Seong Joo Kang. Let u be a smooth enough solution to a quasilinear hyperbolic mixed problem:

Transcription

1 Comm. Korean Math. Soc , No. 2, pp THE ENERGY INEQUALITY OF A QUASILINEAR HYPERBOLIC MIXED PROBLEM Seong Joo Kang Abstract. In this paper, e establish the energy inequalities for second order quasilinear hyperbolic mixed problems in the domain R n Introduction Let u be a smooth enough solution to a quasilinear hyperbolic mixed problem: 1 P 2 t, x, u, Du, Du = ft, x, u, Du in, ut, x, 0 = gt, x on, u0, x = u 0, u t 0, x = u 1, here = R n+1 + = {t, x, x n t, x R n, x n > 0} and = {t, x, x n x n = 0} is the boundary of, and 2 P 2 t, x, u, Du, D 2 t a i t, x, u, Du xi x i, 0,0 i,=0 is strictly hyperbolic ith respect to {t = constant}, f and g are smooth functions of its arguments and a i is symmetric. In this paper, e consider appropriate energy inequalities for equation 1. This result ill be effectively used in the proof of the regularity of conormal solutions to quasilinear hyperbolic mixed problems. Received September 1, Mathematics Subect Classification: 35L70. Key ords and phrases: quasilinear hyperbolic equation, mixed problems, Gronall s inequality. This research is supported by the Institute of Natural Science Fund 2000, Duksung Women s University.

2 226 Seong Joo Kang 2. Preliminaries Before e prove the energy inequalities e need the folloing remark and lemmas. We note that strictly hyperbolicity of P 2 t, x, u, Du, D implies the folloing inequality : 3 ξ 2 Cξ 2 + a 0 i t, x, u, Duξ i ξ, here C is a positive constant and ξ = ξ 0, ξ 1,, ξ n R n+1. Throughout this thesis, e use instead of for notational convenience. i, 0,0 i, 0,0 i,=0 For the proof of Schauder s lemma, see Rauch [9], and for the Gagliardo- Nirenberg inequalities, see Nirenberg [8]. Lemma 2.1. Schauder s lemma If u, v H s R n and s > n 2, then uv H s R n and uv H s C u H s v H s. Lemma 2.2. Gagliardo-Nirenberg inequalities Let u L q R n and its derivative of order m, D m u, belong to L r R n, 1 q, r. For the derivatives D u, 0 < m, the folloing inequalities hold 4 D u p constant D m u a r u 1 a q, here 1/p = /n + a1/r m/n + 1 a1/q, for all a in the interval /m a 1 the constant depending only on n, m,, q, r and a, ith the folloing exceptional cases: 1. If = 0, rm < n, q =, then e make the additional assumption that either u tends to 0 at infinity or u L q R n for some finite q > If 1 < r <, and m n/r is a nonnegative integer, then 4 holds only for a satisfying /m a < 1. Throughout this paper e treat the case in hich the regularity indices s and s are integers; analogous results hold in the general case.

3 The energy inequality 227 Lemma 2.3. Let 0 s s and suppose that L R n H s R n. Then for α = s, it follos that D α L 2p R n and D α L 2p C L 1 1 p H s 1 p, here p = s/s and C is a constant depending only on s, s and n. Corollary 2.4. Suppose that L loc Rn H s loc Rn for s 0. If f C R, then f H s loc Rn. 3. Energy inequality Since the equation 1 satisfies the uniform Lopatinski condition, the equation is ell-posed. See Chazarain-Piriou [5], or Sakamoto [10] for the definition of the uniform Lopatinski condition and the existence of solutions of the equation 1. No e state and prove the main theorem for equation 1. Theorem 1. Energy Inequality Let P 2 t, x, u, Du, D be a partial differential operator of order 2 on, as in 2, strictly hyperbolic ith respect to the planes {t = constant} and let ut, x CR; Hloc s C 1 R; H s 1 loc, s > n 2 + 2, satisfy the equation 5 P 2 t, x, u, Du, Du = ft, x, u, Du in, ut, x, 0 = gt, x on, here f is smooth function of its arguments. If u0, x Hloc s and u t 0, x H s 1 loc and the boundary of is not characteristic hypersurface of P 2, then, for all t, ut, x Hloc s. Moreover, if u has compact support in x for each time t, then e have 6 ut, x H s C t u0, x H s + u t0, x H s 1 for all t, here C t = Ct is independent of u and f. Proof. By finite propagation speed and an analysis local in time, it may be assumed that u has compact support in x for each t. Let α = s 1. We apply D α α 1 x 1 α n x n to equation 1 obtaining an

4 228 Seong Joo Kang equation, by the Leibniz formula, 7 P 2 D α u = D α f + β<α C β D α β a i D β u xi x h, i, 0,0 here C β are constants depending only on multiindices β. The use of the Leibniz formula is ustified belo, using Lemma 2.3. Let t, x = D α ut, x. Then 7 can be ritten as 8 P 2 t a i Du xi x = h. i, 0,0 Let Et be energy for equation 8 defined by 9 E 2 t = 2 t, x + t 2 t, x + a i Du xi t, x t, x. By differentiating 9 ith respect to t and integrating by parts, e have 2Et det dt = + 2 2t + 2 t + a i xi t x ai t xi = 2 + tt a i xi x i, 0,0 + 4 a 0 x 2 + ai t xi ai x i a n.

