Í I (udt<t>-vxum-va)dxdt

Transcription

1 proceedings of the american mathematical society Volume 107. Number I, September 1989 ON THE REGULARITY PROPERTIES FOR SOLUTIONS OF THE CAUCHY PROBLEM FOR THE POROUS MEDIA EQUATION KAZUYA HAYASIDA (Communicated by Walter D. Littman) Abstract. We consider the Cauchy problem for the equation d u = àum in RN x (0, T). We assume that 1 < m < 3N/(3N - 2) and the initial data u0 is in Ci{RN) and Uq > 0 in RN. Then we prove that the second derivatives of u'" with respect to the space-variable are in L2(RN x (0, 7")). We consider the Cauchy problem for the porous media equation (d,u dtu = Aum 1 u(x,0) = m0(x) on R inrnx(0,t) where m > 1 and u0(x) > 0. The problem (1.1) has been studied by many authors. For a detailed account of (1.1) we refer to the work of Peletier [7]. We say that u(x,t) is a solution of (1.1), if and -T / / [u(x, Jr* t)2 + \Vum(x,t)\2]dxdt <oo Í I (udt<t>-vxum-va)dxdt Jr* + / u0(x)4>(x,0)dx = 0 for any continuously differentiable function <f>(x, t) with compact support in R x [0, T). The existence of such a solution is due to Sabinina [8] under some condition on u0. We are concerned with the regularity of u in (1.1). The Holder regularity of u was shown by Caffarelli and Friedman [6]. For N = 1 the precise Received by the editors September 21, 1987 and, in revised form, June 30, Mathematics Subject Classification (1985 Revision). Primary 35K55, 35K65, 35K15. Key words and phrases. Porous media, initial data. This research was partially supported by Grant-in-Aid for Scientific Research, Ministry of Education American Mathematical Society /89 $1.00+ $.25 per page

2 108 KAZUYA HAYAS1DA Holder exponent with respect to the space-variable was obtained by Aronson [1]. Similar results for the time-variable were studied by di Benedetto [4], when TV > 1. There arises a question whether the derivative dtu is a function or not. Concerning this there are results such as Aronson and Bénilan [2], Bénilan [5], where the assumption on u0 is very weak. For a function space A the assertion "dju G A " is alsmot equivalent to "dx dx um G A, 1 <i,j < N". According to [5] dtuglp([s,t]xbr) for any I < p < I + l/m,0 < ô < T and R > 0, where BR = {x G RN ; \x\ < R}. Further if N = I and m > 2 particularly, dtu G L (ô, T;LP(BR)) for any 1 < p < 1 + \/(m - 2). In this connection we note also the results in [3], [9] and [10]. Our theorem is stated in the following section. Let us denote by (, ) and the inner product and the norm in L (R ), respectively. Let us write GT = RN x (0, T). Our aim is to prove Theorem. Suppose 1 < m < 3N/(3N-2). Let w0 be in C0(R ), and let w0 > 0 in RN. Let u be a solution of (1.1). Then dxdx um G L2(GT), \<i,j <N. More precisely, if un< M in RN, then for a = 1 - \/m (2.1) m,,2, n m(l-a/2),2 f \\dxdxum\\2dt<c\\\u \\' ' '.,,ttm,_ m.,2, am,_, m,2xl + M VM0 +(u0,1vu0 )], where C depends on m,n,t and not on u0,m. Our method is to derive a uniform energy inequality for each solution of nondegenerate parabolic equations, which are the regular approximation of ( 1.1 ) appearing in [6]. We prove our theorem. For n > 0 let un(x,t) be the solution of f dru" - A(uT (3.1) I ' [ w''(x,0) = M0(X) + 3 n in R onrn. x (0,oo) It is known that n <un < M + n and un is classical (cf., e.g., [8]). If we set vn = (u'')m and y/n - (u0 + n)m, (3.1) becomes Í (vn)~"d,v'' = mavn inrnx(q,<x>) (3.1 ) \ K, \ v"(x,0) = y/"(x) onrn,

