1. INTRODUCTION 3
Next, we consider a parabolic viscous perturbation of (1.1)
(1.4) L(6, u, d)u-e ] T d3 (Bjfk(b, u)dku) = F(6, u).
lj,kd
with Dirichlet boundary conditions:
(1.5) u\dn = Q-
Note that nonhomogeneous boundary conditions reduce to homogeneous ones, chang-
ing u into UQ + v, and adding i^o to the parameters 6Q- F ° r l n e parabolic problem
(1.5), (6, n) is allowed to range in a possibly larger open set O* containing O. For
small amplitude boundary layers, we may take 0 * = O BxU; however, in general
we wish to allow the situation that the solution of (1.5), within the boundary layer
near 90, may take on values of (6, u) that lie outside the domain of well-posedness
(i.e., hyperbolicity) of equation (1.1).
A S S U M P T I O N 1.1.
(HO) The Aj and Bj^ are N x N real matrices, C°° for (b,u) in O*; F is a
smooth function from O* to E ^ .
(HI) There is c 0 such that for all (6, u) G O* and all ( G l d the eigenvalues
°fYl(j,k=i€3€kBj,k(u) satisfy Re// c|£|2.
(H2) For all (6, u) G O, the eigenvalues ofJ2^jAj(b, u) are real and semi-simple
and have constant multiplicities for (b,u) G O :— B x U and $ G l d \ {0}.
(H3) There is c 0 such that for all (b,u) G O and ( G l r f the eigenvalues of
^ E j t i ^ j W + Ej,fc=i£j&Bj\fc(w) satisfy Re fi c|£|2.
(H4) For all (b,u) G Od, there holds det An(b,u) ^ 0.
The Assumption (HI) means that the perturbation
B(b,u,d):=Yldj(Bj,k(b,u)dk-)
is uniformly parabolic. (H2) means that L is hyperbolic, at least when the state
(6, u) remains in the domain O. The important Assumption (H4) means that the
boundary dtt is noncharact eristic for L. The Assumption (H3) is a compatibility
condition between L and B. For example, when B Ax is the Laplacian, (HI) is
trivial and (H3) follows immediately from (H2). When (1.1) is a system of conser-
vation laws which admits a strictly convex entropy rj(u), the system is symmetric
hyperbolic. If in addition, Re (j]n (u) YHj^Bj^i^)) is definite positive for all £ ^ 0,
then the assumptions (HI) and (H3) are satisfied.
The first step is to find the correct limiting boundary conditions for equation
(1.1). These come from the study of a natural "inner layer" o.d.e equation, for
b G Bd, (see [Gi-Se], [Rou], ):
(1.6)
An
(b,w)^-^(Bn(b^^)=0, u;(0 ) = 0,
where Bn(b,u) Yvj(x)vk(x)Bj,k(p,u). In (1.6) z stands for a fast variable in
the direction v. If (y,xn) *— » y + xnu(y) with y G 9Q and xn small parametrizes
a neighborhood of dfl, then z is a placeholder for xn/e. In what follows, what we
call a solution of (1.6), is a solution on [0, oo[ such that (b,w(z)) G 0 * for all z.
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