

(t

-

z),

r)

-

ii:

+

(t

-

T), T)

-

u;(x

(x

-

(t

-

r),

t)]

dr,

s0 uJx,

t)

=

g&x

+

t)

+

’ (u;(x

+

(t

-

t),

r)]

dt.

s Thus, for A and B sufficiently large positive we have IA,&) = (j:;_t-

JB;jgl(u)dc

+ ;I;+-

- J-^“)g&)d~

+ /=; {(j_y;;;;-j;l;I;)

[n&r, r) - u:(g, r), do} dr

and, consequently, Ir,,.(t)l

d tsup{g,(cJ):0

d -A,a

3 Bj + tsup{g&):a

+ r2 supfu&,

+(a,

~1E

Q,( -4

4 u Q,P, t,)

+ t2 SUP{U:(O,

9:@,

4 E

Qt(-4

t) u Q,P, 0).

-I -A,o

3 B!

By Theorem 3.1 we can estimate the last two terms, r2 SUP{@‘(~,

t): (a, 4 E

Q,( -A

< 4t2SUp{g;(X):XE[-A-tt,

t) u Q,@, 0) -A + t] u [B - t,B + t]!.

Since gi is integrable, each term goes to zero as A, B + co. Thus,

lim A,B-+m

G

I, B(t) = 0 and, as each ’

H.

352

ui is nonnegative,

we can conclude

G. KAPIZR and G. K. LEAF

that each ui(. .,t) is integrable,

u EL+, and

It follows from the above theorem that u is a strong solution of the Carleman bining this result with Theorem 2.3 we see that it coincides with the semigroup -A. t 3 0. u(t) = S(t) 9

4. DECAY

OF

SOLUTIONS

NEAR

A UNIFORM

STEADY

equation. ComS generated by (3.1)

STATE

In this section we employ the methods developed by Inoue and Nishida [11] to estimate the asymptotic decay rate for solutions of the Carleman equation (2.1) which are close to a spatially uniform steady state solution. If g = (a, a), where a is some positive constant, then the initial value problem (2.1), (2.2) admits the solution u = (a, a), at least in the space of continuous functions. In physical terms, this solution corresponds to the absolute Maxwellian distribution function for the Carleman equation. Consider now the perturbed problem

(4.1)

for the vector-valued

r~ = (q,, q2) = (u, - a, uz - a), subject

function

vl,(x,O) = b,(x),

to the initial

conditions (4.2)

92(x, 0) = 62(x),

where +i = gi - a, i = 1,2. Equations (4.1), (4.2) give rise to an abstract initial value problem which is conveniently studied in the complex Hilbert space H = L,(R) x L,(R). The topology in H is the product topology; 1.IH d enotes the norm in H, IL& = (lurl~, + 1~~1~~~‘~for any u = (u,, ~2) E H. Let B be the linear operator

defined

in H by the expression

Bu = (u; + 2u(u, - u,), -u; with maximal Furthermore,

u = ($7 uz),

domain D(B). Here, u1 and u2 are functions of x alone and ’ denotes let I be the bilinear operator defined in H x H by the expression I(% 0) = (u2u2 - UiVi’U,U,

with maximal

- 2u(u, - u,)),

domain

- u*v*),

D(T). Then 4(.1) and (4.2) lead to the following 2

+ h(t)

= WI(t), r](t)),

yI(O)= 4. The linear operator

u = (u,, u*),

-B

generates

a semigroup

t > 0,

differentiation.

u = (2’1’u,), initial

value problem

in H, (4.3) (4.4)

e --lB in H, which can be most easily expressed

in

353

Initial value problems for the Carleman equation

terms of Fourier transforms, ikxexp( - t&(k)) G(k) dk,

x E R,

where B(k) is a 2 x 2 Hermitian matrix,

For fixed k, the eigenvalues of B(k) are cc(k) and /3(k), a(k) = 2a - s(k),

with

B(k) = 2a + s(k),

s(k) = (4~’ - k2)1’2.

