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-di- mensional 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 different- iable 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 differ- entiable 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<oo, {ul<u i, s ]xL,<~, lul< u 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<R}, f((Qo)= n U Q~, h<ho O~<t-~<T where Q~--{(t, x) ilxl< R--at--(i + l)h}(t~< t < ti+l), a = sup max 0qD~ x[<R, u]<U,t~<T ~ Ou ' and the domain 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 .