A particle-in-Burgers model: theory and numerics - Laboratoire de ...

Model and motivation Auxiliary steps Results h = 0: coupling h = 0: definition, uniqueness h = 0: numerics, existence The coupled problem
...Frozen particle: definition(s)...
Let us provide alternative characterizations of entropy solutions:
Proposition (equivalent definitions)
A function u ∈ L ∞ (R + ×R) is an entropy solution if and only if it satisfies any
of the following assertions:
A. The function u verifies the adapted entropy inequalities with
(c−, c+) ∈ G 1 λ ∪ G 2 λ.
B. The function u verifies the Kruzhkov entropy inequalities for all
nonnegative test function ϕ ∈ C ∞ c (R + ×R) such that ϕ|x=0 = 0,
moreover,
for a. e. t > 0 ((γ−u)(t), (γ+u)(t)) ∈ Gλ.
D. There exists C = C(λ,u∞, c±) such that the function u verifies
∀ϕ ∈ C ∞ c (R +
×R), ϕ 0
R +
[|u−c(x)|∂tϕ+Φ(u, c(x))∂xϕ] dx dt
R
+ |u0 − c(x)| ϕ(0, x) dx −C(ϕ)dist
R
(c−, c+), Gλ)
for all (c−, c+) ∈ R×R.