The projections onto the corresponding

eigenspaces are

and

respectively. We observe that both a(k) and B(k) are real for Ikl 6 2a, and complex conjugate with real part equal to 2a for Ikl 3 2a. We proceed to establish a bound for the function eezBu in the Sobolev space H,, H,=

m (1 + k2)’ IIii(k)l12dk < co .

ugH: i

I

s

Here, II .I[ denotes the Euclidean norm in R2. The norm in H, is denoted by 1.IHr, IIu HI =(j

m (1 + k2)’ Ilii(k)l12dk li2, --m

LEMMA4.1. Let u E H, n L for some nonnegative

le -“&,

UEH,.

1.Then eKtBu E H, and

< CE(u)(l + L)-“~

where E(u) = lulHl + lul,;C is a positive constant which depends on a and 1. Prooj

By definition, leeiB I&, = d

m (1 + k’)’ Ile-‘B’k’ C(k)jl’ dk s -m OD(1 + k’)’ e-21Rea’k) IIP,(k) ii(k dk s -cc co (1 + k’)’ e-2tRep’k) IIP&k) li(k)l12 dk. + s --co

H. G. KAPER and G. K. LEAF

354

The last integral

is easily estimated

that Re p(k) 3 2a for all k, and IIP&k)ll = 1,

by observing

03 (1 + k’)’ e-2tReB’k) IIP,(k) ii(k

dk < Iu[&.

(*)

s We split the first integral Re a(k) = 2a, so

ini

two parts and estimate

each part separately.

s s

(1 + k2)’ e-2tRea(k) lip,(k) ii(k 1k132a

For Ikl 3 2a we have

dk d &,.

(**)

For Ikl < 2a we have Re a(k) 3 k2/2a, so (1 + k2)’ e- 2tRea’k)IIf’,

ii(k

dk

IklC2a

20

2(1 + 4a’)’ sup{(lii(k)l12:lkl

d

d 2aJ

e-tk2sadk. s

0

Now,

I14k)l12 = Iu,(k)/2+

and, furthermore,

s

lu,(k)l’

20

e -tk2Jadk

<

C(1

+

t)-1’2

0

for some positive

constant

C which depends

only on a. Thus,

(1 + k2)’ e-2tRea(k) lip,(k) ii(k s lk:

6

< -. The statement

of the lemma

dk

20

c

(l +Ipa2J1 lul; (1 +

t)-

112.

now follows from the estimates

(***)

(*), (H) and (***).

If u = (u,, u2) E H, n L is such that

sm

(u, + u2) (x) dx = 0,

--m

the estimate LEMMA

of Lemma

4.2. Let u E H,

4.1 can be improved n

upon.

L be such that (4.5) is satisfied. le-teuIH~ d CE’(u)(l

If xu E L, then + t)-3’4

(4.5)

355

Initial value problems for the Carleman equation

where E’(u) = I&, + InIL + Ixz&; C is a positive Proof

As in the proof of Lemma le-“.1;,

constant

which depends

on a and 1.

4.1. we have (1 + k*)’ e-2a’k)’ I/P,(~) li(k)l12 dk.

< 21~1;, +

(*)

s lh1<20 We split the integral into two parts and estimate E(k) 3 cr(a) = (2 - J5) a, so

s

each part separately.

(1 + k’)’ e-2a’k)r lip,(k) ii(k)

For a < Ikl < 2a we have

dk d I$,.

(**)

osikl$?a

For Ikl < a, we first observe data. If xu E L, we have

that Pm(O)fi(0) = 0, because

ii(k) = ii(O) + kii’(tlk), P, is analytic

near the origin;

of the constraint

(4.5) on the initial

0<8<1.

hence,

P,(k) ii(k) = kP”([k) (C(O)+ kii,(ek)) + kP,(O) a’@k),

O<[
i.e. P,(k) ii(k) = k(P=([k) ii(k) + P,(O) ii’(e The operator norm of P#k) tion. One verifies that

can be estimated

p2(ik)=

&

by the spectral radius p([k) of its matrix representa-

(16~~ - i’k’)

d L a2

for

Ikl d a.

Thus,

s s

(1 + k*)‘e-

2a’k)zI\P,(k) ii(k) //* dk

Ikl
(1 + k*)’ e-2”‘k”k2[IIPh([k) /!%]
d 2(1 + uqr

$suplljfi(k)~~*:lkl

ii(k)

+ I/P,(O) ii’(Bk)

< u! + sup{IIii’(k)l12:lkl

dk

< u)]Sa

k* eArk”“dk. 0

As we observed

in the proof of Lemma

4.1,

sup{Ilii(k)l12:jkj

G a! G&/U/t.

