2

LARGE VISCOUS BOUNDARY LAYERS

case [HoZ] that the L1 norm of the Green's function is not necessarily bounded

in multi-dimensions, but in general may grow time-algebraically. This is a con-

sequence of focusing and spreading in the underlying hyperbolic propagation, the

effects of which are even more dramatic without parabolic regularization. Indeed,

examples given by Rauch [Ral] of

Lp

instability, p ^ 2, of smooth perturbations

of constant states give reason to believe that

L2

is the only LP norm in which we

can expect that multi-dimensional hyperbolic problems be well-posed.

Thus, we are restricted in multi-dimensions by the hyperbolic (or "outer") part

of the solution to seeking

L2

—

L2

bounds, analogous to but (even in the constant-

coefficient case) distinct from the

L1

Green's function bounds found in [Gr-Ro].

Moreover, we must obtain these bounds by a method suitable for the analysis

of both hyperbolic boundary-value problems and their parabolic regularizations.

To satisfy these requirements, we follow Kreiss' analysis of hyperbolic equations.

Our basic estimate concerns the L2 stability of the linearized equations, and is

proved using symmetrizers and a suitable extension of Kreiss' analysis to parabolic-

hyperbolic problems.

It can be seen by comparison with explicit representations of the resolvent in

the planar case, carried out respectively in [Ag] and [Z], that this basic estimate

is sharp for both hyperbolic and parabolic parts of the equations. Moreover, we

significantly relax the structural assumptions under which the parabolic results of

[Z] were obtained, just as the Kreiss analysis relaxed the assumptions necessary for

the hyperbolic results of [Ag] .

Consider a first order quasilinear system

d

(1.1) L(6,u,d)u := dtu + ] T A3(b,u)d3u = F(b,u)

J= I

The equation holds on R x ft where ft is a regular bounded domain in Rd. The

unknown u is valued in RN and b — b(t, x) will be a given function which represents

the variables (£, x) and the various possible source terms:

(1.2) &(*, x) = (£, x, 60(t, x)) e R 1+ d x RM° = R M .

For example, one can think of (1.1) as a system of conservation laws with unknown

bo+u where bo is some background variable state and u a perturbation. The function

6o can also appear as a forcing term in the right hand side. Since we will use change

of coordinates, we also include the variables (£, x) in the coefficient b. The parameter

b will vary in a domain B — [—T,T] x ft x BQ where BQ is a bounded open set in

RM°. The pair (6, u) will vary jointly within an open set O C B x R^, having the

form of a graph U6G;gZY(6) over B, where U(-) is a continuous set-valued function

from B to open sets in R^; in the simplest case, U(b) = U and O = B x U. (The

latter suffices to discuss small amplitude boundary layers, i.e., u small, or, more

generally, for boundary layers with small variation; see discussion below Definition

1.3.) We denote by Bd the set B = \-T,T\ x dft x B0, and Od := UbeBdU(b) the

restriction of O to dft. For (6, u) = (t,x,bo,u) G OQ, we denote by

d

(1.3) An(b, u) = Yl Vj(x)Aj{b, u)

i = i

the boundary matrix where v(x) = (^i(x),..., Ud(x)) is the inner unit normal vector

to dft at x.