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

D. Serre / Journal of Functional Analysis 236 (2006) 409–446 441
boundaries in a van der Waals fluid. Such a fluid is described by the usual Euler equations of an
inviscid, isentropic compressible fluid. The unknowns are the density ρ and the velocity v. The
equations govern the conservation of mass and momentum:
∂tρ + div(ρv) = 0, (51)
∂t(ρv) + Div(ρv ⊗ v) +∇xp = 0. (52)
The pressure is given as a non-monotone equation of state p = π(ρ). The function π is increasing
on (0,ρ−) (gas phase) and (ρ+, +∞) (liquid phase), while being decreasing over (ρ−,ρ+).
Notice that (51), (52) imply formally the conservation of energy
1
∂t
2 ρ|v|2
1
+ ρε(ρ) + div
2 ρ|v|2
+ ρε(ρ) + π(ρ) v = 0, (53)
where ε is the function defined by
dε
dρ
π(ρ)
= .
ρ2 The gas and liquid phases are the states for which system (51), (52) is hyperbolic. The sound
speed c is then the real number √ π ′ (ρ). Given a discontinuity across a smooth hypersurface, it
fulfills the Lax shock condition if the normal velocity V of the shock front satisfies the inequalities
(v · ν + c)+,(v· ν − c)−