Small time global exact null controllability of the incompressible - PowerPoint PPT Presentation

Small time global exact null controllability of the incompressible Navier-Stokes equation with Navier slip-with-friction boundary condition Joint work with Jean-Michel Coron and Franck Sueur Fr´ ed´ eric Marbach Laboratoire Jacques-Louis Lions, UMR 7598, UPMC Univ Paris 06, Sorbonne Universit´ es IHP, June 22, 2016 Work partially supported by the ERC advanced grant 266907 (CPDENL) of the 7th Research Framework Programme (FP7). Fr´ ed´ eric Marbach (LJLL) Navier-Stokes slip-with-friction IHP, June 22, 2016 1 / 25

Small time global results in fluid mechanics implies to study boundary layers Small time: T ≪ 1 Global: large states | u ( t , · ) | L 2 (Ω) ≫ 1 ∂ Ω \ Γ Ω Boundary condition Γ Our goal: null controllability u ( T , · ) = 0.+ Fr´ ed´ eric Marbach (LJLL) Navier-Stokes slip-with-friction IHP, June 22, 2016 2 / 25

Boundary layer profiles When Re ≫ 1, the fluid behaves like the solution of an inviscid equation inside the domain. However, near the boundary, viscous effects prevail. towards inner domain free flow perturbed fluid fluid at rest boundary condition u = 0 Figure: Blasius speed profile for the Dirichlet boundary condition Fr´ ed´ eric Marbach (LJLL) Navier-Stokes slip-with-friction IHP, June 22, 2016 3 / 25

Incompressible Navier-Stokes equation inside the domain Ω u : Ω → R d is the velocity, p : Ω → R is the pressure,  u t + ( u · ∇ ) u − ∆ u + ∇ p = 0 on Ω ,    div ( u ) = 0 on Ω ,   (NS) on ∂ Ω \ Γ , BC     u (0 , · ) = u ∗ on Ω .  No boundary condition on Γ (control region): under-determined system. Fr´ ed´ eric Marbach (LJLL) Navier-Stokes slip-with-friction IHP, June 22, 2016 4 / 25

Navier slip-with-friction boundary condition on the uncontrolled part of the boundary On ∂ Ω \ Γ, we assume: u · n = 0 and [ D ( u ) n + Au ] tan = 0 , (Navier) where D ij ( f ) := 1 2 ( ∂ i f j + ∂ j f i ) , [ f ] tan := f − ( f · n ) n and A : ∂ Ω → M d ( R ) is smooth (but not necessarily constant, neither signed / coercive). Fr´ ed´ eric Marbach (LJLL) Navier-Stokes slip-with-friction IHP, June 22, 2016 5 / 25

Special cases of boundary condition u · n = 0 and [ D ( u ) n + Au ] tan = 0 perfect slip , when A is the Weingarten map: curl u = 0 in 2D , u · n = 0 curl u ∧ n = 0 in 3D . slip condition when A = 0: u · n = 0 and [ D ( u ) n ] tan = 0 . scalar case A = 1 β Id for flat boundaries : u · n = 0 and u tan = β∂ n u tan . Fr´ ed´ eric Marbach (LJLL) Navier-Stokes slip-with-friction IHP, June 22, 2016 6 / 25

Slip length β Imagine that the profile is displaced by a length β inside the boundary. towards inner domain free flow perturbed fluid β Fr´ ed´ eric Marbach (LJLL) Navier-Stokes slip-with-friction IHP, June 22, 2016 7 / 25

Our result Theorem (Coron, M., Sueur) Let Ω be a smooth connected bounded domain in R 2 or R 3 . Let Γ ⊂ ∂ Ω intersecting all connected components of ∂ Ω . Let A be a smooth matrix-valued function on ∂ Ω . Let u ∗ ∈ L 2 (Ω) be divergence free, tangent to the boundary. For any T > 0 , there exists a trajectory u (in an appropriate functional space) solution to (NS) and (Navier) such that: u ( T , · ) = 0 . Fr´ ed´ eric Marbach (LJLL) Navier-Stokes slip-with-friction IHP, June 22, 2016 8 / 25

