Weak Solution of the Cauchy Problem for a Multi-Dimensional Quasi-Linear Equation

WEAK SOLUTION OF THE CAUCHY PROBLEM FOR A MULTI-DIMENSIONAL QUASI-LINEAR EQUATION

N. N. Kuznetsov

1. In this article we prove the existence of a weak solution of the Cauchy problem for the multi-dimensional quasi-linear equation (1) and investigate certain properties of this solution. We shall say, as is usual [1], that the function u(t, x 1..... Xn) =u(t, x), which is measurable and bounded in ~2 and which satisfies Eq. (1) in ~ in the sense of distribution theory, is the weak solution of the equation

0-T0u+ ~]~=ln 0~ (u, t, x~,Oz~x~..... xO = 0 (1)

in the domain ~ ~ En+ 1. If u ~ (x) is measurable and bounded in En, we shall say that in the strip S = { 0- t < T, ] x I < oo } the weak solution u (t, x) is the solution of the Cauchy problem with initial conditions

u (0, z) = uO (z), (2)

when u (t, x) -- u ~ (x) converges weakly to zero as t ~+0. If u (t, x) is the weak solution of the problem (1)- (2) in the strip S, then clearly for any smooth function g (t, x) with support lying in the half-space t < T, we have

Og (3)

The one-dimensional case (n = 1) of the problem (1) -(2) has been studied extensively in the literature. The most general existence theorem was proved in [1] for an equation in which the function ~v (u, t, x) was con- vex in u. But the same method can be used to prove the existence of a weak solution without the convexity condition with the additional assumption that the initial function is of locally bounded variation. Little is as yet known about the multi-dimensional equation. In the only paper [2] on the subject it was shown (by Oleinik's method) that a weak solution exists when ~vi = ~vi (u), and the function u ~ (x) is of locally bounded variation in the Tonnelli-Cesaro sense. We shall prove here a general existence theorem and investigate certain properties of the solution. We note that in our proof we use the method of "smoothing," the basic idea of which belongs to B. L. Rozh- destvenskii [3]. The method is as follows. We introduce a nonnegative infinitely differentiable, weighting function coh (x) = h -n co (h-ix) (co (y) = 0 for I Y I > 1, fw(y)dy = 1) and put u~(x) = f w h (x I x') u ~ (x')dx'. We consider the problem (1) -(2) with initial function u~ (x). Since grad u~ = O (h-l), the continuously differentiable solution u(1)(t, x) of this problem is defined in the strip 0 -< t ~ tl, where t I = O (h). We next introduce the function u~ (x) = fo~ h (x --x') u(l)(tl, x') dx' and define the continuously differentiable solution u(2)(t, x) of Eq. (1) with initial condition u(2)(ti, x) = u~ (x) in the strip t 1 s t -< t 2. The process is then repeated. We shall prove below that the function u(i)(t, x) is uniformly bounded (this is obviously true if certain assump- tions are made about the functions ~vi); hence, ti+ 1 --t i = O (h) for all i, and so the number of strips into which S is split is N h = O (h-i).

Putting

uh(t, x) = u(0 (t, x) (ti-1 ~ t < t~, i = 1, 2 ..... Nh, to : 0),

M. V. Lomonosov Moscow State University. Translated from Matematicheskie Zametki, Vol. 2, No. 4, pp. 401-410, October, 1967. Original article submitted January 26, 1967.

733 we obtain the "approximate" weak solution of the problem (1) -(2). Let

i s {ytOg u~ + ~-~.~,(uh, t, x))dtdx + IEngCO, x) U~ ) (4) be the discrepancy between this solution and the exact one.

THEOREM 1. If the functions u h (t, x) are uniformly bounded in S and the weighting function is sym- metric, c0(y) = co("y), then 5h(g ) -~ 0 as h ~ 0 for any function g (t, x) which is twice continuously differentiable and has a support lying in t < T.

Proof. Since in each strip ti < t < ti.+l, u h (t, x) is a continuously differentiable solution of (1) and satisfies the initial conditions Uh (ti, x) = u~(x) = fw h (x- x')uh(t i --0, x')dx', then

5hOg)= ~iLhl I g(t,, x)Cuh(ti--O, x)--U~(x))dx "4- I gCO, x)Cu~176

From the symmetry of w(y)

~h, = I g(q' x)(uh(~-- 0, x)-- ui(x))dx Uh (X)t dx )ti-o u h (t~ -- O, x) dx. = i i{

But

g (x + hy) -- 2g (x) + g (x -- by) ~ Cxh2 { Y t2' 2 so that { 6hi I -< C2h2 fsuppg(ti, x) {Uh (ti, x) { dx. Since the u h are uniformly bounded, Nh -< C3/h, so that

which was required to be proved.

