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 ∂ t ρ + ∂ x ( ρ u ) = 0 , t ≥ 0 , x � = h ( t ) ρ ( ∂ t u + u ∂ x u ) − ∂ xx u + ∂ x ( ρ γ ) = 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 ) = [ ∂ x u − ρ γ ]( 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) = h 0 , h (0) = ℓ 0 , ( x ∈ [ − 1 , 1] \ { h 0 } ) . u (0 , x ) = u 0 ( x ) , ρ (0 , x ) = ρ 0 ( x ) (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) , (1.4) ρ ( t , − 1) = ρ − 1 ( t ) ( t � 0) , or  u ( t , − 1) = u − 1 ( t ) , u ( t , 1) = − u 1 ( t ) ( t � 0) ,    u − 1 ( t ) > 0 , u 1 ( t ) > 0 ( t � 0) , (1.5) ρ ( t , − 1) = ρ − 1 ( t ) ( t � 0) ,    ρ ( t , 1) = ρ 1 ( t ) ( t � 0) .

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 : u 0 ∈ C 2+ α , ρ 0 ∈ 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 of 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 L p − L q 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 h 0 ∈ ( − 1 , 1) , ℓ 0 ∈ R • u 0 ∈ H 1 ( − 1 , 1) and u 0 ( h 0 ) = ℓ 0 • ρ 0 ∈ H 1 ( − 1 , h 0 ) ∩ H 1 ( h 0 , 1) and ρ 0 ( x ) > 0 ∀ x ∈ [ − 1 , 1] \ { h 0 } . • u − 1 , u 1 ∈ H 1 (0 , T ) , ρ − 1 , ρ 1 ∈ H 1 (0 , T ) and ρ − 1 ( t ) > 0 , ρ 1 ( 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 h 0 ∈ ( − 1 , 1) , ℓ 0 ∈ R • u 0 ∈ H 1 ( − 1 , 1) and u 0 ( h 0 ) = ℓ 0 • ρ 0 ∈ H 1 ( − 1 , h 0 ) ∩ H 1 ( h 0 , 1) and ρ 0 ( x ) > 0 ∀ x ∈ [ − 1 , 1] \ { h 0 } . • u − 1 , u 1 ∈ H 1 (0 , T ) , ρ − 1 , ρ 1 ∈ H 1 (0 , T ) and ρ − 1 ( t ) > 0 , ρ 1 ( 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 � x − 1 � x � 1) . (3.1) ξ = Ψ( t , x ) = ρ ( t , η ) d η ( t � 0 , h ( t ) 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 ) , ξ 1 ( t )), where � h 0 � t ξ − 1 ( t ) = ρ 0 ( η ) d η + ρ − 1 ( s ) u − 1 ( s ) d s , − 1 0 � 1 � t ξ 1 ( t ) = ρ 0 ( η ) d η + ρ 1 ( s ) u 1 ( s ) d s , h 0 0 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 ) , ξ 1 ( t )) , ξ � = 0 ρ 2 ∂ ξ � ∂ t � ρ + � u = 0 ρ γ ) = 0 , ∂ t � u − ∂ ξ ( � ρ ( ∂ ξ � u )) + ∂ ξ ( � u ( t , 0) = ˙ � h ( t ) , m ¨ ρ γ ] ( t , 0) , h ( t ) = [ � ρ ( ∂ ξ � u ) − � u ( t , − ξ − 1 ( t )) = u − 1 ( t ) , � u ( t , ξ 1 ( t )) = − u 1 ( t ) , � ρ ( t , − ξ − 1 ( t )) = ρ − 1 ( t ) , ρ ( t , ξ 1 ( t )) = ρ 1 ( t ) � � ρ (0 , ξ ) := � ρ 0 ( ξ ) = ρ 0 (Φ(0 , ξ )) � u (0 , ξ ) := � � u 0 ( ξ ) = u 0 (Φ(0 , ξ )) ˙ h (0) = h 0 , h (0) = ℓ 0 .

System in fixed known domain: we define � ξ for ξ ∈ [ − ξ − 1 ( t ) , 0] , ξ − 1 ( t ) y = Γ( t , ξ ) = ξ for ξ ∈ [0 , ξ 1 ( t )] . ξ 1 ( t ) and by setting � � − 1 , u ( t , y ) = � ρ ( t , Γ − 1 ( t , y )) u ( t , Γ − 1 ( t , y )) , ζ ( t , y ) = � we have the following system in (0 , T ) × ( − 1 , 1) , y � = 0 ∂ t ζ + β∂ y ζ − α∂ y u = 0 � 1 � α � � ∂ t u + β∂ y u − α∂ y ζ ∂ y u + α∂ y = 0 ζ γ u ( t , 0) = ˙ h ( t ) , � α � ζ ∂ y u − 1 m ¨ h = ( t , 0) ζ γ u ( t , − 1) = u − 1 ( t ) , u ( t , 1) = − u 1 ( t ) , 1 1 ζ ( t , − 1) = ρ − 1 ( t ) , ζ ( t , 1) = ρ 1 ( t ) + initial conditions

where  1   for y ∈ [ − 1 , 0) ,  ξ − 1 ( t ) α ( t , y ) = 1    for y ∈ (0 , 1] ξ 1 ( t )  − y ˙  ξ − 1 ( t )   for y ∈ [ − 1 , 0) ,  ξ − 1 ( t ) β ( t , y ) = − y ˙  ξ 1 ( t )   for y ∈ (0 , 1] ,  ξ 1 ( t ) • β = 0 , in case of homogeneous boundary condition.

Construction of Fixed point map Let f 1 ∈ L 2 (0 , T ; L 2 ( − 1 , 1)) and f 2 ∈ L 2 [0 , T ] and consider the following system ∂ t ζ + β∂ y ζ − α∂ y u = 0 , � α 0 � ∂ t u − α 0 ∂ y ∂ y u = f 1 , ζ 0 u ( t , ± 0) = ˙ h , � α 0 � m ¨ h = ∂ y u ( t , 0) + f 2 , ζ 0 + initial conditions and boundary conditions • Linearization preserves the coupling of the equations of the fluid and of 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 f 1 ∈ L 2 (0 , T ; L 2 ( − 1 , 1)) and f 2 ∈ L 2 [0 , T ] and consider the following system ∂ t ζ + β∂ y ζ − α∂ y u = 0 , � α 0 � ∂ t u − α 0 ∂ y ∂ y u = f 1 , ζ 0 u ( t , ± 0) = ˙ h , � α 0 � m ¨ h = ∂ y u ( t , 0) + f 2 , ζ 0 + initial conditions and boundary conditions • Linearization preserves the coupling of the equations of the fluid and of the structure. • essential in our approach for obtaining the local existence result for initial data less regular than other works