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220-HW3

# 220-HW3 - MATH 220 PROBLEM SET 3 DUE THURSDAY so 0(x = 0 if...

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MATH 220: PROBLEM SET 3 DUE THURSDAY, OCTOBER 15, 2009 Problem 1. Let ψ C ( R ) be given by ψ ( x ) = 0 , x< 1 , 1 + x, 1 <x< 0 , 1 x, 0 <x< 1 , 0 , x> 1 , so ψ 0, ψ ( x ) = 0 if | x | ≥ 1 and integraltext R ψ ( x ) dx = 1. Let ψ j ( x ) = ( jx ), so ψ j ( x ) = 0 if | x | ≥ 1 /j , and integraltext ψ j ( x ) dx = 1 for all j . (1) Show that ψ j δ 0 in D ( R ) (i.e., to be pedantic, ι ψ j δ 0 ). (Hint: this is essentially written up in the notes on distributions!) (2) Let φ C c ( R ) be such that φ ( x ) = 1 for | x | < 1. Show that { ι ψ 2 j ( φ ) } j =1 is not a convergent sequence in R . Use this to conclude that { ι ψ 2 j } j =1 does not converge to any distribution. (3) Show that there is no continuous extension of the map Q : f mapsto→ f 2 on C ( R ) (so Q : C ( R ) C ( R )) to D ( R ), i.e. there is no map ˜ Q : D ( R ) → D ( R ) such that ˜ Q ( ι f ) = ι Qf for every f C ( R ) and u j u in D ( R ) implies ˜ Qu j ˜ Qu in D ( R ). Problem 2. Consider the conservation law u t + ( f ( u )) x = 0 , u ( x, 0) = φ ( x ) , with f C 2 ( R ). Let v = f ( u ). Show that if

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220-HW3 - MATH 220 PROBLEM SET 3 DUE THURSDAY so 0(x = 0 if...

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