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By Vinay D.S., Kwon B.S.

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A)v{d\{r) in f; b) g[tf](r) in ^ - ; c) T (r) in units of 300K; d) p(r) -p(re). T h e azimuthal heat flux q [6] has a boundary layer t o match the boundary layer of the velocity field. Note t h a t , by (8)3, b o t h b o u n d a r y layers essentially compensate each other so as t o ensure the vanishing of the shear stress. For an appreciation of the values of q [9] ( r ) , shown in Fig. 2, we compare t h e m with q [r] (rj) which, by ( l l ) i , amounts t o 1 0 3 ^ . T h u s the azimuthal heat flux is mostly smaller t h a n 1 ° / o o of the radial one in the circumstances considered.

Arch. Rational Mech. , 133:1-75, 1995. 11. A. Bressan, G. Crasta, and B. Piccoli. Well-posedness of the Cauchy problem for n x n conservation laws. Mem. Amer. Math. Soc, 694, 2000. 12. A. Bressan and P. Goatin. Olenik type estimates and uniqueness for n x n conservation laws. J. Differential Equations, 156:26-49, 1999. 13. A. Bressan and P. LeFloch. Uniqueness of weak solutions to systems of conservation laws. Arch. Rational Mech. , 140:301-317, 1997. 21 14. A. Bressan and M. Lewicka. A uniqueness condition for hyperbolic systems of conservation laws.

Fy the k - t h field is boundary characteristic k - 1 exiting i 1 * 4 1_ ^ entering fields Figure 1. The assumption on characteristic fields and boundary speed in the neighborhood of u. Next we consider the hyperbolic limit of the equation (4) under the hyperbolic rescaling (t,x) »—> (t/e,x/e). 1), which is a viscosity solution to the hyperbolic system with boundary ut+A{t,u)ux=0. (6) Note that the system is not in conservation form, and the flux matrix A depends explicitly on time. As a particular case, we construct the solution to the boundary Riemann problem u0(x) = u0, ub(t) — up.

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4-1BB by Vinay D.S., Kwon B.S.

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