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

Small time global exact null controllability

• f 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, ·)|L2(Ω) ≫ 1

Ω ∂Ω \ Γ Γ Boundary condition

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.

free flow fluid at rest perturbed fluid towards inner domain 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 → Rd is the velocity, p : Ω → R is the pressure,

          

ut + (u · ∇)u − ∆u + ∇p = 0

• n Ω,

div(u) = 0

• n Ω,

BC

• n ∂Ω \ Γ,

u(0, ·) = u∗

• n Ω.

(NS) 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

• n the uncontrolled part of the boundary

On ∂Ω \ Γ, we assume: u · n = 0 and [D(u)n + Au]tan = 0, (Navier) where Dij(f ) := 1 2 (∂ifj + ∂jfi) , [f ]tan := f − (f · n)n and A : ∂Ω → Md(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

• f boundary condition u · n = 0 and [D(u)n + Au]tan = 0

perfect slip, when A is the Weingarten map: curlu = 0 in 2D, u · n = 0 curlu ∧ n = 0 in 3D. slip condition when A = 0: u · n = 0 and [D(u)n]tan = 0. scalar case A = 1

βId for flat boundaries:

u · n = 0 and utan = β∂nutan.

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.

free flow perturbed fluid towards inner domain 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 R2 or R3. Let Γ ⊂ ∂Ω intersecting all connected components of ∂Ω. Let A be a smooth matrix-valued function on ∂Ω. Let u∗ ∈ L2(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

1

Controllability of the Euler equation by means of the return method [Coron 1993], [Coron 1996], [Glass 1997], [Glass 2000].

2

Vanishing viscosity asymptotic boundary layer expansion for the Navier boundary condition [Iftimie, Sueur 2011].

3

Well prepared dissipation of the boundary layer [M. 2014].

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: uε(t, x) := εu(εt, x), pε(t, x) := ε2p(εt, x). These are now defined for t ∈ (0, 1). Moreover, the initial data is now small: uε(0, ·) = εu∗. Expand: uε(t, x) = u0(t, x) + εu1(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.

                    

∂tu0 +

• u0 · ∇
• u0 + ∇p0 = 0

[0, 1] × Ω, div u0 = 0 [0, 1] × Ω, u0 · n = 0 [0, 1] × ∂Ω \ Γ, u0(0, ·) = 0 Ω, u0(T, ·) = 0 Ω. (1) You can choose u0(t, x) = ∇θ0(t, x). In 2D, you can even choose u0(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.

              

∂tu1 +

• u0 · ∇
• u1 +
• u1 · ∇
• u0 + ∇p1 = 0

[0, 1] × Ω, div u1 = 0 [0, 1] × Ω, u1 · n = 0 [0, 1] × ∂Ω \ Γ, u1(0, ·) = u∗ Ω. (2) At the final time u1 = 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:

      

uε

t + (uε · ∇)uε − ε∆u + ∇pε = 0,

div(uε) = 0, uε(0, ·) = εu∗. However, it is not sufficient to conclude. Indeed, although this method yields good controllability inside the domain, we only have weak estimates

• f 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 uε of:

                

uε

t + (uε · ∇)uε − ε∆u + ∇pε = 0

(0, T) × Ω, div(uε) = 0 (0, T) × Ω, uε · n = 0 (0, T) × ∂Ω, [D(uε)n + Auε]tan = 0 (0, T) × ∂Ω, uε(0, ·) = u∗ Ω

• n (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: uε(t, x) = u0

Euler(t, x) + √εv

• t, x, ϕ(x)

√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

      

∂tv +

• (u0 · ∇)v + (v · ∇)u0

tan + u0 ♭ z∂zv − ∂zzv = 0,

∂zv(·, ·, 0) = g0 at z = 0, v(0, ·, ·) = 0 at t = 0, where we introduce the following definitions: u0

♭ (t, x) = u0(t, x) · n(x)

ϕ(x) , in [0, T] × Ω, g0(t, x) = 2χ(x)

• D
• u0(t, x)
• n(x) + Au0(t, x)
• tan

in [0, T] × Ω. 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: uε(t = 1, ·)L2(Ω) ≈

• √εv
• t = 1, ·, ϕ(·)

√ε

• L2(Ω)

≈ ε3/4. But this is not enough... scaling back, this yields: u(ε, ·)L2(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∗ ∈ L2(0, 1) and T > 0.

        

ut + uux − uxx = qint(t)

• n

(0, T) × (0, 1), u(t, 0) = qbc(t)

• n

(0, T), u(t, 1) = 0

• n

(0, T), u(0, x) = u∗(x)

• n

(0, 1) (3) using two scalar controls qint and qbc.

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) x = 1 u(t, x) u∗(x) H

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) u(t, x) H 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

• III. Well-prepared dissipation of the boundary layer

Natural lazy-control smoothing effect

u(t, x) h(x) − H −3 x = 0.5 x = 1

Figure: Relax. Choose null controls. Watch the diffusion kill the residue.

Fr´ ed´ eric Marbach (LJLL) Navier-Stokes slip-with-friction IHP, June 22, 2016 21 / 25

Proof of the main theorem

Two time scales

Let T be the (possibly small) time in which we want to prove global exact null controllability. Introduce ε ≪ 1. Consider two time intervals: [0, εT] and [εT, T]. After usual scaling, these will be [0, T] and [T, T/ε].

Fr´ ed´ eric Marbach (LJLL) Navier-Stokes slip-with-friction IHP, June 22, 2016 22 / 25

Proof of the main theorem

Dissipation estimates for the boundary layer

In the second phase [T, T/ε], u0 = 0. Thus

      

∂tv +

• (u0 · ∇)v + (v · ∇)u0

tan + u0 ♭ z∂zv − ∂zzv = 0,

∂zv(·, ·, 0) = g0 at z = 0, v(0, ·, ·) = 0 at t = 0, So we have a heat equation on the half line z ≥ 0, where x ∈ Ω is merely a parameter. Decay properties depend on vanishing moments of the initial data (at time T). If you assume enough vanishing moments, you can prove any polynomial decay in L2(R+) (or stronger spaces).

Fr´ ed´ eric Marbach (LJLL) Navier-Stokes slip-with-friction IHP, June 22, 2016 23 / 25

Proof of the main theorem

How to ensure vanishing moments?

Use the transport term like in this model system:

      

∂tv + (u0 · ∇)v − ∂zzv = 0, ∂zv(·, ·, 0) = g0 at z = 0, v(0, ·, ·) = 0 at t = 0, to make sure the moments vanish at t = T.

Fr´ ed´ eric Marbach (LJLL) Navier-Stokes slip-with-friction IHP, June 22, 2016 24 / 25

Perspectives

Possible improvements: controllability to the trajectories, remove the assumption on Γ, strong solutions, . . . The main perspective is to attempt to apply this method to the Dirichlet boundary condition u = 0 on ∂Ω \ Γ in order to prove the same result in this more difficult case1.

1note: try to finish before Jean-Michel’s birthday; could make a great gift. Fr´ ed´ eric Marbach (LJLL) Navier-Stokes slip-with-friction IHP, June 22, 2016 25 / 25