THEOREM 2. Let the functions 0qgi (u, t, x) be bounded for t u [ < U in every compact part of S. 0u loc Then the limit of the sequence of approximate solutions uhi(t, x) which converges in the topology of L 1 (S) and is uniformly bounded (by U) is a weak solution of the problem (1) - (2). Proof. For the twice continuously differentiable trial function g(t, x) the proof is obtained by passing to the limit in (4), for from the conditions of the theorem it follows that ~vi (Uhk,t , x) ~ ~vi (u, t, x) in L~~ For the continuously differentiable function g(t, x) the truth of the assertion is established bypass- ing to the limit in (3) from the functions gm (t, x), which have second derivatives. The theorem is proved. By this theorem the problem of the existence of a weak solution of the problem (1) -(2) is reduced to the problem of the compactness of the family of functions {u h (t, x)}. 2. Let the function cpi (u, t, x) be twice continuously differentiable in {lul < U l, 0 -< t ~ T1, t x l < r162 We assume further that

sup ~ o~ f 02q~i ~, ,~,<~, ~ ~1 < ~ (v, t), sup max Ou0% ~ A (t), s., max I <. (,), ]xl

734 where U < Ui, the function ~(v, t) is continuous and does not vanish in its first argument and A(t), Bit ) are continuous in [0, T1]. We denote by Z(t; U0) any solution of the equation

dZ dt - r (z, t), satisfying the condition Z(0, Uo) = Uo(U0 < U), and let T2(Uo) be the solution of the equation Z(t; Uo) = U. We choose the strip S by putting T = rain (Tt, T2). Then on every integral curve of the system

dxt (t) O$i (u, t, x) du (t) ~ "K~ OcPi dt ~ Ou ' -~ ~ X.Ji Oxi ' passing through the point (u ~ 0, x ~ (I u ~ I < U0), the inequality I u (t) I -< Z (t, U0) is satisfied and the pro- jection of this integral curve on the space (t, x) belongs to a finite part of S. From the last inequality it clearly follows that the functions u h (t, x) are uniformly bounded (with respect to h):

I u~ (t, x) l< z (t; uo) < u. (5)

This is a direct consequence of the properties of the classical solutions of (1) and of the weighting functions. Below we shall make use of the domain

Qo={Xilxl

a = sup max 0qD~ x[

Qt = K(Qo) fl (t = const).

3. In this section we discuss the function u~ of locally bounded variation according to Tonnelli- loc Cesaro (i.e., its gradient is a measure, cf. for example [2]). In terms of the space L 1 this is equivalent to the function u~ being "Lipschitz continuous in the mean"

Ia l u~ x + &) --z,o(x)ldx~ K(O)16 [. (h)

In fact since f lu~ (x + 6) --u~(x)l dx = fdx t f01(6 grad U~)x+Ssds, (A) fonowsfrom the 'boundedncss of. f I grad u~ l dx. Conversely if (A) is satisfied, then it is also satisfied for u~ (with a constant independent of h) and f l n grad u~ I dx is uniformly bounded for any vector n, whence follows the boundedness of the variation (it is sufficient to take the coordinate vectors for n).

THEOREM 3. If the function u~ is measurable and bounded and has locally bounded variation, then there is a weak solution of the problem (1)- (2) in the strip S which has locally bounded variation for everyt. Proof. We put q0 = rues Q0, V0 = var u~ We shall prove the compactness of the family of functions Q0 {uh}in L 1 (K (Q0))* By (5) it is uniformly bounded. We shall show that for any domain ~2t ~ Qt var a h ~ M (t), (6) ~t where M(t) is uniformly bounded for 0 -< t -< T.