Key ideas of the proof Controllability of the Euler equation by means of the return method 1 [Coron 1993], [Coron 1996], [Glass 1997], [Glass 2000]. Vanishing viscosity asymptotic boundary layer expansion for the 2 Navier boundary condition [Iftimie, Sueur 2011]. Well prepared dissipation of the boundary layer [M. 2014]. 3 Fr´ ed´ eric Marbach (LJLL) Navier-Stokes slip-with-friction IHP, June 22, 2016 9 / 25

I. Controllability of Euler Usual scaling argument We let T = ε ≪ 1 be a small time and u ∗ be a large initial data. We introduce: u ε ( t , x ) := ε u ( ε t , x ) , p ε ( t , x ) := ε 2 p ( ε t , x ) . These are now defined for t ∈ (0 , 1). Moreover, the initial data is now small: u ε (0 , · ) = ε u ∗ . Expand: u ε ( t , x ) = u 0 ( t , x ) + ε u 1 ( t , x ) + . . . Fr´ ed´ eric Marbach (LJLL) Navier-Stokes slip-with-friction IHP, June 22, 2016 10 / 25

I. Controllability of Euler Choice of a reference trajectory Build a return method like reference trajectory.  ∂ t u 0 + u 0 · ∇ u 0 + ∇ p 0 = 0 � � [0 , 1] × Ω ,     div u 0 = 0   [0 , 1] × Ω ,     u 0 · n = 0 (1) [0 , 1] × ∂ Ω \ Γ ,   u 0 (0 , · ) = 0  Ω ,      u 0 ( T , · ) = 0  Ω .  You can choose u 0 ( t , x ) = ∇ θ 0 ( t , x ). In 2D, you can even choose u 0 ( t , x ) = α ( t ) × ∇ θ ( x ). Fr´ ed´ eric Marbach (LJLL) Navier-Stokes slip-with-friction IHP, June 22, 2016 11 / 25

I. Controllability of Euler Flushing of the initial data The initial data u ∗ is flushed by the reference trajectory. ∂ t u 1 + u 0 · ∇ u 1 + u 1 · ∇ u 0 + ∇ p 1 = 0  � � � � [0 , 1] × Ω ,     div u 1 = 0   [0 , 1] × Ω ,  (2) u 1 · n = 0 [0 , 1] × ∂ Ω \ Γ ,      u 1 (0 , · ) = u ∗  Ω .  At the final time u 1 = 0. Fr´ ed´ eric Marbach (LJLL) Navier-Stokes slip-with-friction IHP, June 22, 2016 12 / 25

Is it enough? for Navier-Stokes In, [Coron 1996], Jean-Michel tried to apply this method to the Navier-Stokes equation in 2D with Navier boundary conditions. With the same scaling: t + ( u ε · ∇ ) u ε − ε ∆ u + ∇ p ε = 0 ,  u ε    div ( u ε ) = 0 ,  u ε (0 , · ) = ε u ∗ .   However, it is not sufficient to conclude. Indeed, although this method yields good controllability inside the domain, we only have weak estimates of the final state in W − 1 , ∞ (Ω) near the boundaries. ⇒ We need to compute what happens near the boundaries. Fr´ ed´ eric Marbach (LJLL) Navier-Stokes slip-with-friction IHP, June 22, 2016 13 / 25

II. Vanishing viscosity expansion of Navier-Stokes Convergence of Navier-Stokes to Euler? Do the solutions u ε of: t + ( u ε · ∇ ) u ε − ε ∆ u + ∇ p ε = 0 u ε  (0 , T ) × Ω ,    div ( u ε ) = 0 (0 , T ) × Ω ,     u ε · n = 0  (0 , T ) × ∂ Ω ,  [ D ( u ε ) n + Au ε ] tan = 0 (0 , T ) × ∂ Ω ,      u ε (0 , · ) = u ∗  Ω  on (0 , T ) × Ω converge to the corresponding solution of Euler? Can we write an asymptotic expansion? Fr´ ed´ eric Marbach (LJLL) Navier-Stokes slip-with-friction IHP, June 22, 2016 14 / 25

