By M. Mimura, T. Nishida

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**Additional info for Recent Topics in Nonlinear Partial Differential Equations: v. 1 **

**Example text**

Now we proceed t o consider the expansions for 6 = @(t,x,r(t,x;6);6). D. 2. rJ 8 , and qx E with respect t o 6 E [0,11. and the radius of convergence 5, X-, P 0< for 5 < Fo, P is detemined by are Cm-functwns w i t h respect t o 6 E 10,ll o pp It\ < Proof. - p), unifonnty i s the same Banach space a8 X Here X- which is the inverse function of = S(t,x;6) It1 < a ( ; , p P and t h a t of x = x(t,[,1;6). They w i t h vatues i n XPS < Since xE(t,E,1;6) T3 V - > 0 (cf. 1;6): r(t,x;6) = ,1;6) i(t,x;6) = $(t,E,,l;6).

5). The velocity potential (x,Sy), (x,y) E Q ( t ) ,It1 @ It1 < a(pl - p), and i s the real p a r t o f the Hence we have cPlt,x,y;GI a[pg-pl, i s a harmonic function 6 E CO,~], i . e . , it s a t i s f i e s equation f l . l ) , and it i s i n f i n i t e Z y many times d i f f e r e n t i a b t e s with respect t o 6 E CO,11 w i t h values i n analytic functions of (x,y) in Water Waves and Friedrichs Expansion The d e r i v a t i v e s 0 Ox, Y and s i m i l a r expansions t o ( 4 . 7 1 , Proof.

And/or at x=O i n the i n t e r v a l T U" = [0,1], and - - the sdme except a t one o r two p o i n t ( s ) 1, t h e d e p t h o f N - s l i t s a r e determined by t h e genttalized V a ~ ~ ' e v a - F i 6 e - M h w rel. a o l . colzdLtion k, = k+( q ) LdKe x=l. As i s suggested i n [ 7 where U" a r e f u n c t i o n s o f rl c A,, : determined by See, F i g . l . 3 . We s h a l l show i n t h e n e x t s e c t i o n t h a t such s o l u t i o n s a c t u a l l y e x i s t , and can be c o n s t r u c t e d u s i n g t h e s i n g u l a r p e r t u r b a t i o n technique.