3 REGULARITY PROPERTIES FOR SOLUTIONS OF THE CAUCHY PROBLEM 109 where a= 1- l/m. For simplicity we denote vn(x,t) and ^(x) by v(x,t) and y/(.x), respectively. From the proof in Sabinina [8] we easily see that (3.2) \\v"(-,t)-rim\\<c(\\y/-rim\\ + (M + t1)"m,2\\vy,\\), 0<t<T, where C depends on T and not on y/, n and M. Let C e C2(RN) and C > 0 in RN. From (3.1 ' ) we have Hence -! (Ç,dtv2-") + m(ç,\vxv\2) = ^(AÇ,v2). I a ' 1 Combining this with (3.2) we obtain -(C,^(-,r)2-")] + i (AC,v2)dt. (3.3) / (C, Vvü V'<C[(CV~'*) + lk->,'"l 2 +(M+//riiv^ii2+//2m]. Let w = -Çdv v for 1 < j < N. By integration by parts we have (3.4) -(Av,w) = {CVxdxv, Vxdxv) + ^-Vmv,Vxdxv)-(Vi;.Vxv,d2xv), where we have assumed that v(-,t) G C3(RN). But (3.4) is valid for any v(-,t) G C (R ) by taking an approximating sequence of v. Similarly we have (3.5) (v~"d,v,w)= -a(v-"~[dxv-dtv,t;dxjv) It is easy to see that + (v~"d.dx v, Cdx v) + (v~"d,v,dxç-dxv). I Xj Xj - I Xj X, and (v~' ' dx v dtv, Çdxv) = m(çv~lav, (dx v) ), (v'"d,dxv,cdxiv) = {2(Çv-n,d!(dxv)2) (v "d,v,dxç-dxv) = m(av,dxt;-dxv). I Xj Xj Xj Xj Combining these equalities with (3.5), we have r-t / (v~"d.v,w)dt> -^ / (Cv~[Av,(dxv)2)dt ' rt - \&V~", (dx V)2) + m [ (Av,dxC-dx v)dt ' ' -1 i

4 110 KAZU YA HAY ASIDA From this and (3.4), (3.1 ' ), it follows that (3-6) [T\\Cl/2Vxd v\\2dt ' - (C^~a'(V)2) + f/ (C«T,At;,(^t;)V - f\*v,axjc dxv)dt - j\dxjt; vxv,vxdxv)dt + [ (VÇ-Vv,d2v)dt. ' Now by integration by parts (Cv-ld2v,(dxv)2)= -(dxc-v7l,(dxv)3) + (Cv 2,(dxv)4)-2(Cv ld2v,(dxv)2) Thus we have (Çv-ld2v,(dxv)2) so that (3.6) becomes = ±[(Çv 2,(dxv)4)-(dXjC-v-\(dxv)3)], (3.7) T\\C'\xdxv\\2dt ' I (VC-V^t/.ôJi/)^. f/o ' " - ^ Here we estimate the quantity (if-2, Hence (Ctr2ô which implies that v, (dxv)3) = 2((v~2dx v, (dx vf) (cl w)4). By integration by parts - 3(Ci> f6 «, ( \*/)2) - (dxjç-v~\ (dxv)3). (Çv 2,(dxv)4) = 3(Çv ldxv,(dxv)2) + (dx)c-v l,(dxv)3) <92(C,(d2xV)2)-r{(Cv-2,(dxV)4) + (dxjc-v-l,(dxvf), (Cv~2, (dx v)*) < 9(C, (d2v)2) + 2(dxC tt1, (dx v)').

