Order Form - FENOMEC

34 CHAPTER II. THE RIEMANN PROBLEM
wave, that is, the shock speed coincides with the right-hand characteristic speed:
f(u l ) − f ( ϕ ♮ (u l ) )
u l − ϕ ♮ = f ′( ϕ ♮ (u l ) ) , (2.9)
(u l )
that is, the rarefaction is “attached” to the shock.
It is obvious that, in Theorem 2.2 (as well as in Theorem 2.3 below) the Riemann
solution is monotone and, when it contains two waves, the intermediate state (specifically
here ϕ ♮ (u l )) depends continuously upon the data u l and u r and converges to
u l or to u r when passing from one case to another. These important properties of
classical solutions will no longer hold with nonclassical solutions. (See the weaker
statement after Theorem 4.1, below.)
Proof. Observe that in Case (c) above, after a right-contact wave one can add a
rarefaction fan, precisely because the left-hand of the rarefaction fan travels with
a speed faster than or equal to (in fact, equal to) the shock speed; see (2.9). In
view of (2.7) and (2.8), the function described in the theorem is an admissible weak
solution of the Riemann problem. To establish that this is the unique solution made
of elementary waves, we make the following observations:
• After a shock connecting u − to u + , no other wave can be added except when
u + = ϕ ♮ (u − ). (The shock is then a right-contact and can be followed with a
rarefaction preserving the monotonicity of the solution.)
• After a rarefaction connecting u − to u + , no other wave can be added except
another rarefaction.
We conclude that a Riemann solution is monotone and contains at most two
elementary waves. This establishes the desired uniqueness result.
□
When the flux is a convex-concave function, in the sense that
uf ′′ (u) < 0 (u ≠0), f ′′′ (0) ≠0,
lim f ′ (u) =−∞
|u|→+∞
and all of the entropies are enforced, we obtain
{ (
−∞,ϕ −♮ (u − )] ∪ [u − , +∞ ) , u − ≥ 0,
S(u − )= (
−∞,u− ] ∪ [ϕ −♮ (u − ), +∞ ) , u − ≤ 0,
(2.10)
(2.11)
and
⎧
⎪⎨ [0,u − ], u − > 0,
{ }
R(u − )= 0 , u− =0,
⎪⎩
[u − , 0], u − < 0.
We state without proof:
(2.12)
Theorem 2.3. (Riemann problem – Convex-concave flux.) Suppose that the function
f is convex-concave (see (2.10)) and fix some Riemann data u l and u r . Then,
the Riemann problem (1.1) and (1.2) admits a unique classical entropy solution in
the class P, made of shock waves satisfying all of the entropy inequalities (1.3) and
rarefaction waves which, assuming u l ≥ 0, is given as follows:
(a) If u r ≥ u l , the solution u is a (classical) shock wave connecting u l to u r .