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: uniqueness, comparison, L 1 contraction.
Theorem (L 1 contraction+comparison, analogous to Kruzhkov theory)
Let u0 and v0 be two initial data in L ∞ (R) and let u and v be the associated
entropy solutions. Then for all R > 0,
for a.e. t > 0
R
R
(u − v) + (t, x) dx
R+Lt
(u0 − v0)
−R−Lt
+ (x) dx
where L = max{u∞,v∞}. Consequently, if (u0 − v0) + ∈ L 1 (R), we have
for a.e. t > 0 (u − v)
R
+
(t, x) dx (u0 − v0)
R
+ (x) dx.
In particular, for all u0 ∈ L ∞ (R), there exists at most one solution and the map
S(t) : u0 ↦→ u(t,·) on its domain is an order-preserving L 1 contraction.
The proof is straightforward using
– the Kato inequality away from the interface (standard Kruzhkov)
– the characterization B. (“with traces” ) of entropy solutions
– and the dissipativity of Gλ .