Weak Solutions for a System Modeling the Movement of a Piston in a Viscous Compressible Gas
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Journal of Mathematical Fluid Mechanics
Weak Solutions for a System Modeling the Movement of a Piston in a Viscous Compressible Gas Julien Lequeurre Communicated by M. Hieber
Abstract. We first study the global-in-time existence of strong solutions to a one-dimensional system modeling the movement of a piston in a viscous compressible gas. Moreover, we prove the asymptotic stability of the solution toward a chosen constant state (in the sense that we can impose the final position of the piston, the final densities being fixed by the conservation of mass and the choice of the final position) thanks to a constant force acting in the equation of the point mass whose expression depends explicitly of the chosen final position. The norm of the solution in the function space of the initial data decays exponentially toward this constant state. Then, we prove the existence of weak solutions to this system for initial velocity in the energy state and for the initial density with bounded total variation. The weak solution is unique and also decay exponentially toward the chosen constant state thanks to the same constant force acting on the point mass. We use the result of existence of strong solutions to prove the existence of weak solutions, whereas the result on exponential decay of weak solution is independent of the one for the strong solutions.
1. Introduction 1.1. System We study the motion of a piston in a compressible viscous gas. The physical domain Ω = (−1, 1) is split into two parts. First the piston which is described by its constant mass m > 0 and its position h (depending on time t) and the gas domain Ωh(t) = Ω\{h(t)}. The gas is characterized by its density ρ > 0, its velocity v and its (isentropic) pressure p(ρ) = aργ (with a > 0 and γ > 1 two constants). We study the following system ρ(v˙ + vvx ) = (μvx − p(ρ))x ρ˙ + (ρv)x = 0 ˙ v(t, h(t)) = h(t) v(t, ±1) = 0 ¨ mh(t) = [μvx (t) − p(ρ(t))]h(t) + f (t) ˙ (h(0), ρ(0), v(0), h(0)) = (h0 , ρ0 , v 0 , g 0 )
in in in in in
QhT , QhT , (0, T ), (0, T ), (0, T ),
(1.1)
where μ > 0 is the constant dynamic viscosity of the gas, (h0 , ρ0 , v 0 , g 0 ) are the initial data of the problem and f a right-hand side (possibly a control function). The non cylindrical domain QhT is defined, for h : (0, T ) → (−1, 1) a smooth function by {t} × Ωh(t) QhT = t∈(0,T )
and, for a function θ : Ωh(t) → R, the jump [θ]h(t) is defined as [θ]h(t) = θ(h(t)+ ) − θ(h(t)− )
where
θ(h(t)+ ) =
0123456789().: V,-vol
lim
x>h(t), x→h(t)
θ(x),
θ(h(t)− ) =
lim
x 0) to (1.12)–(1.13) for initial data h0 ∈ (−1, 1), ρ0 in BV(Ωh0 ), v 0 in L2 (Ωh0 ) and g 0 ∈ R, right-hand side f in L2 (0, T ; R), if g 0 belongs to L2 (0, T ; R), vˆ to L4 (QhT ), ρˆ satisfies inf
0
(t,x)∈Qh T
ρˆ(t, x) > 0
and
sup
0
ρˆ(t, x) < +∞
(t,x)∈Qh T
and if (1.15) is satisfies in the following sense
T T u ˆ(t, x ˆ) ˙ 0,κ − ˆ) dˆ xdt − (−κ)g(t)ψ(t, h ) dt = ψ(0, x ˆ)dˆ x ψ(t, x ˆ) − vˆ(t, x ˆ)ψxˆ (t, x u0 (ˆ x) Ωκ0 Ωκ0 0 0 h
h
(1.16) and if (1.14) is satisfied in the following sense T T 0
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