Similarly, sup{ Ilfi’(k)ll*:lkl d u) d & 1~~1;.

H. G. KAPERand G. K. LEAF

356

Furthermore, a kZ e-‘k’!“dk

< C(1 + r)-3’2

s0 for some positive

constant

s

2a’k)t III’,(k) ii(k)

(1 + k’)‘e-

C which depends

only on a. Thus,

dk d C(l

+,“”

’ 1) [IuI,. + /xuIJ’(l max ( a”’

+ t)-3’2.

Iklso

(***) The statement

of the lemma now follows from the estimates

If the constraint

(4.5) is strengthened

(*), (**), and (***).

to U1 + u* = 0,

the estimate now show.

of Lemma

4.2 can be obtained

without

(4.6)

the additional

LEMMA 4.3. Let u E H, n L be such that (4.6) is satisfied.

assumption

Then the estimate

xu E L, as we will

of Lemma

4.2 can be

simplified, le-*e~lHI d CE(u) (1 + t)-3’4, where E(u) is defined

as in Lemma

Proof. As in the proof of Lemma le-fBul~,

4.1, C is a positive

constant

which depends

on a and 1.

4.2 we have (1 + k2)’ e-2a’k)f I)P,(k) ii(k

d 3lr&, +

dk.

s VI
(kl < a.

Thus (1 + k2)’ e-2a(k)f IIP,(k) zi(k)l/’ dk s Ikl
= s IklSa

d (’ laf2)l

II(f’,(k)

-

Pa(O)) fi(k) 112 dk

sup{ Ilzi(k)ll’:lkl d uj [

k2 e-tk2’a dk 0

(*) norm One

351

Initial value problems for the Carleman equation

The statement of the lemma now follows from the estimates (*) and (**). The result of Lemma 4.3 should be compared with a similar result obtained for the Broadwell model by Inoue and Nishida, [ll], Theorem 1, under the additional assumption that u’ E L. As the bilinear operator I in (4.3) is such that I(u, u) always satisfies the constraint (4.6), it follows from Lemma 4.3 that the inequality lePrB I(u, u)IH1< CE(I(u, u)) (1 + t)-3’4 is satisfied whenever I’(u, v) E H, n L. Before we proceed to the solution of the nonlinear initial value problem (4.3), (4.4), we establish sufficient conditions for I(u, u) to belong to H, n L. LEMMA4.4. If u, u E H n L, then I(u, u) E L and II(a, u)l, d Cl~Mal~, for some positive C. Proof. From the definition of the norm in L we obtain II(V)

= 21%

- %UllL,

4%lL1 + I%&) d 2(/& IbL + 14L2MLJ d %lE, + l%lfJ””hl2, + l%l2J”’ = 2l4, lblw

d

LEMMA 4.5.

Let u, u E H, for some 1 > $. Then F(u, u) E H, and

ll% 4mG+4fI blff,’ for some positive constant C. Proof. From the definition of the norm in H, we obtain (1 + k’)’ I;;;‘;;(k) - i$,(k)(’ s Using the identity Gi = tii * fii and Young’s inequality we find \I+, U)& = 2

(1 +k’)’ )6,(k))’ dk ) + j- luJk)l dk (j A By the Cauchy-Schwarz

dk.

(1 + k2)’ )Uk)j’

dk)].

inequality, llB,(k)ldk

< j(I

+dkkz)j+(l + k2)%(k)lZdk,

where the frost integral converges for 2 > i. A similar estimate holds for the integral j ]tT,(k)( dk. Thus we find that, for some constant C, II+, u)I;, G +I;, which proves the statement of the lemma.

I&,

H. G. KAPERand G. K. LEAF

358

We now come to the main result of the present section. THEOREM 4.6. If C#J E H, n L for some I > f, and E(4) = I&, + I& is sufliciently small, then the initial value problem (4.3), (4.4) has a solution q such that r(t) E H, n L for all t > 0 and

Iv(t)1H, = O/(t- ’ “)

as

t + co.

