By Michael Reed

Similar science & mathematics books

Haphazard reality: Half a century of science

Casimir, himself a well-known medical professional, studied and labored with 3 nice physicists of the 20th century: Niels Bohr, Wolfgang Pauli and Paul Ehrenfest. In his autobiography, the intense theoretician we could the reader witness the revolution that resulted in quantum physics, whose impression on smooth society grew to become out to be time and again higher than the 1st atomic physicists may have imagined.

Extra resources for Abstract Non Linear Wave Equations

Example text

To see what = that is i n v o l v e d is t h e solution ~(t) E D ( A 2 ) , ~ ' (t) ~ D(A) We must show that ~(t) 9 D ( A 2 ) , J(~(t)) u ( t ) E D ( B 3) - J(~(t))) = <0,- sum of three terms and of w h i c h inequality we have I lu(tl~ Cuct+~hl "H- uCtl ) - u t r <_ el IBuCt) I1~ I [ uCt+hl h - u(t) of p a r t and ut(t) s D ( B 2 ) . u(t+h) 3 _ u(t) h one let us is s t r o n g l y for J, Sobolev are - see Section > is <0, u(t)2(u,(,t+h). - u(t) )> h Thus, be the l (I]~) is s i m i l a r .

One global One then (36) using the in a w e a k sense. applies most tions. Therefore, = f(x) ut(x,o) = g(x) the the regularized inequality) one on D(A) = D ( B 2) ~ sections r We w i l l be clear Define take from Fn(X) are in the go o v e r the a way that approach -uP shows that u(t) of S t r a u s s satisfies which to r e a l - v a l u e d func- onl[: 0 (B L 2 (R n) D(B) of r e a l - v a l u e d and of c o u r s e introduction. initial the a r g u m e n t se- A + ~~ spaces to this to be the then argument section are of the r e g u l a r i z e d side attention for this There all use b a s i c a l l y {Un(t) } b y a c o m p a c t n e s s By a l i m i t i n g a r g u m e n t our L 2 ( R n) past hand (36) in such Un(t) the and given side follow D(B) = e -tA all n.

B of Section Note the h y p o t h e s e s ~o G D = ~ (r), the of C o r o l l a r y integral the solution of T the family t is u n i f o r m l y is g l o b a l 2, we can c o n c l u d e equa- {Mr} w h i c h and since is now equicontinuous. for all the cases w h e r e we got global that 2 interval w h i c h has the same bound. ( - T,T), is i n d e p e n d e n t same p r o o f works ( - T,T) for every on the w h o l e on all of ~ of strong interval (35) in the case that M t m = o, considered is u n i f o r m l y continuous existence in part for each - t, but only equicontinuous estimate corresponding to 55 on finite Finally, we make two remarks.