735 We consider the strip ti_ 1 --< t < t i. In it u h (t, x) is continuously differentiable and satisfies (1). Dif- ferentiating the latter with respect to Xs, multiplying the result by Ux s and summing the identity obtained over s, we obtain the following differential inequality (strictly speaking, it is only true for [ grad Uh I ~ 0, but more detailed analysis shows that (7) holds independently of this assumption) :

dlgddUhl Jr- ~]ikO--~b-~i-t-~)Igradua[{0~$r Ou h . < nA(t)lgradun] -]- lfnB(t), where

d 0 ~ 0(pi O d'-t : ~ 2f_ X..h Ou Oxi"

Let x =f (t, ~ ) be the equation of the characteristic of (1) passing through the point (ti_i,~) in the domain Qti_l. Let ~2ti_i c Qti_l and let ~2tbe the image of t2ti_l under the transformation ~ ~ x =f (t, ~ ) (ti_ 1 -< t < ti). Then, by a well-known theorem from the theory of dynamical systems

d (det 1r ( Ouh O~i dt ~ 2i O~(piOu~ Ox~ q- Ou Ox~) (det/~).

Hence

d [ grad u h I det 15 dt nA [grad uh I det/r -I- lfnB det/~.

Integrating the above with respect to ~ over the domain 12ti_l we obtain

d---latlgraduh[dx~< nA fatlgraduh[dX CnBmesq,, so that for ti-1 -< t < t i

var a h % var uhena(ti_l) --~ qo if ~ [t B (~) ear d~, (7) a (~) = f t~ A (q) ar I.

Since, from the properties of mean functions,

var uh ~ var u h (re -- O, x), Q tr ~t~-o then (7) gives

e "~(~ + qo t f~- It. B ('~) e~(') dv = M (t), var Uh d Vo ~t

and (6) is proved.

Let 5 be an arbitrary vector. Let Qts = {t, x It = const, x~Qt, z ~-8~ Ql}-

Since Qt is a sphere in En, then rues (Qt \ Qt5) <- CI6 ]. Hence, from (6),

Iotlu~(t, x + 5)-uh(t, x)ldz= fo~lUhl(t, x + 5)

- uh (t, x) I dx -~ I ot\Q~ I uh (t, X ~- 5) -- Uh (t, X) I dx ~ M] 01 + 2UC I 5 [" (8)

736 Let 0--

S Qe I Uh (t", x) -- ul~ (t', x) I dx.

In each strip ti < t < ti+l,uh(t, x) is a continuously differentiable solution of (1); hence, assuming that t k - t' < tk+l, tm -< t" < tm+l (m - k), by integrating (1) with respect to t and using (6), we obtain

IQt. lUh(~",X)--uh(t',x)ldx

(if m = k,the sum term is absent). Since uh(ti, x) = fw h (x --x')uh(ti -- 0, x')dx', then (6) gives

I Qt" I Uh (ti, X) -- Uh (t4 -- O, X) [ dx ~ Mh, (lO) since the sum in (9) is bounded by Mh(m --k). Since ti+ 1 --t i -> const h, then (m --k) s const h -1 (t"--t'), and (9) takes the form

I qt" ] ah (t", x) -- a h (t', x) I dx < M2 (t" -- t'). (ii)

Hence, for every sufficiently small ~-

SK(Qo) I Uh (t + ~, x) -- uh (t, x) l dx ~"-I*1 dt ~^ ah (t + % x) -- u h (t, x) [ dx + 4Uqo 1~ l dl~l Jwt '

~1.[ jQ,+I.I] Uh (t + ~, x) -- ah (t. X) I dx -t- Ms I "r [ ~ (Ms -4- TM2) ] "r I"

Finally, taking this result together with (8), we obtain the bound:

IK(Qo)[uh(t +'r X + b)--uh(t, x) ldt dx ~ M41TI A- MsIbr, (12) whence follows the compactness of the family {Uh} in LI(K(Q0) ).

Let Uhi --* u. Then Uhi ~ u in any domain lying inside K(Q0). On the other hand, choosing a non-van- ishing family of domains Q0(k) , covering En, we can select by a diagonal process from {Uhi} a subsequence {Uhi, }, converging in any of the domains K(Q0(k)). Thus, Uhi,~u in the topology of L~~ Then by Theo- rem 2,u is a weak solution. The assertion of the theorem about the boundedness of the variation is established, for example, in the following way: u clearly satisfies (8) (with constants M and C which depend on Q0); hence, from the results at the beginning of this section, the variation of u is bounded for each value of t. The theorem is proved.