Proof. The solution of the initial value problem (4.3) (4.4) satisfies the integral equation q(t) =

em@

rep't-s)B l-(?(s),q(s))ds.

qf~+

(4.7)

s0 Let X be the Banach space X = {u:[O, m) -+ H,

n

L:sup{(l

+ t)lj4 lu(t)lHl:t 3 01 < co}

with norm 1.Ix, (t& = sup{(l + tp4~u(t)l,,: t 3 01,

UEX.

We propose the following iterative process for the solution of (4.7) in X. Let #O)(t) = epte c$, [‘O’(t) =

f e-"-"'"

T(q'"'(s),q'o)(s)) ds

s0

and define for

n =

1,2,.

.. q'"'(t)=

?jJn-l'(t)+

p"(t),

z e-(t-s)B {T(q'")(s),~(n-l)(s)) +

i'"'(t)=

r(['n-l)(~), q'n-l)(~))j ds.

s0

prove that this process is well defined in X if 4 E H, n L for some 1 > $. From Lemma 4.1 it follows that q(‘) E X . Then, q(‘)(t) E H, n L and, by Lemmas 4.4 and 4.5, r(P)(t), P(0) E H, n L for all t 2 0, with

We

E(r(P)(r),

P(t))

= I~WW,

PW

H1 + IwP)w,

P(t))

IL

< Cl~‘“‘(t)l~*.

Thus, (1 + t)1’4poyt)lH,

< (1 +

d Ie-ct-S)B T(~'O)(~),~~(O)(S))I~, ds

tp4 s

< C(1 +

<

ty4

' w(~(O)(s), s0

C(1 + ty s

G

~(0)(s))(1 +

f I~‘“‘wIl (1+

clPwup~u + I)“f

pS

s)3,4

p” s)3/4

(1 + s,1,2(;s+

t _ s)3,4.t 2 0).

(*)

359

Initial value problems for the Carleman equation

It follows that sup{(l

+ t)“41[‘O’(t)lH,: t 3 01 < @z

and, therefore, i”’ E X. Suppose now that q (R- ‘) E X and [(“-l) E X. Th en, trivially, $“) E X. Furthermore cc”- l)(t)) E H, n L and r(ccn- ‘j(t), q’“-“(t)) E 23, n L for all t 2 0, with

r(rl’“)(t),

Thus (1 + t)l’4)P’(t)l,,

d C(1 +

ty

f [IP(4IH, s0

(“-yS)IHl x 15

+

IYI("-Y&l

ds

(1 + t - s)3’4

(**I It follows that sup{(l

+ ty4~p’(t)lH,:

and, therefore, cc”)E X. We now proceed to establish a convergence From Lemma 4.1 we obtain the estimate

t 3 O] < cc

criterion

for the sequence

((q’“), [“‘))] in X.

ll’“‘lx < CE(6) and from the inequality (*) above, constant C, we have the estimate

so possibly

li’O)lx d C’1$“($,

after a readjustment

of the

11(O)), d C2E2(&. From the definition

of q(“’ and the triangle

inequality

Iyl’“)Ix d IYyl)Ix and from the inequality (**) above, readjustment of the constant C,

JPlx

we conclude

that for n = 1,2, . . .

+ Iyqx

< C’((yl(n)(y + ($“-‘)IJ(<‘n-l’IX,

l[‘“‘lx d c(21?/‘“-‘)Ix

so possibly

after a

+ I[‘“-l)I&‘n-l)Ix.

In the Appendix we show that these estimates imply the convergence of the sequence {(l$“)lX, in R2 and, in particular, l[‘n)lx + 0 as n + 00, provided E(4) is sufficiently small. Thus, (q’“), [‘“)) -+ (q,, 0) in X for some qa E X, provided E(4) is sufficiently small. It remains to show that q, is a solution of the integral equation (4.7). But this is straight~[‘“‘JJ]

H. G. KAPBRand G. K. LWF

360

forward,

since ? ‘“+l)(t) = v]‘“‘(t) + i’“‘(t) f =

rp'(t) +

e-'f--s)B[l+fcn)(s),q'")(s)-

Y]'"-~)(s))

s0 +

Using the bilinearity

T(rf"(s) -

+-l)(s),

q'"-"(s))]

ds.

of I we find that rl‘n+l)(r) = r’“)(r) +

' e-'t-s)B [r(r]'")(s), y]'")(s)) s0

-

and, by repeated

r “+i)(t) Letting

n +

r(fyl)(s),

$-l)(s))]

ds

application,

cc we

see

= i+‘)(t) +

’ e-‘l-s)B r+p)(s), p)(s)) ds. s0

that the limit qm satisfies (4.7).

