Abstract differential equations and nonlinear mixed problems by Tosio Kato

By Tosio Kato

The current article relies at the Fermi Lectures I gave in may perhaps, 1985, at Scuola Normale Superiore, Pisa, during which i mentioned quite a few tools for fixing the Cauchy challenge for summary nonlinear differential equations of evolution style. right here I current a close exposition of 1 of those equipment, which bargains with “elliptic-hyperbolic” equations within the summary shape and which has purposes, between different issues, to combined initial-boundary worth difficulties for yes nonlinear partial differential equations, akin to elastodynamic and Schrödinger equations.

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To be able to state the regularity theorem for (LW), we have to formulate the compatibility condition. 2) into the language of the scale {JTy,]/;}. , = a r 7 ( 0 ) - £ ^ 01 - 0 ^ l/s) i-k Amk k=0 r = 1, 2 , . . , 5. 1). The compatibility condition of order s - 1 for (LL) is met if 4>r G Xg+i-r is actually in l/i = if' (hence in t/s+i-r) successively for r = 0 , 1, . . , s. 3, these considerations lead immediately to the following regularity theorem. 2. Assume conditions (LLl) to (LL4). Let 4>G i/s+i ond 4>G I/«.

T G /'). ABSTRACT EVOLUTION EQUATIONS. ETC. 41 Proof. 8) and 5¿г¿(0) = (j>r by straightforward computation; recall that the Pr are polynomials. Details may be omitted. 6. 5) to - v, with A' = etc. and with (j) replaced by 0i, etc. 0) + K T'sup{\\\idtA ” - 5i^)(i)|||(i:s,o);i e /'} + K sup{||(C7"(i, 0) - U(t, llo; t G /'} + i ||(C7"(i,r)-C/(t,r))«;y(r)||odr. >=> 0 The first term on the right tends to zero as n —►oo by hypothesis, since (jp in Ys+\. 11). 9 and the bounded convergence theorem. 13).

6. (continuous dependence) Let u E C(5+i,o)(/; (5^)) ^ solution to (N) with г¿(0) = (j). +i,o)(/;(F)). 7. (a) If г¿ e C (I\W ) is a solution of (N), then u e G^s+iMLAY)) and u(t) E E. 1). + xH to E. As shown above, 5 is a closed subset of c Ys+i. It is not a C*-manifold in the usual sense, but it accomodates the solution curve u E C(s+\,o)(LAY)) starting from each point 0 E H, which is smooth (up to order s+ 1) in the lower norms || ||;. 7. - Nonlinear evolution equations. 1. - The linearized equation.

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