Partial Differential Equations - Modelling and ... - ResearchGate

62 E.J. Dean and R. Glowinski
Assuming that Ω is simply connected, we introduce:
u = {u 1 ,u 2 } = {∂ψ/∂x 2 , −∂ψ/∂x 1 },
v = {v 1 ,v 2 } = {∂ϕ/∂x 2 , −∂ϕ/∂x 1 },
ω = ∂u 2 /∂x 1 − ∂u 1 /∂x 2 ,
θ = ∂v 2 /∂x 1 − ∂v 1 /∂x 2 ,
V g = {v | v ∈ (H 1 (Ω)) 2 , ∇ · v =0, v · n = dg/ds on Γ },
V 0 = {v | v ∈ (H 1 (Ω)) 2 , ∇ · v =0, v · n =0onΓ },
( )
0 1
L = .
−1 0
Above, n is the unit vector of the outward normal at Γ and s is a counterclockwise
curvilinear abscissa on Γ . The formulation (71) is equivalent to
⎧
∫
Find u(t) ∈ V g for
∫
all t>0 such that
∫
⎪⎨
∂ω/∂t θ dx + ∇u:∇v dx = Lp : ∇v dx, ∀v ∈ V 0 ,
Ω
Ω
Ω
(72)
∂p/∂t + p + ∂I Qf (p)+L∇u =0,
⎪⎩
{u(0), p(0),ω(0)} = {u 0 , p 0 ,ω 0 }.
The problem (72) has a visco-elasticity flavor, −Lp playing here the role
of the so-called extra-stress tensor. Ast → +∞, we obtain thus at the limit a
(kind of) viscosity solution.
Acknowledgement. The authors would like to thank J. D. Benamou, Y. Brenier,
L. A. Caffarelli and P.-L. Lions for assistance and helpful comments and suggestions.
The support of NSF (grant DMS-0412267) is also acknowledged.
References
[Aub82] Th. Aubin. Nonlinear Analysis on Manifolds, Monge–Ampère Equations.
Springer-Verlag, Berlin, 1982.
[Aub98] Th. Aubin. Some Nonlinear Problems in Riemanian Geometry. Springer-
Verlag, Berlin, 1998.
[BB00] J.-D. Benamou and Y. Brenier. A computational fluid mechanics solution
to the Monge–Kantorovich mass transfer problem. Numer. Math.,
84(3):375–393, 2000.
[Cab02] X. Cabré. Topics in regularity and qualitative properties of solutions of
nonlinear elliptic equations. Discrete Contin. Dyn. Syst., 8(2):331–359,
2002.
[CC95] L. A. Caffarelli and X. Cabré. Fully Nonlinear Elliptic Equations.
American Mathematical Society, Providence, RI, 1995.
[CH89] R. Courant and D. Hilbert. Methods of Mathematical Physics, Vol. II.
Wiley Interscience, New York, 1989.