Order Form - FENOMEC

18 CHAPTER I. FUNDAMENTAL CONCEPTS AND EXAMPLES
Given any convex function U let F be a corresponding entropy flux (Example 1.9).
Multiplying (4.1) by U ′ (u) we obtain the entropy balance
∂ t U(u)+∂ x F (u) =εU(u) xx − εU ′′ (u) u 2 x,
which, formally as ε → 0, leads to infinitely many entropy inequalities
∂ t U(u)+∂ x F (u) ≤ 0.
□
Example 4.2. Conservation law with cubic flux revisited. The equation in Example
1.5 may be augmented with diffusive and dispersive terms, as follows
∂ t u + ∂ x u 3 = εu xx + δu xxx . (4.2)
where ε>0andδ ∈ I R. Using the quadratic entropy U(u) =u 2 we obtain
∂ t u 2 (3 u 4 ) (
+ ∂ x = ε u
2 )
2
xx − 2εu2 x + δ ( 2 uu xx − u 2 x)
x
= ε ( u 2) xx − 2εu2 x + δ ( (u 2 ) xx − 3 u 2 )
x
which, in the limit ε, δ → 0, yields the single entropy inequality
∂ t u 2 3 u 4
+ ∂ x ≤ 0. (4.3)
2
We will see later on (Theorem III-2.4 in Chapter III) that for solutions generated by
(4.2) the entropy inequality (3.8) does not hold for arbitrary entropies ! □
Example 4.3. Diffusive-dispersive conservation laws. Consider next the model
∂ t u + ∂ x f(u) =ε ( b(u) u x
)x + δ (c 1(u)(c 2 (u) u x ) x ) x
, (4.4)
where b(u) > 0 is a diffusion coefficient and c 1 (u),c 2 (u) > 0 are dispersion coefficients.
Let (U ∗ ,F ∗ ) a (strictly convex) entropy pair satisfying
U ∗ ′′ (u) = c 2(u)
c 1 (u) , u ∈ I R.
(U ∗ is unique up to a linear function of u.) Interestingly, the last term in the righthand
side of (4.4) takes a simpler form in this entropy variable û = U ∗(u), ′ indeed
δ (c 1 (u)(c 2 (u) u x ) x ) x
= δ (c 1 (u)(c 1 (u)û x ) x ) x
.
Any solution of (4.4) satisfies
∂ t U ∗ (u)+∂ x F ∗ (u) =ε (b(u) U ∗(u) ′ u x ) x
− εb(u) U ∗ ′′ (u) |u x | 2
+ δ ( c 1 (u)û (c 1 (u)û x ) x −|c 2 (u) u x | 2 /2 ) x .
In the right-hand side above, the contribution due to the diffusion decomposes into a
conservative term and a non-positive (dissipative) one. The dispersive term is entirely
conservative. In the formal limit ε, δ → 0 any limiting function satisfies the single
entropy inequality
∂ t U ∗ (u)+∂ x F ∗ (u) ≤ 0.
□
x