THEOREM 4. If the functions u~ and v~ are bounded (by U0) and measurable and have locally bounded variation, then there exist solutions u(t, x) and v(t, x) of the problem (1) - (2), such that

IQt I u (t, x) -- v (t, x) I dx ~< IQ~ I uo (x) -- vo (x) I dx. (13)

Proof. It can clearly be assumed that for the functions uhj(t, x) and vhj(t, x) the numbers ti coincide and both sequences converge. Let w h(t,x)=u h(t,x)-v h(t,x), A~ (t, x) = [9~ (uh, t, x)- ~ (vh, t, x)I/w~ (t, x).

737 Then in every strip t i < t < ti+l, we have

dwl, + wr, ~ OAi d'-7- ~ -- O, (14) where d/dt = a/at + "~,i A~~ Oxi Let x = r (t, ~ ) be the equation of the characteristic of (14) passing through the point (ti, ~ ). Further, let ~1t, ~ Qt' for any fixed t', and let ~lt be the image of a t, under the transform- ationx=r ~)-~x=r ~). Then

d(det%) .~ 04~ (15) ~t = det ~p~ 2J~ and hence,

Since sup max I Ai I -< a (cf. the notation introduced in Section 2), then from ~t' c Qt' it follows Ixl

Note. Although the assertion of the theorem concerns a pair of functions, it does not follow, because they are arbitrary, that there is a set of solutions, any two of which satisfy (13), for any set of initial functions. However, if the family of initial functions tl is countable, then from the proof given above, and using a diagonal process of choosing a convergent subsequence from the countable family of sequences, it follows that there is a set of solutions ~* (11) , such that (13) is satisfied for any pair of functions from ~* (u) 4. Let the function u~ be measurable and bounded: [u ~ (x) I < U0. Then u~ ~ (En) and as is well- known, Ialu~(x-]-6)--uo(x)]dx--~O as I S] -~ 0 for any finite domain a. We introduce the modulus of con- loc tinuity in the mean for the function u(x) EL 1 (En) , putting

(p; a) = sup I~ 1 ~ (x + ~) -- u (x) ] dx.

THEOREM 5. There is a weak solution of the problem (1) -(2) in the strip S which is bounded in it and if 99i is independent of xl, ..., Xn, then its modulus of continuity X t (p; ~2) for any ~ ~ Qt satisfies ~ (v; ~) < ~ (v; Q0), (16) where X (p; Q) is the modulus of continuity of the initial function.

Proof. Since the continuously differentiable functions (which are, obviously, functions of bounded loc variation) are dense in L 1 (En) , then there exists a sequence of functions u(n)(x) which converges to u~ loe in the topology of L 1 (En) and is uniformly bounded: [ u n [ < U 0. Let u (n) (t, x) be the corresponding weak solutions of the problem (1) -(2). Then the sequence u(n)(t, x) is uniformly bounded ([ u n [ -< U) in S and converges to the function u(t, x) which is bounded by the same number in the topology of L~~ This follows from the completeness of the space LI(K(Q0)), and

Ig(qo) [ U (n) (f~, X) -- U(m) (t, Z)] dx < T I qo [ un (x) -- u(m) (x) [dx

(cf. the note on Theorem 4). By Theorem 2, u is a weak solution of the problem (1) -(2).

738 To prove the last assertion of the theorem we note that by closure the inequality (13) is extended over arbitrary initial functions (bounded by U0). Hence, applying it to v~ = u~ + 6), we obtain

fQtlu(t, x A-8)--u(t, x)Jdx~< IQoluo(x-b 6)--uo(x)ldx, whence (16) follows. The theorem is proved. We note that in the case when ei depends on x, the property of conserving the modulus of continuity may not hold (provided it is nonlinear, which corresponds to the case of bounded variation).

LITERATURE CITED lo O. A. Oleinik, Discontinuous solutions of non-linear differential equations, Usp. Matem. Nauk, 1_~2, No. 3 (1967), pp. 3-73. 2. E. Conway and J. Smoller, Global solutions of the Cauchy problem for quasi-linear first order equations in several space variables, Comm. Pure and Appl. Math., 19_, No. 1 (1966), pp. 95-105. 3. B. L. Rozhdestvenskii, A new method of solving the Cauehy problem in the large for quasi-linear equations, Dokl. Akad. Nauk SSSR, 138, No. 2 (1961), pp. 309-312.

739