Analysis of a system modelling the motion of a piston in a viscous - - PowerPoint PPT Presentation

Analysis of a system modelling the motion of a piston in a viscous gas

Debayan Maity

Institute De Mathematiques de Bordeaux

June 21, 2016 Joint work with Tak´ eo Takahashi and Marius Tucsnak.

Outline

1 Introduction 2 Known results 3 Local in time existence 4 Global Existence 5 Further Comments

Setting up the problem

• We consider a one dimensional model for the motion of a particle

(piston) in a cylinder filled with a viscous compressible gas. −1 1 h(t)

• Gas-piston system evolves in the interval (−1, 1) and

h : [0, ∞) → (−1, 1) denotes the position of the particle.

• The extremities of the cylinder are fixed, but the gas is allowed to

penetrate inside the cylinder.

• The gas is modelled by the 1D compressible Navier-Stokes

equations, whereas the piston obeys Newton’s second law.

Governing equations

Motion of the gas: described in the Eulerian coordinate system by its density ρ = ρ(t, x) and the velocity u = u(t, x), which satisfy the one dimensional compressible Navier-Stokes system in ∂tρ + ∂x(ρu) = 0, t ≥ 0, x = h(t) ρ(∂tu + u∂xu) − ∂xxu + ∂x(ργ) = 0, t 0, x = h(t) (1.1) where γ 1. Motion of the Piston: u(t, h(t)) = ˙ h(t) (t 0), m¨ h(t) = [∂xu − ργ](t, h(t)) (t 0), where m is the mass of the piston and the symbol [f ](t, x) stands for the jump at instant t of f at x, i.e., [f ](t, x) = f (t, x+) − f (t, x−). The position of the piston (and, consequently, the domain occupied by the gas) is one of the unknowns of the problem, we have a free boundary value problem.

Initial Condition:

• h(0) = h0,

˙ h(0) = ℓ0, u(0, x) = u0(x), ρ(0, x) = ρ0(x) (x ∈ [−1, 1] \ {h0}). (1.2) Boundary Condition: u(t, −1) = 0, u(t, 1) = 0 (t 0), (1.3)

• u(t, −1) = u−1(t) > 0,

u(t, 1) = 0 (t 0), ρ(t, −1) = ρ−1(t) (t 0), (1.4)

• r

       u(t, −1) = u−1(t), u(t, 1) = −u1(t) (t 0), u−1(t) > 0, u1(t) > 0 (t 0), ρ(t, −1) = ρ−1(t) (t 0), ρ(t, 1) = ρ1(t) (t 0). (1.5)

Outline

1 Introduction 2 Known results 3 Local in time existence 4 Global Existence 5 Further Comments

Known Results in 1D

• Shelukhin, , 1978 :
• Global in time existence of classical solutions, with initial

conditions : u0 ∈ C 2+α, ρ0 ∈ C 1+α, 0 < α < 1 and homogeneous boundary condition.

• Shelukhin, 1982:
• Similar result as above when gas and the piston are supposed

to be heat conducting, and homogeneous boundary conditions.

• Antman and Wilber, 2007 :
• asymptotic behavior of solutions as the ratio of the mass of the

gas and of the mass of the piston tends to zero.

Known results in higher dimension:

Rigid Structure immersed in Compressible fluid :

• Desjardins and Esteban, 2000 :
• Global existence of a weak solution for γ ≥ 2 and upto

collision.

• Feireisl, 2003 :
• Global existence of a weak solution for γ > N/2 and regardless
• f possible collisions of two or more rigid bodies and/or a

contact of the bodies with the boundary

• Boulakia and Guerrero,2009 :
• Global in time strong solution for small initial data.
• Hieber and Murata 2015 :
• Local in time strong solution in Lp − Lq setting.

Goal

Existence and uniqueness of global in time strong solutions of the initial and boundary value problem.

• non homogeneous boundary conditions.
• less regular initial data.

Theorem (D.M, Tak´ eo Takahashi and Marius Tucsnak)

• Let T > 0 and assume that h0 ∈ (−1, 1), ℓ0 ∈ R
• u0 ∈ H1(−1, 1) and u0(h0) = ℓ0
• ρ0 ∈ H1(−1, h0) ∩ H1(h0, 1) and ρ0(x) > 0 ∀ x ∈ [−1, 1] \ {h0}.
• u−1, u1 ∈ H1(0, T), ρ−1, ρ1 ∈ H1(0, T) and

ρ−1(t) > 0, ρ1(t) > 0 Then, the initial and boundary value problem formed by (1.1), (1.2) and (1.5) admits a unique strong solution on [0, T].

Goal

Existence and uniqueness of global in time strong solutions of the initial and boundary value problem.

• non homogeneous boundary conditions.
• less regular initial data.

Theorem (D.M, Tak´ eo Takahashi and Marius Tucsnak)

• Let T > 0 and assume that h0 ∈ (−1, 1), ℓ0 ∈ R
• u0 ∈ H1(−1, 1) and u0(h0) = ℓ0
• ρ0 ∈ H1(−1, h0) ∩ H1(h0, 1) and ρ0(x) > 0 ∀ x ∈ [−1, 1] \ {h0}.
• u−1, u1 ∈ H1(0, T), ρ−1, ρ1 ∈ H1(0, T) and

ρ−1(t) > 0, ρ1(t) > 0 Then, the initial and boundary value problem formed by (1.1), (1.2) and (1.5) admits a unique strong solution on [0, T].

