4 LARGE VISCOUS BOUNDARY LAYERS
One introduces the set
C {(6, u) G OQ ' (1.6) has a solution
(1.7)
such that u = lim w(z)\
Following [Gi-Se], [Rou], [Gr-Gu] the correct limit boundary conditions for (1.1)
read
(1.8) VxedQ, (b(t,x),u(t,x))eC.
Constants are solutions of (1.6) and Assumption (H4) implies that for all (b,u)
there is a family of solutions wt^v(z), depending on parameters v, such that WbjU,v
converges to u as z tends to infinity at an exponential rate (stable-central man-
ifold theorem). That one of these solutions connects u to zero is a condition on
(6, u) which defines C. That the connection can be chosen smooth with respect to
parameters is a transversality assumption.
To start the discussion, we first assume that for all b G BQ a family of solutions
of (1.6) is chosen, connecting 0 to a set of end states Cb C U(b).
ASSUMPTION
1.2. We are given a smooth manifold C C
OQ
such that for all
b G Be, Cb := {u G U(b) : (6, u) G C} ^ 0, and a smooth function W from C x [0, oo[
such that for all (b,u) G C, Wb)U W(b,u,-) is a solution of (1.6) and Wb,u(z)
converges to u when z tends to +oo? at an exponential rate, which can be chosen
uniform on compact subsets of C.
Assumption 1.2 is the natural analog of assumption (H4), [Z], made in the
planar shock theory.
The properties of C depend on the stability of w&}U solutions of ODE (1.6).
Consider the linearized equation of (1.6) around WblU
(1.9)
. , .dw , ., ,
N
,,dw d (
x
dw\
A ^ ) ^ + (A'n(w).w)---(Bn(W)-)dz\dz
dz n dz)
s ( c - « - ) ) S -
[w(0) - 0 .
These equations depend on the parameters b G Bg and the function w(z), with
w G
Coo([0,oc[;RiV).
The unknown is w.
DEFINITION
1.3. We say that the limiting boundary conditions (1.8) are transver-
sal if:
i) For all b G BQ, Cb is a smooth manifold of dimension 7V_ equal to the number
of negative eigenvalues of An(b, u).
ii) For (b,u) G C, the tangent space of Cb at u is the set of u such that the
linearized equation (1.9) with w = Wb,u n a s a (unique) solution w such that u =
lim^oo w(z).
In [Gr-Gu], it is proved that in the small amplitude case, i.e., for u in a suitably
small neighborhood of 0, there is a unique manifold C and connection W having
the properties above; moreover, the transversality condition is satisfied. They also
prove that the boundary conditions (1.8) are maximally dissipative, when u is
small and the parabolic term is the Laplace operator. In the large, we substitute
for maximal dissipativity the more general uniform Kreiss-Lopatinski condition.
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