Weak Convergence Methods for Nonlinear Partial Differential ...

Exercise: Prove that any smooth integral solution is a classical solution of (3.1).
We consider the special situation in 1 + 1 dimensions, where where u is a
smooth solution on R × [0, ∞)\C for a (smooth) curve C. Along C u is supposed
to hava a single jump discontiuity. Let V ⊂ R × (0, ∞) be open and such that C
bisects V into two parts V 1 (left of C) and V 2 (to the right of C): V = (C ∩ V ) ∪
·
·
V 1 ∪ V 2 .
Figure 3.2: Test volume around a shock.
For v ∈ C ∞ c
(V ) the integral solution u satisfies
∫ ∞ ∫ ∞
0 = u · ∂ t v + F(u) · ∂ x v dxdt
0 −∞
∫
∫
= u · ∂ t v + F(u) · ∂ x v dxdt + u · ∂ t v + F(u) · ∂ x v dxdt
V 1 V 2
Noting that v vanishes on ∂V , partial integration gives
∫
∫
∫
(
u · ∂ t v + F(u) · ∂ x v = − (∂ t u + ∂ x F(u)) · v + u
(V 1
) ) ν 2 + F(u (V1) )ν 1 · v.
V 1 V 1 C
Here u (V1) denotes the trace of u| V1 on C, ν = (ν 1 , ν 2 ) is the outer normal of V 1
along C. As ∂ t u + ∂ x F(u) = 0 within V 1 (cf. the above exercise), we obtain
∫
∫
(
u · ∂ t v + F(u) · ∂ x v = u
(V 1
) ) ν 2 + F(u (V1) )ν 1 · v.
V 1 C
Similarly,
∫
∫
u · ∂ t v + F(u) · ∂ x v = −
V 2
C
(
u
(V 2 ) ν 2 + F(u (V 2) )ν 1
)
· v.
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