- 2 Let Us Fix T 0 Consider The Square 12 X R2 0 X1 1 0 X2 A And The Linear Evolution Problem Ux2x2 1 (199.5 KiB) Viewed 21 times
= 2. Let us fix T > 0. Consider the square 12 = {x € R2 : 0 < x1 <1,0 < x2 < A} and the linear evolution problem • Ux2x2 = f , in (0,T) x 12 Ut - UXX1 = u(0, x) = g(x) = , for χεΩ (1) for all t € [0,T] and x € 212, Oy, u(t, x) = = 0 where geL?(12) and f e L?((0,7), L?(32)). a) Let us set (5 marks] B(u, v) = Sou Vu(x).Vv(x)dx, Vu, v € H'(92). E Show that u u e L?([0, T], H'(92)) n H'([0,T], H'(12)*) (u'(t), 4)H1" ,H1 + B(u(t), 4) = (f(t), 4)2, VŲ € H'(12), for a.e. t € (0,7) = (2) u(0) = 9 is a weak formulation of the linear evolution problem (1), where g in the last equation is identified through the embedding L?(12) CH (12)* with an element of H(92)*.