Strategy:

We follow a classical strategy:

• Existence and uniqueness of local in time strong solution.
• To use fixed point argument.
• We first rewrite the system in a fixed domain.
• We rewrite the system in Lagrangian mass co-ordinate.
• Derive a priori estimates to show we do not have “contact,”

“vaccum” or “blow-up” in finite time.

Outline

1 Introduction 2 Known results 3 Local in time existence 4 Global Existence 5 Further Comments

Lagrangian-mass transformation

Let ξ = Ψ(t, x), where Ψ(t, x) is the signed mass of the gas filling the domain between h(t) and x at instant t. More precisely, we set ξ = Ψ(t, x) = x

h(t)

ρ(t, η) dη (t 0, −1 x 1). (3.1) Assume (ρ, u) is a smooth enough solution of (1.1), (1.2) and (1.5) (this means, in particular, the function ρ is positive and bounded away from zero). Then, for every t 0 the map x → Ψ(t, ·), is a C 1 diffeomorphism which is strictly increasing from (−1, 1) to (−ξ−1(t), ξ1(t)), where ξ−1(t) = h0

−1

ρ0(η) dη + t ρ−1(s)u−1(s) ds, ξ1(t) = 1

h0

ρ0(η) dη + t ρ1(s)u1(s) ds, for every t ∈ [0, T]. Moreover, Ψ(t, h(t)) = 0 (t ∈ [0, T]).

System in moving known domain

Let Φ(t, ·) = Ψ−1(t, ·) and we set

• ρ(t, ξ) = ρ(t, Φ(t, ξ)),
• u(t, ξ) = u(t, Φ(t, ξ)),

We have the following system in (0, T) × (−ξ−1(t), ξ1(t)), ξ = 0 ∂t ρ + ρ2∂ξ u = 0 ∂t u − ∂ξ ( ρ(∂ξ u)) + ∂ξ( ργ) = 0,

• u(t, 0) = ˙

h(t), m¨ h(t) = [ ρ (∂ξ u) − ργ] (t, 0),

• u(t, −ξ−1(t)) = u−1(t),

u(t, ξ1(t)) = −u1(t),

• ρ(t, −ξ−1(t)) = ρ−1(t),
• ρ(t, ξ1(t)) = ρ1(t)
• ρ(0, ξ) :=

ρ0(ξ) = ρ0(Φ(0, ξ))

• u(0, ξ) :=

u0(ξ) = u0(Φ(0, ξ)) h(0) = h0, ˙ h(0) = ℓ0.

System in fixed known domain:

we define y = Γ(t, ξ) =

• ξ

ξ−1(t)

for ξ ∈ [−ξ−1(t), 0],

ξ ξ1(t)

for ξ ∈ [0, ξ1(t)]. and by setting ζ(t, y) =

• ρ(t, Γ−1(t, y))

−1 , u(t, y) = u(t, Γ−1(t, y)), we have the following system in (0, T) × (−1, 1), y = 0 ∂tζ + β∂yζ − α∂yu = 0 ∂tu + β∂yu − α∂y α ζ ∂yu

• + α∂y

1 ζγ

• = 0

u(t, 0) = ˙ h(t), m¨ h = α ζ ∂yu − 1 ζγ

• (t, 0)

u(t, −1) = u−1(t), u(t, 1) = −u1(t), ζ(t, −1) = 1 ρ−1(t), ζ(t, 1) = 1 ρ1(t) + initial conditions

where α(t, y) =        1 ξ−1(t) for y ∈ [−1, 0), 1 ξ1(t) for y ∈ (0, 1] β(t, y) =          −y ˙ ξ−1(t) ξ−1(t) for y ∈ [−1, 0), −y ˙ ξ1(t) ξ1(t) for y ∈ (0, 1],

• β = 0, in case of homogeneous boundary condition.

Construction of Fixed point map

Let f1 ∈ L2(0, T; L2(−1, 1)) and f2 ∈ L2[0, T] and consider the following system ∂tζ + β∂yζ − α∂yu = 0, ∂tu − α0∂y α0 ζ0 ∂yu

• = f1,

u(t, ±0) = ˙ h, m¨ h = α0 ζ0 ∂yu

• (t, 0) + f2,

+ initial conditions and boundary conditions

• Linearization preserves the coupling of the equations of the fluid and
• f the structure.
• essential in our approach for obtaining the local existence result for

initial data less regular than other works

Construction of Fixed point map

Let f1 ∈ L2(0, T; L2(−1, 1)) and f2 ∈ L2[0, T] and consider the following system ∂tζ + β∂yζ − α∂yu = 0, ∂tu − α0∂y α0 ζ0 ∂yu