We observe that Theorem 4.6 is concerned with the existence of a solution of the initial value problem (4.3), (4.4) and does not pertain to the question of uniqueness. If we wish to establish a uniqueness result we may refer to the results of Section 3 on classical solutions. In particular, if 4 = (41, b2) E H, n L for some 1 > 5, then 4i E C’(R), i = 1,2, by Sobolev’s imbedding theorem. Moreover, since sup{ 14i(x)I : x E R} 6 E(4), we can choose E(4) sufficiently small to satisfy the conditions of Theorem 4.6 as well as the inequality a + 4i(x) 3 0 for all x E R, i = 1,2. Under these circumstances, Theorem 3.2 is applicable and uniqueness follows. If the initial data 4 = (41, 4,) E H, n L are such that m (4, s -m

+ &)(x)dx

= 0

(4.8)

then the decay rate estimate of Theorem 4.6 can be improved upon. In physical terms, the constraint (4.8) implies that the initial perturbation C#J represents a zero total density. THEOREM 4.7. Let C#J E H, n L for some 1 > i be such that (4.8) is satisfied. If XC$E L and E’(4) = iciently small, then the initial value problem (4.3), (4.4) has a solution l&f, + q such that r(t) E H, n L for all t 2 0 and

l4%+ I4’ ISsuff

Iq(t)l,, = O(t-“‘“)

as

t -+ a.

Proof The proof is analogous to the proof of Theorem 4.6, the only difference definition of the function space X, which in this case is given by X = (u: [0, co) + H, n L: sup{(l and the use of Lemma

is finite.

4.2, instead

of Lemma

being

the

+ t)3’41U(t)l,,: t 3 O! < a!

4.1. The decay estimate

follows from the fact that

Initial value problems

for the Carleman

equation

361

If the constraint (4.8) is strengthened to 41 + 42 = 0

(4.9)

the decay rate estimate of Theorem 4.7 can be obtained without the additional assumption x4 E L. In physical terms, the constraint (4.9) implies that the initial perturbation 4 represents a zero total density at every point x E R. THEDREM 4.8. Let 4 E H, n L for some 1 > t be such that (4.9) is satisfied. If E(4) = 1blH, + 141L is sufficiently small then the initial value problem (4.3), (4.4) has a solution yesuch that s(t) E H, n L for all t 3 0 and

(~(t)l,~ = 0(te3j4)

as

t + 00.

Proof: The proof is identical to the proof of Theorem 4.7 if one uses Lemma 4.3 instead of Lemma 4.2. 5. SUMMARY

The results of Sections 2, 3 and 4 summarize the current state of knowledge about the initial value problem for the Carleman equation. Basically, the results relate to two different types of problems. In Sections 2 and 3 we considered the pure initial value problem subject to appropriate constraints on the initial data. We showed that in the positive cone of the space of integrable functions the solution is obtained from the initial data via a contraction semigroup. By suitably restricting the range of initial data the semigroup generates a classical solution, the existence of which was established earlier by Kolodner who used fixed point arguments and special properties of solutions of Riccati equations. For constant initial data of the type ui(x, 0) = u,(x, 0) = a for all x E R, where a is some positive constant, the Carleman equation admits the solution ul(x, t) = UJX, t) = a for all x E R, t 3 0, at least within the realm of classical solutions. A small perturbation of the initial data, u,(x, 0) = a + 4i(x), u2(x, 0) = a + 4Z(x), x E R, say, will lead to a perturbation of this uniform steady state. If the solution is represented in the form ui(x, t) = a + qi(x, t), uZ(x, t) = a + qZ(x, t), x E R, t 2 0, we expect the perturbations ~~ and Y/~to decay to zero as t increases. This, then, is the second type of initial value problem related to the Carleman equation. We considered it in detail in Section 4. We proved that in the Hilbert space of square integrable functions the perturbations decay algebraically to zero as t + co, provided the initial perturbations (4r, 4,) are sufficiently smooth and sufficiently small. If the initial perturbations represent a zero net contribution to the total particle density (i.e. m (4, + 4,) (x) dx = 0), then the decay is O(t-“I”), otherwise it is 0(t-“4). s -03 Acknowledgement-The authors wish to express their appreciation to Drs. S. Reich (U. of Southern California) Schechter (Duke U.) for many valuable remarks and suggestions on an earlier draft of the manuscript.

and E.

