Nonlinear Evolution Equations. Global Behavior of Solutions by Alain Haraux

By Alain Haraux

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Additional info for Nonlinear Evolution Equations. Global Behavior of Solutions

Example text

What we intend to do now is to show the connection between this problem and the notion of entropy. 9) with the Brownian motion {~(s), 0 < s

In the above example one has h(pl~) = - S(p), and generally h has the meaning of a negative entropy. 38) an exact meaning we have to endow M(X) with a topology. 1. (cf. 21)). X) and Ve CO0. 38) is the following. Pronosltion 1, Let G c M(X) be open in the weak topology. X) be closed in the weak topology. 41) lim sup 1n l o g Qn(Lne A) < n--*** - inf h(pln). pEA We give the main steps of the proof of Proposition 1 in such a way that they can be extended immediately to much more general situations.

25) follows, since in the discrete case too positivity of the Ljapunov exponent implies absence of absolutely continuous spectrum (the proof is similar as in the continuous case). 26) is the following. e. ~ R. Subtle resonances occur which lead to a singularly continuous spectrum. e.. The resonances, which can occur in the case of the Mathieu operator on Z, are therefore excluded, if the boundary condition 0 is fixed, and one has indeed localization for Lebesgue - almost all 0e [0,~). 5. E x p l a n a t i o n of T h e o r e m 1 and introduction to an extended B o l t z m a n n theory of entropy This section is devoted to an explanation of Theorem 1.

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