5 REGULARITY PROPERTIES FOR SOLUTIONS OF THE CAUCHY PROBLEM 111 Since 2(dXjC-v'l,(dxv)i) < (Cv~l,(dxvf) + -\c\dxjq2,(dxv)2) for sufficiently small e > 0, we obtain (3.8) (Cv-2,(dxv)4)< T^-(C,(d2Xjv)2) + C(e)(rl(dX)C)2,(dxv)2). From this and Cauchy's inequality it follows that (3.9) (^C-^\(^t/)3) <277^y(C,(ö>)2) Now we define + C'(e)(C[(dXjC)2,(dxv)2). + ^ Z(Cv~ld2xv,(dxv)2), l<j<n. Using (3.9) and Cauchy's inequality we have for Ô > 0 ^e(i.(e^,»)v(íñ)k»-2.<v)4) 3a,,a2..X2N.,,.-1, ^r_(c,(a;yt/n + c(c:(dxjt;r,(dxvr), Easily, (.^ d2 v)2 < (N - 1) E/^(^v)2. Thus by (3.8) it follows that (3.10) Er ^ \ a,»r,x2 9 faô a\ 3ae 1 / v^/c.2 x2\. '^l^-1* +T^(T + 6j + 4Tr^)J(c'Ç(V)j + c^(cl(dxj02,(dxv)2). 1 Here we put ô (N - l)/3. Then From our assumption on m we see that 3aN/2 < 1. Next we estimate the remaining terms on the right-hand side of (3.7). For sufficiently small e' > 0 2\{^,dx i;-dxv) j + (dxjr_.vxv,vxdxv)-(vt;.vxv,d2xv)\ < Z IK1/2v * VII2 + C( )(Cl I vci2,1 v^ 2). j Therefore combining (3.7), (3.10) with the above, we conclude that 2 f IIC1/2v A^ 2 í < C[(Cw~a, iv^l2) J 0 + /'V1 VC 2, Vxi; 2)dt].

6 112 KAZUYA HAYASIDA Let (x) be in C^ (R ), and let us put =. Then from the above inequality and (3.3) it follows that for 1 < i, j < N (3.11) [\\C'\dxy\\2dt < C[(ayynr\\vv"\2),, _i; 2, n,2 r»%, h n m,,2,,.,,,amn_ b.,2, 2m, + ( V<S,(>') )-r- ^'-// -r-(m+//) IIV^'H +// ]. As is well known, vn \ v (r\ \ fs) in G, where t> = um and «is the solution of (1.1). For each positive integer n we put (x) = Ç (x), where '. (*) = {ô \x\ < n) (\x\>2n) and they are uniformly bounded in R up to the third derivatives. Then the constant C on the right-hand side of (3.11) are independent of n. Letn 7 ting n > 0 in (3.11), we see that dx dx i> -^ dxdxv weakly in L]oc(G) and /0r \\Çl/2dx dx v\\ dt < the right-hand side of (2.1). This completes the proof. References 1. D. G. Aronson, Regularity properties of flow through porous media, SIAM J. Appl. Math. 17 (1969), D. G. Aronson and P. Bénilan, Régularité des solutions de l'équation des mileux poreux dans RN, C. R. Acad. Sei. Paris. Sér. A 288 (1979), S. Benachour and M. S. Moulay, Régularité des solutions de l'équation des mileux poreux en une dimension d'espace, C. R. Acad. Sei. Paris, Sér. I 298 (1984), E. di Benedetto, Regularity results for the porous media equation, Ann. Mat. Pura Appl. (4) 121 (1979), P. Bénilan, A strong regularity V for solution of the porous media equation, Res. Notes in Math. 89, L. A. CafTarelli and A. Friedman, Regularity of the free boundary of a gas flow in a n- dimensional porous medium, Indiana Univ. Math. J. 29 (1980), L. A. Peletier, The porous media equation, in Application of nonlinear analysis in the physical sciences (H. Amman et al., editors), Pitman, New York, 1981, pp E. S. Sabinina, On the Cauchy problem for the equation of nonstationary gas filtration in several space variables, Soviet Math. 2 (1961), F. Simondon, Effect régularisant local pour u = (<p(u))xx, C. R. Acad. Sei. Paris, Sér. I 299 (1984), N. I. Wolanski, A diffusion problem with a measure as initial datum, J. Math. Anal. Appl. 102 (1984), , Department of Mathematics, Faculty of Science, Kanazawa University, Kanazawa 920, Japan