Order Form - FENOMEC

30 CHAPTER II. THE RIEMANN PROBLEM
for some constants u − , u + ,andλ with u − ≠ u + , satisfies the infinite set of entropy
inequalities (1.3) if and only if Oleinik entropy inequalities
f(v) − f(u − )
≥ f(u +) − f(u − )
for all v between u − and u + (1.6)
v − u − u + − u −
are satisfied. Moreover, (1.3) and (1.6) imply Lax shock inequalities
f ′ (u − ) ≥ λ ≥ f ′ (u + ). (1.7)
According to the Rankine-Hugoniot relation (derived in Theorem I-2.3), the speed
λ in (1.5) is determined uniquely from the states u − and u + :
λ = a(u − ,u + ):= f(u +) − f(u − )
u + − u −
∫ 1
(1.8)
= a(u − + s(u + − u − )) ds,
0
where
a(u) =f ′ (u), u ∈ I R.
The (geometric) condition (1.6) simply means that the graph of f is below (above,
respectively) the line connecting u − to u + when u + u − , resp.). The
condition (1.7) shows that the characteristic lines impinge on the discontinuity from
both sides. The shock wave is said to be compressive and will be referred to as a
classical shock.
Theorem 1.2. (Lax shock inequality.) When the function f is convex all of the
conditions (1.3), (1.4), (1.6), (1.7), andLax shock inequality
are equivalent.
u − ≥ u + (1.9)
Proofs of Theorems 1.1 and 1.2. For the function in (1.5) the inequalities in
(1.3) are equivalent to (see Theorem I-5.1)
that is,
E(u − ,u + ):=−a(u − ,u + ) ( U(u + ) − U(u − ) ) + F (u + ) − F (u − ) ≤ 0,
E(u − ,u + )=
= −
= −
≤ 0,
∫ u+
u −
∫ u+
u −
∫ u+
U ′ (v) ( −λ + f ′ (v) ) dv
U ′′ (v) ( −λ (v − u − )+f(v) − f(u − ) ) dv
u −
U ′′ (v)(v − u − )
( f(v) − f(u− )
− f(u +) − f(u − )
)
dv
v − u − u + − u −
(1.10)
where (1.8) was used to cancel the boundary terms in the integration by parts formula.
Since U ′′ is arbitrary (1.10) and (1.6) are equivalent.
On the other hand, it is geometrically obvious that (1.6) is also equivalent to
f(v) − f(u + )
v − u +
≤ f(u +) − f(u − )
u + − u −
for all v between u − and u + . (1.11)