Estimation optimale du gradient du semi-groupe de la chaleur sur le ...

440 D. Serre / Journal of Functional Analysis 236 (2006) 409–446
with ν := c0 min{λ,λ+ μ} and c0 is some suitable positive number, independent of the other
parameters.
The steady elliptic BVP. Because of the coincidence between ωP and ωS when τ = 0, the
calculation above does not give an answer for the steady BVP. This is due to our choice of the
modes, because we do not obtain the affine modes in the limit; it has the effect that becomes
trivial in this limit case. A more careful analysis would give us a non-trivial function .
Here, the differential equations imply
2
∂d −|η| 22 w = 0,
and the same for vd. The modes that decay at +∞ thus satisfy
∂d +|η| 2
w = 0, ∂d +|η| 2 vd = 0.
In other words, we have w = (αxd + β)exp(−|η|xd), with an analogous formula for vd. Going
back to the very ODEs, we find the general formula for the decaying modes:
w = (λ + μ)axd + b |η| 2 e −|η|xd ,
vd = (λ + μ)a|η|xd + (3λ + μ)a + b|η| e −|η|xd .
Inserting this into the boundary conditions, we obtain again a linear 2 × 2 system, whence a
correct Lopatinskiĭ determinant
(0,η)=−4λ(λ + μ)|η| 4 .
This shows that the elliptic BVP is well-posed if and only if λ + μ = 0.
Remark that the ill-posed elliptic BVPs form a hypersurface in the space of parameters (λ, μ),
despite the dimension three of the physical domain. This is due to the isotropy assumption. For
a non-isotropic energy density, we expect that these ill-posed steady problems form a set of
non-void interior. Then the transition between well-posed and ill-posed hyperbolic IBVPs would
occur along a boundary of this set.
6.4. Phase transition in a van der Waals fluid
Kreiss’ and Sakamoto’s theory of hyperbolic IBVPs has been adapted by A. Majda [12] to
treat the linearized stability of shock waves in systems of conservation laws (a second paper
[13] treats the non-linear stability). The context differs slightly, in the sense that the boundary
condition is coupled with a PDE that describes the evolution of the shock front. These boundary
conditions come from the linearization of the Rankine–Hugoniot condition. Despite these
technical differences, the same notions of Kreiss–Lopatinskiĭ condition, UKL and Lopatinskiĭ
determinant remain relevant.
Majda’s method is relevant for Lax shocks, where the Rankine–Hugoniot condition provides
the right number of boundary conditions. It has been adapted by H. Freistühler [7] to
so-called undercompressive shocks, when additional jump conditions are given in complement
of the Rankine–Hugoniot condition. This extension of the theory is particularly relevant to phase