By Bertrand Mercier

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1) u(x,0) = u0(x) where u 0 e D(L) is given. 1) admits u ~ C0(0,r;H~(1)). ,t)ll < Cllu01Ls, for t s [0,T], S where T is positive and given. Proof: The proof of this result is an elegant applieaton of the theory of pseudo-differential operators (see M. Taylor, [17], pp. 62-65). content ourselves with establishing the a priori estimate in solution assuming it exists. 1) we have then, by setting K = [AS,L] --- ASL - LA s d 2 d 2 ~u ASu) + (hSu, A s ~u d'-~ llu(t)l]s = d'~ t]hSu(t)110 = (AS ~ ' ~-t) = - (ASLu, ASu) - (ASu,ASLu) = - (LESu, hSu) - (Ku, hSu) - (hSu ,LASu) ~ (hSu, Ku) = - (Ku, ASu) - (ASu, Ku), where we have utilized the antisymmetry of Since order K L.

Be the solution of the approximate adjoint problem aW N , (~--~--+ L WN,VN) = 0, for all v N ~ SN (WN(O) - ¢,VN) = 0, for all v N e SN. 2, if ~ e lip (I), lIW(t) - WN(t)li 0 < C N-Sll~lls+I for t < T. 7) (¢,UN(t) - u(t)) < C N-SH~Hs+itlu0fl 0. 7) as an error estimate in the Sobolev space of negative indices. 7) converges to zero as ~ is regular). This explains intuitively the success of the Fourier method with smooth- 41 ing, consisting Osher, [12]). of smoothing the initial solution By that we mean the following; let u 0 (see Majda-McDonoughp be a positive regular function with a compact support such that: / p(x)dx = I.

Furthermore, from the definition of the norm li. 1) (applied with r = T'I) IIY-PcYll0 < C(I+N2)-(T-I)/211ylIT_I C(l+N2)-(T-l)/2tiull . that 60 We have II(Lc-L)]NIi 0 < II(Pc-I)yll0 + H(Pc-I)a]NII I C(l+N2)(l-~)/2iluIIT + ;i(Pc-l)a]Nil I. 5) we have liauNiIy < CIIUNII~ This establishes ilauNflT, Cil uil T. 6) ilu(t)ll . 8) z ~u )--f- ~u 11~_~ii0 = il(l_PN) ~-t li0 ~ C(I+N 2 U~IIT-I < C(I+N 2) 2 l-r i-~ liullr' 61 where the last inequality is gotten by noting ~U B--~ = -Lu, IIL u II~ - 1 < Cllul} .