• = f1,

u(t, ±0) = ˙ h, m¨ h = α0 ζ0 ∂yu

• (t, 0) + f2,

+ initial conditions and boundary conditions

• Linearization preserves the coupling of the equations of the fluid and
• f the structure.
• essential in our approach for obtaining the local existence result for

initial data less regular than other works

Construction of Fixed point map

Let f1 ∈ L2(0, T; L2(−1, 1)) and f2 ∈ L2[0, T] and consider the following system ∂tζ + β∂yζ − α∂yu = 0, ∂tu − α0∂y α0 ζ0 ∂yu

• = f1,

u(t, ±0) = ˙ h, m¨ h = α0 ζ0 ∂yu

• (t, 0) + f2,

+ initial conditions and boundary conditions

• Linearization preserves the coupling of the equations of the fluid and
• f the structure.
• essential in our approach for obtaining the local existence result for

initial data less regular than other works

BT = f1 f2

• ∈ L2(Q0,T) × L2[0, T]
• f1L2(Q0,T∗) + f2L2[0,T∗] 1
• ,

We consider the map    N : BT → L2(Q0,T) × L2[0, T], f1 f2

• →

F1(ζ, u) F2(ζ, u)

• .

where F1(ζ, u) = α∂y α ζ ∂yu

• − α0∂y

α0 ζ0 ∂yu

• − α∂y

1 ζγ

• − β∂yu,

F2(ζ, u) = α ζ − α0 ζ0

• (∂yu)
• (t, 0) −

1 ζγ

• (t, 0).
• To show, there exists T∗ small enough such that N is a strict

contraction in BT∗

Outline

1 Introduction 2 Known results 3 Local in time existence 4 Global Existence 5 Further Comments

Global in time Existence

Main steps to prove global existence:

• Assume all the hypothesis and (h, ρ, u) is a strong solution solution

defined on [0, ˜ τ] for every ˜ τ ∈ (0, τ).

• To prove global in time existence of solution, enough to show the

following :

• No contact: −1 < h(t) < 1, for all t ∈ [0, τ)
• No Vacuum:

inf

t∈[0,τ)

x∈[−1,1]\{h(t)}

ρ(t, x) C.

• No blow-up:

supt∈[0,τ)

• u(t, ·)|H1(−1,1)+ρ(t, ·)H1(−1,h(t))+ρ(t, ·)H1(h(t),1)
• C

Mass and energy estimates

• For every t ∈ [0, τ) we have

h(t)

−1

ρ(t, x) dx C, 1

h(t)

ρ(t, x) dx C.

• Integrating the density equation.
• Energy estimate: There exists a strictly positive constant C such

that 1

−1

ρ(t, x)u2(t, x) dx + 1

−1

ργ(t, x) dx + ˙ h2(t) C (t ∈ [0, τ)).

• Multiplying the velocity equation by u and integrating by parts.

bound of ρ

We define the auxiliary function B(t, x) in [−1, 1] \ {h(t)} satisfying the following properties ∂xB = ρu, ∂tB = ∂xu − ργ − ρu2, B(0, x) = B0(x) = x

h0

ρ0(y)u0(y)dy.

• for every t ∈ [0, τ) we have

sup

x∈[−1,1]\{h(t)}

|B| C

• 1 +

1

−1

ρ(t, x)u2(t, x) dx 1/2 + t 1

−1

(ργ(σ, x) + ρ(σ, x)u2(σ, x)) dx dσ

• .
• In particular

sup

x∈[−1,1]\{h(t)}

|B| C

Upper Bound of density and no contact

• ∂

∂t (ρeB) + u ∂ ∂x (ρeB) + ργ+1eB = 0,

• We first get

sup

−1x<h(t)

ρ(t, x) Mexp   2 sup

σ∈[0,τ)

−1x<h(σ)

|B(σ, x)|    ≤ C,

• From the relation:

h(t)

−1

ρ(t, x) dx = h0

−1

ρ0(x) dx+ t ρ−1(σ)u−1(σ) dσ (t ∈ [0, τ))

• we get

1 + h(t) ≥ C(1 + h0) > 0

• h(t) − 1 < 0.

Lower Bound of density

• ∂

∂t (1 ρe−B) + u ∂ ∂x (1 ρe−B) − ργ−1e−B = 0. sup

−1xh(t)

1 ρ(t, x) M exp   2 sup

σ∈[0,τ)

−1x<h(σ)

|B(σ, x)|    +C exp   4 sup

σ∈[0,τ)

−1x<h(σ)

|B(σ, x)|   

Norm estimates

For every t ∈ [0, τ) the following estimates hold: h(t)

−1

(∂xu)2 dx + t h(s)

−1

• (∂tu)2 + (∂xxu)2

dx ds C

• 1 +

t h(s)

−1

(∂xρ)2 dx ds

• ,

sup

t∈[0,τ)

h(t)

−1

(∂xρ)2(t, x) dx C,

Outline

1 Introduction 2 Known results 3 Local in time existence 4 Global Existence 5 Further Comments

Future direction of work and open questions

• Global in time strong solution for higher dimensional models.
• Study of Fluid-Structure system involving temperature equation.
• Contollability and stabilizabilty of compressible fluid-structure

models.

Thank you.