Order Form - FENOMEC

6 CHAPTER I. FUNDAMENTAL CONCEPTS AND EXAMPLES
which implies
U(u) − U(v) −∇U(v) · (u − v) > 0, u ≠ v in U.
Definition 1.7 is illustrated now with some examples.
Example 1.9. Scalar conservation laws. When N =1andU = I R (1.15) imposes
no restriction on U, so that any (strictly convex) function U : U→R I is a (strictly
convex) mathematical entropy. The entropy flux F is given by F ′ (u) =U ′ (u) f ′ (u),
i.e.,
∫ u
F (u) :=F (a)+ U ′ (v) f ′ (v) dv
with a ∈U fixed and F (a) chosen arbitrarily.
Example 1.10. Decoupled scalar equations. Consider a system (1.1) of the form
u =(u 1 ,u 2 ,... ,u N ) T , f(u) =(f 1 (u 1 ),f 2 (u 2 ),... ,f N (u N )) T .
(For example, the linear systems in Example 1.6 have this form if they are written in
the characteristic variables.) Such a system is always hyperbolic, with λ j (u) =f j ′(u j),
and the basis { r j (u) } can be chosen to be the canonical basis of R
. The
1≤j≤N
system is non-strictly hyperbolic, unless for some permutation σ of { 1, 2,... ,N } we
have
f σ(1) ′ (u σ(1))