Steady compressible Navier-Stokes equations with inflow boundary - PowerPoint PPT Presentation

Steady compressible Navier-Stokes equations with inflow boundary condition Tomasz Piasecki 1 , joint work with Piotr B Mucha 2 1 Institute of Mathematics, Polish Academy of Sciences 2 University of Warsaw Institute of Applied Mathematics and Mechanics Mathflows 2015 Porquerolles, France

The system ρv · ∇ v − µ ∆ v − ( µ + ν ) ∇ div v + ∇ π ( ρ ) = 0 in Ω , div ( ρv ) = 0 in Ω , n · 2 µ D ( v ) · τ + fv · τ = b, on Γ , (1) n · v = d on Γ , ρ = ρ in on Γ in , v - velocity of the fluid ρ - density π ( ρ ) - pressure (given function) T Piasecki Steady compressible NSE 2/11

The boundary ∂ Ω = Γ = Γ in ∪ Γ out ∪ Γ 0 where Γ in = { x ∈ Γ : v · n < 0 } , Γ out = { x ∈ Γ : v · n > 0 } , (2) Γ 0 = { x ∈ Γ : v · n = 0 } , Γ ∗ = Γ 0 ∩ Γ in ∪ Γ out . T Piasecki Steady compressible NSE 3/11

Known results - inflow condition (1) Regular solutions for small data: 1 A.Valli, W.M.Zaj¸ aczkowski (1986) 2 J.R.Kweon, R.B.Kellogg (1996): bounded domain, to be discussed more precisely 3 J.R.Kweon, R.B.Kellogg (1997): unbounded domain with Γ in and Γ out separated 4 P.B. Mucha, T.P. (2014): Solutions close to Poiseuille or constant flow in a cylindrical domain, barotropic case 5 T.P., M.Pokorný (2014): As above, complete system with temperature No large data results for inflow condition in stationary case! T Piasecki Steady compressible NSE 4/11

Known results - inflow condition (2) Closer look at [J.R.Kweon, R.B.Kellogg (1996)]: Γ in = ( x 1 ( x 2 ) , x 2 ) Γ out = ( x 1 ( x 2 ) , x 2 ) Two singularity points x ∗ , x ∗ ; Existence of regular solution ( v, ρ ) ∈ W 2 p × W 1 p satisfying the equations a.e., under the assumptions: 2<p<3 Boundary near the singulaity points: x 2 ( x 1 ) ∼ | x 1 | q , q ≤ 2 after obvious translation Our motivation: relax the above assumptions. T Piasecki Steady compressible NSE 5/11

Definition of solutions (1) We look for ( v, ρ ) ∈ W 1+ s × W s p close to (¯ v, ¯ ρ ) ≡ ([1 , 0] , 1) where p � 1 /p | f ( x ) − f ( y ) | p ��� � | f | p dx ) 1 /p + � f � W s p (Ω) = ( dxdy | x − y | 2+ sp Ω 2 Ω 1 No longer satisfy the equations a.e.; we need a weak formulation. 2 We need to get rid of inhomogeneity on the boundary ⇒ we construct u 0 ∈ W 1+ s : p u 0 · n | Γ = d − n (1) , ≤ C � d − n (1) � W 1+ s − 1 /p � u 0 � W 1+ s (Γ) p p T Piasecki Steady compressible NSE 6/11

Definition of solutions (2) Introducing the perturbations: u = v − ¯ v − u 0 σ = ρ − ¯ and ρ we get ∂ x 1 u − µ ∆ u − ( ν + µ ) ∇ divu + γ ∇ σ = F ( u, σ ) in Ω , div u + ∂ x 1 σ + ( u + u 0 ) · ∇ σ = G ( u, σ ) in Ω , n · 2 µ D ( u ) · τ + f u · τ = B on Γ , (3) n · u = 0 on Γ , σ = σ in on Γ in , where γ = π ′ (1). Definition A regular W s p solution to the system (1) is a couple ( v, ρ ) ∈ W 1+ s × W s p such that v = ¯ v + u + u 0 and ρ = ¯ ρ + σ where p ( u, σ ) ∈ W 1+ s × W s p is a solution to the system (3). p T Piasecki Steady compressible NSE 7/11

Main result (T.P., P.B. Mucha 2015) Assumptions: 1 | x 2 ( x 1 ) − x 2 ( y 1 ) | ≥ C | x 1 − y 1 | N 2 b ∈ W s − 1 /p (Γ) , d ∈ W 1+ s − 1 /p (Γ) , π ∈ C 2 , 1 p p 3 ρ in ∈ W s p (Γ in ) ∩ W r p (Γ s ) where 0 < s < r < 1 depend on ∂ Ω and sp > 2. 4 ( d − n (1) ) x 1 ′ ( x 2 ) and ( d − n (1) ) x 1 ′ ( x 2 ) are bounded around the singularity points. 5 The data is close enough to (¯ v, ¯ ρ ) Then ∃ a solution ( v, ρ ) ∈ W 1+ s × W s p to the system 1 such that p � v − ¯ v � W 1+ s + � ρ − ¯ ρ � W s p ≤ C ( DATA ) . (4) p This solution is unique in the class of solutions satisfying (4) T Piasecki Steady compressible NSE 8/11

Idea of the proof (1): A priori bounds 1 Energy estimate for the linear system; 2 Estimate for the steady transport equation w + w x 1 + U · ∇ w = H, w | Γ in = w in : � w � W s p ≤ C [ � H � W s p + � w in � W s p (Γ s ) ] . p (Γ in ) ∩ W r Here we need the constraints on the boundary and boundary data. 3 Estimate in W 1+ s × W 1 p : p Equation for curl u ⇒ estimate for � curl u � W s p Helmholtz decomposition of the velocity ⇒ estimate for � div u � W s p 4 Estimate for � u � W 1+ s + � w � W s p . p T Piasecki Steady compressible NSE 9/11

Idea of the proof (2): Lagrange-type coordinates , U · n = d − n (1) we define a transformation For given small U ∈ W 1+ s p x = ψ ( y ) which straigthens the characteristics of the continuity equation: ∂ x 1 + U · ∇ = ∂ y 1 Warning: ψ (Ω) � = Ω, therefore we need a second transformation z = A ( y ): [ z 1 − x 1 ( z 2 )] = I E ( y 2 ) � I L ( y 2 ) [ y 1 − x 1 ( z 2 )] , z 2 = y 2 T Piasecki Steady compressible NSE 10/11

Idea of the proof (3): Iteration Define sequence ( u n , w n ): L ( u n +1 , σ n +1 ) = R ( u n , σ n ) Solution of the linear system ⇒ existence of { ( u n , w n ) } . Convergence of the sequence (we need to compare two solutions so we need the second transformation to come back to the original domain). Uniqueness results from the method of the proof. T Piasecki Steady compressible NSE 11/11