5 We note that 4 a 0 x = 4 so that 4 The energy inequality 229 = 4 n [ a 0 x = 2 a 0 x + 4 a 0n xn a0 x a 0 x a 0n a 0 x t + a 0n ]. It follos from 8 that Et det ai = dt t + h x x + 1 ai 2 t xi a0 x t 10 a n a n0. By letting ξ = =, x1,, xn in 3, e have 2 t + 2 x x n C 2 t + a i Du xi and so, by 9, i=1 2 x C E 2 t. i

6 230 Seong Joo Kang No e estimate the third term in 10. inequality imply that ai Du t xi 11 Ct Hölder s and Scharz s ai Du t t a i Du CtE 2 t, x 2 i xi L 1 xi 2 x 2 since D 2 ut, x H s 2 L, for all t, and s 2 > n 2. Similarly, for the second and fourth terms, e have ai Du a Du i x x x xi and a0 Du x Ct i=1 CtE 2 t, CtE 2 t. x 2 i a 0 Du x L 1 L 2 t t 2 For the fifth and sixth terms, by Scharz s inequality and trace theorem, e have a n Du a n Du L x 1 1 Ct x t 2 14 CtE 2 t, 1 2

7 and 15 The energy inequality 231 a n0 Du an0 Du L t 2 CtE 2 t, since Dut, x H s 1 L, for all t, and s 2 > n 2. For the first term, Scharz s inequality yields that + h = + h 16 CtE 2 t + h. It remains to estimate h. We ill shohat for 1 l s 1, D l a i Du D s+1 l u L 2. By the chain rule and the Leibniz formula, D l a i Du may be ritten as a sum of terms of the form b γ Du D α 1 Du D α m Du ith b γ smooth and α α m = l, here α k 1 for k = 1,, m. Let D 2 ut, x = vt, x. Then v L H s 2 since s 2 > n/2, and D l a i Du is ritten as a sum of terms of the form b γ DuD β 1 v D β m v ith β1 + + β m l 1, here β k 0 for 2s 2 k = 1,, m. As e kno b γ Du L and D β β k v L k by Lemma 2.3. Thus Hölder s inequality implies that the use of the chain rule as ustified, and since β β m /2s 2 l 1/2s 2, b γ DuD β 1 v D β 2s 2 m v L l 1 R n. Since D s+1 l u = D s 1 l v, 2s 2 D s+1 l u L s 1 l by Lemma 2.3. Therefore, by Hölder s inequality and Lemma 2.3, D l a i Du D s+1 l u L 2 and D l a i Du D s+1 l u L D l a 2 i Du D L s+1 l u p L q Ct v L v H s 2 since 1/p + 1/q = 1/2, here p = 2s 2/l 1 and q = 2s 2/s 1 l. D α fdu may be ritten as a sum of terms of the form f α Du D α 1 Du D α m Du ith f α smooth, α 1 + +α m = α and f α Du D α 1 Du D α m Du L 2. Moreover, by Lemma 2.3, D α fdu L 2 Ct Du H s 1 Ct u H s.

8 232 Seong Joo Kang By Scharz s inequality, Minkoski s inequality and the facts above, h = D α f + C β D α β a i D β u xi x β<α i, 0,0 1 { 1 t 2 2 D α f C β D α β a i 2 D β u xi x 2 2 β<α i, 0,0 CtEt D α f + v L 2 v L H s 2 CtEt u H s CtE 2 t, and so 17 + h CtE 2 t. From 11 15, 17 and by dividing 10 by Et, e have det 18 CtEt. dt By applying Gronall s inequality to 18, 19 Thus, for given t, Et CtE0, here Ct = e R t0 Cλdλ. ut, x H s Ct u0, x H s + u t 0, x H s 1. References [1] S. Alinhac, Interaction d ondes simple pour des équations complétement nonlineaires, Sém. d E.D.P., École Polytechnique, Paris, no. 8, [2] M. Beals, Propagation and interaction of singularities in nonlinear hyperbolic problems, Birkhäuser, Boston, [3] J. M. Bony, Interaction des singularités pour les équations aux dérivées partielles non-linéaires, Sem. Goulaouic-Meyer-Schartz, exp. no. 22, [4], Interaction des singularités pour les équations aux dérivées partielles non-linéaires, Sem. Goulaouic-Meyer-Schartz, exp. no. 2, [5] J. Chazarain and A. Piriou, Introduction to the theory of linear partial differential equations, North-Holland, [6] F. David and M. Williams, Propagation of singularities of solutions to semilinear boundary value problems, Bulletin of AMS , no. 2.

9 The energy inequality 233 [7] L. Hörmander, Lectures on nonlinear hyperbolic differential equations, Springer, [8] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa , [9] J. Rauch, Singularities of solutions to semilinear ave equations, J. Math. pure et appl , [10] R. Sakamoto, Hyperbolic boundary value problems, Cambridge univ. press, [11] M. Taylor, Pseudodifferential Operators, Princeton University Press, [12] F. Treves, Basic linear partial differential equations, Academic Press, Department of Mathematics Duksung Women s University Seoul , Korea skang@center.duksung.ac.kr