A particle-in-Burgers model: theory and numerics - Laboratoire de ...
A particle-in-Burgers model: theory and numerics - Laboratoire de ...
A particle-in-Burgers model: theory and numerics - Laboratoire de ...
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
Mo<strong>de</strong>l <strong>and</strong> motivation Auxiliary steps Results h = 0: coupl<strong>in</strong>g h = 0: <strong>de</strong>f<strong>in</strong>ition, uniqueness h = 0: <strong>numerics</strong>, existence The coupled problem<br />
...Frozen <strong>particle</strong>: <strong>de</strong>f<strong>in</strong>ition(s)...<br />
Let us provi<strong>de</strong> alternative characterizations of entropy solutions:<br />
Proposition (equivalent <strong>de</strong>f<strong>in</strong>itions)<br />
A function u ∈ L ∞ (R + ×R) is an entropy solution if <strong>and</strong> only if it satisfies any<br />
of the follow<strong>in</strong>g assertions:<br />
A. The function u verifies the adapted entropy <strong>in</strong>equalities with<br />
(c−, c+) ∈ G 1 λ ∪ G 2 λ.<br />
B. The function u verifies the Kruzhkov entropy <strong>in</strong>equalities for all<br />
nonnegative test function ϕ ∈ C ∞ c (R + ×R) such that ϕ|x=0 = 0,<br />
moreover,<br />
for a. e. t > 0 ((γ−u)(t), (γ+u)(t)) ∈ Gλ.<br />
D. There exists C = C(λ,u∞, c±) such that the function u verifies<br />
∀ϕ ∈ C ∞ c (R + <br />
×R), ϕ 0<br />
R +<br />
<br />
[|u−c(x)|∂tϕ+Φ(u, c(x))∂xϕ] dx dt<br />
R<br />
<br />
+ |u0 − c(x)| ϕ(0, x) dx −C(ϕ)dist<br />
R<br />
(c−, c+), Gλ)<br />
for all (c−, c+) ∈ R×R.