We recall here the main definitions and results used in the Coquel-LeFloch methodology [10].
First, we need to define measure-valued solutions which extend the notion of weak solution. Instead of functions inL∞(R×]0, T[), we consider Young measures onR×]0, T[ that are weak-⋆ bounded applications
ν:R×]0, T[−→νx,t ∈Prob(R) such that for all functionsa∈C(R;R)
hνx,t, ai= Z
R
a(λ)dνx,t(λ), a.e.(x, t)∈R×]0, T[
Weak-⋆ boundedness means that (x, t) 7→ hνx,t, ai is in L∞(R×]0, T[). Moreover, it is assumed that all measures under consideration are supported in a compact set K ⊂ R, that is,
hνx,t, ai= 0 (A.1)
for all continuous functionsavanishing onK.
0.001 0.01 0.1
1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1
EL1
ε uni 2
mr 2 uni 4 mr 4 uni 8 mr 8
Figure 11: Relative L1-error between numerical and exact solution for the discontinuous initial condition (6.2). Multiresolution on 8 levels. 2, 4 and 8 cells on the coarsest level
Definition A.1. A Young measure ν:R×]0, T[−→Prob(R) is a measure-valued solution of (2.1) if it satisfies
∂thν, Idi+∂xhν, fi= 0 (A.2) in the usual sense of distributions.
We remark that if u ∈ L∞(R×]0, T[) is a weak solution of (2.1), then the Young measure defined by
νx,t =δu(x,t) (A.3)
is a measure-valued solution of (2.1). As in the weak sense, an entropy condition is necessary to ensure uniqueness of a measure-valued solution.
Definition A.2. A Young measure ν is consistent with the entropy inequality (2.2) asso-ciated with(η, q) if it satisfies
∂thνx,t, ηi+∂xhνx,t, qi ≤0 (A.4) in the sense of distributions. A measure-valued solution ν is an entropy measure-valued solution if it satisfies (A.4) for all entropy/entropy-flux pair (η, q).
By extension of Kruzkov’s uniqueness result [17], DiPerna’s theorem asserts that an entropy measure-valued solution is unique inL∞.
Theorem A.1. (DiPerna). Let u0 ∈L∞(R). Suppose that ν :R×]0, T[−→ Prob(R) is a Young measure satisfying
1. ν is a measure-valued solution of (2.1),
2. ν is consistent with all the entropy inequalities (A.4), 3. for all compact set K in R
limt→0 t>0
Z t
0
Z
K
hνx,s,|Id−u0(x)|idxds= 0. (A.5)
1e-06 1e-05 0.0001 0.001 0.01 0.1
0.001 0.01 0.1 1
ε
∆x0 disc
x1.01 smooth x1.1
Figure 12: Value of thresholdε for which the multiresolution error is of the order of the discretization error
If u∈L∞(R×]0, T[) is the unique weak entropy solution of (2.1), then
νx,t=δu(x,t), a.e.(x, t)∈R×]0, T[ (A.6) The following theorem provides a sufficient condition to ensure condition (A.5) of Theorem A.1
Theorem A.2. (DiPerna [13]). Let u0∈L∞(R). Suppose that ν :R×]0, T[−→ Prob(R) is a Young measure assuming u0 in the weak sense, i.e., for all ϕ∈C1
0(R;R), limt→0
t>0
1 t
Z t
0
Z
R
hνx,s, Idiϕ(x)dxds= Z
R
u0(x)ϕ(x)dx. (A.7)
Furthermore, assume that for allϕ∈C1
0(R;R)≥0and for at least one continuous strictly convex function η:R→R,
lim
t>0
1 t
Z t
0
Z
R
hνx,s, ηiϕ(x)dxds≤ Z
R
η(u0(x))ϕ(x)dx. (A.8) Then,ν assumes the initial condition in L1loc strongly (condition A.5 of Theorem A.1).
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