Order Form - FENOMEC

16 CHAPTER I. FUNDAMENTAL CONCEPTS AND EXAMPLES
Relying on the uniform energy bound (3.13) we now prove (3.4) and (3.6). Relying
on (3.13) we find that for each test-function θ
∫∫
∣
R ε θdxdt∣ =
I +
I R× R
∫∫
∣
∣ε
∫∫
≤ ε
(
B(u ε ) u ε x
I R× I R +
I R× R
) ∣ ∣∣
x θdxdt
I +
|B(u ε ) u ε x||θ x | dxdt
≤ Cε‖θ x ‖ L 2 ( I R× I R +)
≤ C ′ ε 1/2 → 0,
( ∫∫
|B(u ε ) u ε x| 2 dxdt
I R× I R +
) 1/2
which establishes (3.4).
In view of (3.10) the second term in the right-hand side of (3.12) remains nonpositive.
On the other hand, the first term in (3.12) tends to zero since following the
same lines as above
∣
∣ε
∫ +∞
0
∫
IR
(
∇U(u ε )B(u ε ) u ε x)
x θdxdt ∣ ∣∣ ≤ Cε
∫∫
|B(u ε ) u ε x||θ x | dxdt
I R× I R +
≤ C ′ ε 1/2 → 0.
Thus (3.6) holds, which completes the proof of Theorem 3.4.
□
Next, we discuss another general regularization of interest, based on the entropy
variable û = ∇U(u) introduced in the end of Section 1. Recall that u ↦→ û is a change
of variable when U is strictly convex. Consider the nonlinear diffusion-dispersion
model
∂ t u ε + ∂ x f(u ε )=ε û ε xx + δ û ε xxx
= ε ∇U(u ε ) xx + δ ∇U(u ε (3.14)
) xxx ,
where ε>0andδ = δ(ε) ∈ IR
are called the diffusion and the dispersion parameters.
Diffusive and dispersive terms play an important role in continuum physics, as
illustrated by Examples 4.5 and 4.6 below. Understanding the effect of such terms on
discontinuous solutions of (1.1) will be one of our main objectives in this course.
Theorem 3.5. (Zero diffusion-dispersion limit.) Consider a system of conservation
laws (1.1) endowed with a strictly convex entropy pair (U, F ). Letu ε be a sequence of
smooth solutions of the diffusive-dispersive model (3.14) satisfying the uniform bound
(3.3), tending to a constant u ∗ at x →±∞, and such that u ε x and u ε xx decay to zero
at infinity. Suppose also that the initial data satisfy the uniform bound (3.11). Then,
the right-hand side of (3.14) is conservative (see (3.4)) and entropy dissipative (see
(3.6)) in the limit ε, δ → 0 with δ/ε → 0.
Again, combining Theorem 3.5 with Theorem 3.3 we conclude that solutions of
(3.14) can only converge to a weak solution of (1.1) satisfying the entropy inequality
(3.8). Note that (3.8) is derived here only for the entropy U upon which the
regularization (3.14) is based.