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: understanding the coupling.
A “better” (indirect) proof comes from the general theory from AKR .
First, property (i) is actually equivalent to the “Kato inequality” (⇔
L 1 -dissipativity)
+ +
− (u−v) ∂tϕ+Φ (u, v)∂xϕ 0 ∀ϕ ∈ D(Q), ϕ 0.
R +
R
for the solutions
u(t, x) := u−1l{x0}, v(t, x) := v−1l{x0}
of our equation; and the Kato inequality comes by passage to the limit from
the LeRoux approximation case :
ε ε + ε ε + + ε ε
λ(u −v ) (∂xHε)ϕ−(u −v ) ∂tϕ−Φ (u , v )∂xϕ 0.
R +
R
Further, property (ii) means that “G 1 λ ∪ G 2 λ is a definite germ of which Gλ is the
unique maximal extension”. This follows (with some work) from the fact that
Gλ is a complete germ (⇔ the germ allows to solve every Riemann problem).