# Applied and Industrial Mathematics, Venice—2, 1998: Selected by Marco Avellaneda, Robert Buff, Craig Friedman, Nicolas

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0, a reasonable value for TK can be guessed (as in [7]) by taking, instead of b'K in (38) the (limit) solution of a . \lbK = 1 in K with bK = on the inflow part of oK. This design of TK is effective. However, for more general E, even this simple case pre8ents relevant difficulties for the actual computation of (38). A first attempt to approximate in a Ruitable sense the bubble equation has been presented in [10,11]. ° 4 Cond usions The residual-free bubble approach (or equivalently the variational multiscale method) provides a general framework for the stabilization of Galerkin approximations to boundary value problems for PDE's.

Where does this shear banding occur? 1)-those points in (-1,1) where the souiton u(x, t) becomes infinite. For f(u) = e U , we show that T(uo) consists of a single point, the origin. 70) with p < 1 and 6 > 6* predicts shear band localization which occurs at the midpoint of the metal tube. More precisely, assume f(u) = eU , p < 1, 6 > 6*, and uo(x) E 2 C ( -1,1) radially symmetric and decreasing. 1) is radially symmetric, radially decreasing, anb blows up at finite time T > 0. The question of where blowup occurs is answered by the following theorem.

Thus, U2(X,t)::; <5k 1 - p rt supG(x,y,s)ds. ') 1 and supyG(x,y,s) rv G(:r, y, s) is integrable on [0, (0) for each Ilu(" t)lloo ::; Iluolioo + <5k 1 - p sup rcc sup G(x, y, s)ds. x Jo y This implies u(x, t) exists globally and is bounded provided P 2 1. 7s) has a global bounded solution u(x, t) for any E 2 and <5 > 0. Def 1. For a, (3, E G([-I, 1] x [0, T)), a(x, t) ::; (3(x, t), (x, t) E [-1,1] x [0, T), define Sn(x, t) {u E G([-I, IJ x [0, T) : a(y, s) ::; u(y, s) ::; (3(y, s) and u(x, t) = a(x, t)} , and S(3(x, t) {u E G([-1, 1] x [0, T)) : a(y, s) ::; u(y, 8) ::; (3(y, s) u(x, t) = (3(x; t)} .