Order Form - FENOMEC

44 CHAPTER II. THE RIEMANN PROBLEM
• The (second) formulation in Theorem 4.3 below allows us to eliminate the
classical Riemann solution still left out in the (first) formulation given in Theorem
4.1.
For a nonclassical shock connecting some states u − and u + at some speed λ =
a(u − ,u + ) we now write the kinetic relation in the form
{ Φ + (λ), u + u −
where, by definition, the kinetic functions Φ ± : [ f ′ (0), +∞ ) → IR
− are Lipschitz
continuous mappings satisfying
Φ ± (f ′ (0)) = 0,
Φ ±
is monotone decreasing,
Φ ± (λ) ≥ E ± (λ).
(4.6)
In the latter inequality, the lower bounds E ± are the maximum negative entropy
dissipation function defined by
E ± (λ) :=E ( u, ϕ ♮ (u) ) , λ = f ′ (ϕ ♮ (u)) for ± u ≤ 0. (4.7)
Observe that given λ>f ′ (0) there are exactly one positive root and one negative root
u such that λ = f ′ (u). This is why we have to introduce two kinetic functions Φ ±
associated with decreasing and increasing jumps, respectively. Note also that f ′ (0)
is a lower bound for all wave speeds. As we will see shortly, (4.5) is equivalent to a
relation
u + = ϕ ♭ (u − ),
from which we also define ϕ ♯ (u − ) as in (4.3).
Finally, in order to exclude the classical entropy solution we impose the following
nucleation criterion. For every shock connecting u − to u + we have
E(u − ,u + ) ≥ E(u − ,ϕ ♯ (u − )) =: E ♯ (u − ). (4.8)
This condition enforces that a discontinuity having an entropy dissipation larger than
the critical threshold E ♯ (u − ) must “nucleate”, that is, gives rise to nonclassical waves.
Theorem 4.3. (Riemann problem for concave-convex flux – Second formulation.)
Fix some kinetic functions Φ ± : [ f ′ (0), +∞ ) → IR
− (satisfying, in particular, (4.6)).
Then, under the assumptions of Theorem 3.5 the kinetic relation (4.5) selects a unique
nonclassical shock for each left-hand state u − . On the other hand, the nucleation criterion
(4.8) excludes the classical solution. As a consequence, the Riemann problem
admits a unique nonclassical entropy solution (in the class P), described in Theorem
4.1 above.
Proof. For u − > 0 fixed we claim that there is a unique nonclassical connection to
a state u + satisfying the jump relation and the kinetic relation (4.5). Let us write
the entropy dissipation as a function of the speed λ:
Ψ(λ) =E ( u − ,u + (λ) ) , λ = f( u + (λ) ) − f(u − )
u + (λ) − u −
.