REFERENCES 1. CARLEMANT.,/Problemes

mathtmatiques

dans:

theorie cinetique

des gaz;Up$da

(1957).

2. KAPER, H. G., LEAF G. K. & REICH S. Convergence of Semigroups with an Application to the Carleman Equation, submitted for publication. 3. KOLODNW I. I., On the Carleman’s model for the Boltzmann equation and its generalizations, Annali Mat. pura appl., Series 4,63, 1 l-32 (1963).

362

H. G. KAPER and G.K.

LEAF

4. TEMAM R., Sur la r&solution exacte et approch& d’un problkme hyperbolique non lineaire de T. Carleman, Archs Ration. Mech. Analysis 35, 351-362 (1969). 5. MARCHUK G. I., Methodr of Numerical Mathematics, Springer-Verlag, New York (1975). 6. LIONS J. L., Quelques mdthodes de r&solution des problbmes aux limites non lintaires, Dunod, Paris (1969). 7. CRANDALL M. G., Semigroups of nonlinear transformations in banach spaces, in Contributions to Nonlinear Functional Analysis, edited by E. H. Zarantanello, Academic Press (1971). 8. KURTZ T. G., Convergence of sequences of semigroups of nonlinear operators with an application to gas kinetics, Trans. Am. math. Sot. 186,259-272 (1973). 9. TARTAR L., Evolution equations in infinite dimensions, in Dynamical Systems, edited by L. Cesari et al. Vol. I, Academic Press (1976). 10. NISHIDA T. & MIMUIU M., Global solutions to the Broadwell’s model of Boltzmann equation for a simple discrete velocity gas, Lecture Notes in Physics 39, 408412, Springer-Verlag, New York (1975). 11. INOUE K. & NISHIDA T., On the Broadwell model of the Boltzmann equation for a simple discrete velocity gas, Appl. Math. Optim. 3, 27-49 (1976).

APPENDIX In this Appendix

we prove that the inequalities

Itt’“‘lx < Irl’“-‘)lx + li’“-‘)lx, li’“‘lx < C(2)q’“-“I, for n = 1,2, . imply the convergence of the sequence E(4) is sufliciently small. Let c( = CE(4) and define the sequence of ordered

+ li’“-“l&‘“-

l’lx

{ It+“)lx, l~‘“‘lx! in R2 and, in particular, pairs {(a, b,)] recursively

/[‘“‘lx + 0 as n + co, provided

as follows:

(a,, b,) = (a, a’), (a,, b,) = (a,_,

+ b,_,,

C(2a,_,

+ b,_,)b,_,)

n = 1,2 ,...

Then {(a, b,)) is a majorizing sequence for {I$‘)lx, J[‘“)lx! and it suffices to show that {(a, b,)] converges b, -+ 0 as n -+ co, for sufficiently small positive a. Consider the associated map T: R: + R:,

qx, Y) = (x + Y,cm + Y)YX

in RZ and that

x 2 0,y > 0.

T has the following properties: 1. T&O) = (x,O)forallsER+. 2. If([,~)=T(x,y),then~zxand~iyif2x+yO. 3. For any B > 1 and 6 = (B - 1)/(2bC), the triangular region Aa = {(x, y) E R: :x + By < S] is mapped into itself. Therefore, if we choose G(such that the point (a, a’) E Aa, i:e. a + Paz < (p - 1)/(2fiC), then the sequence {(a, b,)! generated from (G(,a’) by repeated applications of T is bounded. Smce the a,‘s are monotone increasing it follows that the sequence {a,) has a limit. Therefore b, = a, - a,_, converges to zero, and we conclude that (a,, b,) + (a, 0) for some positive a.