II. Vanishing viscosity expansion of Navier-Stokes Asymptotic expansion The answer is yes! The expansion looks like: Euler ( t , x ) + √ ε v t , x , ϕ ( x ) � � u ε ( t , x ) = u 0 √ ε + . . . , where ϕ ( x ) is the distance to the boundary ∂ Ω. We see the thickness of the boundary layer and we introduce the fast variable z = ϕ ( x ) / √ ε . In this expansion, v is only tangential: v · n ≡ 0. Fr´ ed´ eric Marbach (LJLL) Navier-Stokes slip-with-friction IHP, June 22, 2016 15 / 25

II. Vanishing viscosity expansion of Navier-Stokes PDE for the boundary layer profile ( u 0 · ∇ ) v + ( v · ∇ ) u 0 � � tan + u 0  ∂ t v + ♭ z ∂ z v − ∂ zz v = 0 ,    ∂ z v ( · , · , 0) = g 0 at z = 0 ,   v (0 , · , · ) = 0 at t = 0 ,  where we introduce the following definitions: ♭ ( t , x ) = u 0 ( t , x ) · n ( x ) u 0 , in [0 , T ] × Ω , ϕ ( x ) � � � � g 0 ( t , x ) = 2 χ ( x ) u 0 ( t , x ) n ( x ) + Au 0 ( t , x ) D in [0 , T ] × Ω . tan Well-posedness and estimates can be proven. Fr´ ed´ eric Marbach (LJLL) Navier-Stokes slip-with-friction IHP, June 22, 2016 16 / 25

Is it small enough. . . to be able to conclude with a local result? At the final time, we have: √ ε v t = 1 , · , ϕ ( · ) � � �� � u ε ( t = 1 , · ) � L 2 (Ω) ≈ ≈ ε 3 / 4 . � √ ε � � � � � L 2 (Ω) But this is not enough... scaling back, this yields: � u ( ε, · ) � L 2 (Ω) ≈ ε − 1 / 4 . Fr´ ed´ eric Marbach (LJLL) Navier-Stokes slip-with-friction IHP, June 22, 2016 17 / 25

III. Well-prepared dissipation of the boundary layer A 1D model with the Burgers equation Here Ω = (0 , 1), u ∗ ∈ L 2 (0 , 1) and T > 0.  u t + uu x − u xx = q int ( t ) on (0 , T ) × (0 , 1) ,    u ( t , 0) = q bc ( t ) on (0 , T ) ,  (3) u ( t , 1) = 0 on (0 , T ) ,    u (0 , x ) = u ∗ ( x ) on (0 , 1)  using two scalar controls q int and q bc . Fr´ ed´ eric Marbach (LJLL) Navier-Stokes slip-with-friction IHP, June 22, 2016 18 / 25

III. Well-prepared dissipation of the boundary layer Use the left boundary control to crush the initial data h ( x ) H u ( t , x ) u ∗ ( x ) x = 1 Figure: After a time of order 1 / H , we almost reach a steady state u ( t , x ) ≈ h ( x ). Fr´ ed´ eric Marbach (LJLL) Navier-Stokes slip-with-friction IHP, June 22, 2016 19 / 25

III. Well-prepared dissipation of the boundary layer Go back down using the inner scalar control h ( x ) H u ( t , x ) 0 x = 0 . 9 x = 1 Figure: . . . but this create a boundary layer residue near x = 1 . . . Fr´ ed´ eric Marbach (LJLL) Navier-Stokes slip-with-friction IHP, June 22, 2016 20 / 25