Order Form - FENOMEC

14 CHAPTER I. FUNDAMENTAL CONCEPTS AND EXAMPLES
In view of (3.2) ( and (3.3)) the left-hand side of (3.5) converges in the weak sense: For
all θ ∈ Cc
∞ I R × (0, +∞) we have
∫∫
(
U(u ε ) ∂ t θ + F (u ε ) ∂ x θ ) ∫∫
(
dxdt → U(u) ∂t θ + F (u) ∂ x θ ) dxdt.
I R× I R + I R× I R +
To deal with the right-hand side of (3.5), we introduce the following definition.
Definition 3.2. (Entropy dissipative regularization.) The right-hand side R ε of
(3.1) is said to be entropy dissipative for the entropy U (in the limit ε → 0) if
∫∫
lim sup ∇U(u ε ) · R ε θdxdt≤ 0, θ ∈ C ∞ ( )
c I R × (0, +∞) , θ ≥ 0. (3.6)
ε→0 I R× I R +
□
We summarize our conclusions as follows.
Theorem 3.3. (Derivation of the entropy inequality.) Let u ε be a family of approximate
solutions given by (3.1). Suppose that u ε remains bounded in the L ∞ norm
as ε → 0 and converges almost everywhere towards a limit u; see(3.2) and (3.3).
Suppose also that the right-hand side R ε of (3.1) is conservative (see (3.4)) andentropy
dissipative (see (3.6)) for some entropy pair (U, F ) of (1.1). Then,u is a weak
solution of (1.1) and satisfies the inequality
∫∫
(
U(u) ∂t θ + F (u) ∂ x θ ) dxdt ≥ 0, θ ∈ Cc
∞
I R× I R +
By definition, (3.7) means that in the weak sense
(
R I × (0, +∞)
)
, θ ≥ 0. (3.7)
□
∂ t U(u)+∂ x F (u) ≤ 0, (3.8)
which is called the entropy inequality associated with the pair (U, F ). In the
following we shall say that a weak solution satisfying (3.8) is an entropy solution.
In the rest of this section we check the assumptions (3.4) and (3.6) for two classes
of regularizations (3.1). The uniform bound (3.3) is assumed from now on. Consider
first the nonlinear diffusion model
where the diffusion matrix B satisfies
∂ t u ε + ∂ x f(u ε )=ε (B(u ε ) ∂ x u ε ) x
, (3.9)
v · D 2 U(u) B(u)v ≥ κ |B(u)v| 2 ,
for all u under consideration and for some fixed constant κ>0.
v ∈ IR
N (3.10)
Theorem 3.4. (Zero diffusion limit.) Consider a system of conservation laws (1.1)
endowed with a strictly convex entropy pair (U, F ). Let u ε be a sequence of smooth
solutions of the model (3.9) satisfying the uniform bound (3.3), tending to a constant
state u ∗ at x → ±∞, and such that the derivatives u ε x decaytozeroatinfinity.
Suppose also that the initial data satisfy the uniform L 2 bound
∫
|u ε (0) − u ∗ | 2 dx ≤ C, (3.11)
IR
where the constant C>0 is independent of ε. Then, the right-hand side of (3.9) is
conservative (see (3.4)) and entropy conservative (see (3.6)).