K. O. Friedrichs's Advanced Ordinary Differential PDF

By K. O. Friedrichs

ISBN-10: 0677009607

ISBN-13: 9780677009605

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Example text

It should be noticed that the energy of v resides in the mean field w since E(v) = E(w) = E(wo) = EO; the fluctuations do not contribute to the energy. On the other hand, the microscopic fluctuations do contribute to the entropy integrals since in general it is not true that Ff(v) has the same value as Ff(w). We might say, therefore, that entropy is not conserved on a macroscopic scale, as part of it is lost to the infinitesimal-scale fluctuations of the vorticity. We have demonstrated that if the parametrized measure v is generated by a sequence of vorticity functions Wj arising from a solution of the Euler equations, then it must satisfy the above energy and entropy constraints.

We would like to show that the measure tfLul + (1 - t)fL U2 belongs to A. Let once again Xt stand for the characteristic function of (0, t) in (0,1) extended by periodicity to all of R, and write Xk(X) = Xt(kx . n) for any non-zero vector n ERN. As we have argued in Chapter 1, Xk ~ t in LaO (n). Consider We claim that fLu(k) ~ tfLul + (1-t)fLu2 in M(D x K). In fact for any continuous 1/J, lim k--+oo J 1/J(x, u(k) (x)) dx = lim r1/J (x, Xk(X)Ul (x) + (1 - Xk(X)) U2(X)) dx r [Xk(X)1/J(X,Ul(X)) k--+oo } 0 = lim k--+oo } II + (1 - Xk(X)) 1/J(x, U2(X))] dx =(tfLul +(1-t)fLu2,1/J)· 3.

13 and the comments after it, the parametrized measure associated to {( Uj, V'Uj)} is {Du(x) Q9 vx } xH! and follow the steps of the previous proof. The two theorems above are concerned with a particular sequence of functions and a particular energy density so that if inequality (3-2) is verified for a given parametrized measure and energy density

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Advanced Ordinary Differential by K. O. Friedrichs


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