Slide 9
Slide 9 text
Implications of the assumptions on v
Well-posedness of the state equation [Dobrushin, Golse, ...]
For given u ∈ Uad
, v(ρ, u) as above and ρ(0, ·) = ρ0 ∈ P2
(Rd )
∂t
ρ(t, x) + ∇x · (v(ρt
, ut
)(x)ρ(t, x)) = 0 in [0, T] × Ω (1)
admits a unique solution ρin ∈ P(Ω) for any T > 0. We define ρ = S(u).
Stability estimate for the state equation
For weak solutions ρ, ¯
ρ of (1) with initial data ρ0
, ¯
ρ0
and controls u, ¯
u,
respectively, one can show that there exists positive constants a, b such
that
W 2
2
(ρt
, ¯
ρt
) ≤ W 2
2
(ρ0
, ¯
ρ0
) + b u − ¯
u 2
L2(0,T;RdM )
exp(at) (2)
for all t ∈ [0, T].
In particular, this shows the weak-continuity of the solution operator.
Example: v(ρt
, ut
) = −K1 ∗ ρt
+ K2
(·, ut
)
C. Totzeck (University of Wuppertal) OC with W2-metric 8/29