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Steady compressible Navier-Stokes equations with inflow boundary condition

Tomasz Piasecki 1, joint work with Piotr B Mucha 2

1Institute of Mathematics,

Polish Academy of Sciences

2University 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 · 2µD(v) · τ + fv · τ = b,

• n

Γ, n · v = d

• n

Γ, ρ = ρin

• n

Γin, (1) v - velocity of the fluid ρ - density π(ρ) - pressure (given function)

T Piasecki Steady compressible NSE 2/11

The boundary

∂Ω = Γ = Γin ∪ Γout ∪ Γ0 where Γin = {x ∈ Γ : v · n < 0}, Γout = {x ∈ Γ : v · n > 0}, Γ0 = {x ∈ Γ : v · n = 0}, Γ∗ = Γ0 ∩ Γin ∪ Γout. (2)

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 Γin and Γout 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)]: Γin = (x1(x2), x2) Γout = (x1(x2), x2) 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: x2(x1) ∼ |x1|q, q ≤ 2 after

• bvious 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

p

× W s

p close to (¯

v, ¯ ρ) ≡ ([1, 0], 1) where fW s

p (Ω) = (

• Ω

|f|pdx)1/p +

• Ω2

|f(x) − f(y)|p |x − y|2+sp dxdy

1/p

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 u0 ∈ W 1+s

p

: u0 · n|Γ = d − n(1), u0W 1+s

p

≤ Cd − n(1)W 1+s−1/p

p

(Γ)

T Piasecki Steady compressible NSE 6/11

Definition of solutions (2)

Introducing the perturbations: u = v − ¯ v − u0 and σ = ρ − ¯ ρ we get ∂x1u − µ∆u − (ν + µ)∇divu + γ ∇σ = F(u, σ) in Ω, div u + ∂x1σ + (u + u0) · ∇σ = G(u, σ) in Ω, n · 2µD(u) · τ + f u · τ = B

• n

Γ, n · u = 0

• n

Γ, σ = σin

• n

Γin, (3) where γ = π′(1). Definition A regular W s

p solution to the system (1) is a couple

(v, ρ) ∈ W 1+s

p

× W s

p such that v = ¯

v + u + u0 and ρ = ¯ ρ + σ where (u, σ) ∈ W 1+s

p

× W s

p is a solution to the system (3).

T Piasecki Steady compressible NSE 7/11

Main result (T.P., P.B. Mucha 2015)

Assumptions:

1 |x2(x1) − x2(y1)| ≥ C|x1 − y1|N 2 b ∈ W s−1/p

p

(Γ), d ∈ W 1+s−1/p

p

(Γ), π ∈ C2,1

3 ρin ∈ W s

p (Γin) ∩ W r p (Γs) where 0 < s < r < 1 depend on ∂Ω and

sp > 2.

4 (d − n(1))x1′(x2) and (d − n(1))x1′(x2) are bounded around the

singularity points.

5 The data is close enough to (¯

v, ¯ ρ) Then ∃ a solution (v, ρ) ∈ W 1+s

p

× W s

p to the system 1 such that

v − ¯ vW 1+s

p

+ ρ − ¯ ρW s

p ≤ C(DATA).

(4) This solution is unique in the class of solutions satisfying (4)

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Idea of the proof (1): A priori bounds

1 Energy estimate for the linear system; 2 Estimate for the steady transport equation

w + wx1 + U · ∇w = H, w|Γin = win : wW s

p ≤ C[HW s p + winW s p (Γin)∩W r p (Γs)].

Here we need the constraints on the boundary and boundary data.

3 Estimate in W 1+s

p

× W 1

p :

Equation for curl u ⇒ estimate for curl uW s

p

Helmholtz decomposition of the velocity ⇒ estimate for div uW s

p

4 Estimate for uW 1+s p

+ wW s

p . T Piasecki Steady compressible NSE 9/11

Idea of the proof (2): Lagrange-type coordinates

For given small U ∈ W 1+s

p

, U · n = d − n(1) we define a transformation x = ψ(y) which straigthens the characteristics of the continuity equation: ∂x1 + U · ∇ = ∂y1 Warning: ψ(Ω) = Ω, therefore we need a second transformation z = A(y):

• [z1 − x1(z2)] = IE(y2)

IL(y2)[y1 − x1(z2)],

z2 = y2

T Piasecki Steady compressible NSE 10/11

Idea of the proof (3): Iteration

Define sequence (un, wn): L(un+1, σn+1) = R(un, σn) Solution of the linear system ⇒ existence of {(un, wn)}. Convergence of the sequence (we need to compare two solutions so we need the second transformation to come back to the

• riginal domain).

Uniqueness results from the method of the proof.

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