Math 444 — Spring 2001 — Test
Total points: 100. Do all questions.
Instructions: Show ALL your working and make your explanations as full as possible,
unless the question says otherwise. Calculators are not allowed on this exam; neither
are books or notes.
If you apply a theorem from class, you need not state the theorem but you must
demonstrate that its hypotheses are satisfied.
Green’s Formulas:
ZΩ
[v∆u+∇v· ∇u]dx =Z∂Ω
v∂u
∂ν dS
ZΩ
[v∆u−u∆v]dx =Z∂Ω·v∂u
∂ν −u∂v
∂ν ¸dS
1: (25 points) Assume Gis a smooth function and u(x, y) is a weak solution of the
conservation law
G(u)x+uy= 0.
(a) Show that if uis in fact a smooth solution except for a jump across the C1-curve
x=ξ(y), then the direction of the curve is related to the jump by
ξ0(y) = G(u`)−G(ur)
u`−ur
.
(b) State the entropy condition, say what it means physically, and illustrate with a
diagram. (If you like, you can assume in part (b) that G(z) = 1
2z2,i.e. Burgers’
equation.)
2: (25 points) Let f(x, t) be smooth.
(a) State Duhamel’s Principle for solving the nonhomogeneous wave equation ztt −
c2∆z=fin Rn(with zero initial conditions) in terms of the solutions to certain
homogeneous problems. (You are not required to solve these homogeneous problems.)
(b) Prove Duhamel’s Principle. That is, show that your formula for z(x, t) really
does solve ztt −c2∆z=f.
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