Nonlinear Partial Differential Equations in Applied Science: by Hiroshi Fujita, etc.

By Hiroshi Fujita, etc.

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Here we shall simply mention one of the problems which the theory addresses: functions g(u) : Rn + characterize the nonlinear which are continuous in the weak topology R when restricted to sequences of functions un(y) : Rm -+ Rn which satisfy linear constant coefficient partial differential constraints. As an example we mention a result from electrostatics which historically motivated the general theory. Consider vector fields z n *' R3 converging weakly in + and R3 wn: R3 -f R3 L2 2 n +z; w n + w .

3 . 2 . The S i n g u l a r Shadow L i m i t Solution Fig. 3. I. = (n 1). 2) ,rl*). ";(x) F i n a l l y , l e t u;(x) { be a f u n c t i o n o f tuated sense. the i n t e r v a l The shadow branch See, F i g . 3 . 2 f o r n 7- discontinuity of (;c~-K,z~+K) has u"*(x) } which h a s n boundary d i s c o n t i n u i t i e s . { v a r i a n t under t h e group a c t i o n D . 1. [-T,T]) (For t h e d e f i n i t i o n of q*, s e e ( O . l ) . ) ( and o f IT1). i s a f u n c t i o n of , t h e even ex- , t h e even e x t e n s i o n of which i s i n - &(x) } We have t h e f o l l o w i n g converges t o (u;(x),n*) = 1.

Roy. S o c . , 8237, 37-72 (1952). This Page Intentionally Left Blank Lecture Notes in Num. Appl. , 5 , 37-54( 1982) Noti/itieLir PDE in Applied Science. -Japrin Seminor, Tokyo, 1982 The Navier-Stokes l n i t i a l Value Problem i n Lp and R e l a t e d Problems Yoshikazu Giga '' Department o f Mathematics Faculty of Science Nagoya University Furo-cho, Chikusa-ku Nagoya 464 JAPAN We discuss the existence of a strong solution of the nonstationary Navier-Stokes system in Lp spaces. Our results generalize Lz results of Kato and Fujita.

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