On stability of scale-critical circular flows in a two-dimensional - - PowerPoint PPT Presentation

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On stability of scale-critical circular flows in a two-dimensional exterior domain

Yasunori Maekawa (Tohoku University)

Mathematics for Nonlinear Phenomena: Analysis and Computation ‐ International conference in honor of Professor Yoshikazu Giga on his 60th birthday ‐ (Sapporo, August 16-18, 2015)

Yasunori Maekawa (Tohoku University) Two-dimensional scale-critical flows

1-1. Two-dimensional Navier-Stokes equations

儀我美一・儀我美保 著； 「非線形偏微分方程式-解の漸近挙動と自己相似解-」 （共立, 1999 年） M.-H. Giga, Y. Giga, and J. Saal; 「Nonlinear partial differential equations. Asymptotic behavior of solutions and self-similar solutions. 」(Birkh¨ auser, 2010) . Incompressible Navier-Stokes equations . . (NS)                    ∂tu + u · ∇u = ∆u − ∇p, t > 0 , x ∈ Ω , div u = 0 , t ≥ 0 , x ∈ Ω , u = 0 , t > 0 , x ∈ ∂Ω , u|t=0 = u0 , x ∈ Ω . u = (u1(t, x), u2(t, x)): velocity field p = p(t, x): pressure Ω = R2 or exterior domain

Yasunori Maekawa (Tohoku University) Two-dimensional scale-critical flows

1-2. Scaling and self-similar solution

If (u, p) solves (NS) in R+ × R2 then (uλ, pλ) also solves (NS) in R+ × R2;

uλ(t, x) = λu(λ2t, λx), pλ(t, x) = λ2p(λ2t, λx), λ > 0

. Lamb-Oseen vortex (forward self-similar solution, circular swirling flow) . .

UG(t, x) = x⊥ 2π|x|2 (1 − e− |x|2

4t ) ,

x⊥ = (−x2, x1)

(i) For each α ∈ R the velocity αUG is a forward self-similar solution to (NS) (with Ω = R2):

UG

λ (t, x) = UG(t, x) , λ > 0 .

|UG(t, x)| ≤ C min { |x|−1 , t− 1

2 }; infinite energy flow

(ii) The vorticity field is the two-dimensional Gaussian:

(rot UG)(t, x) = (∂1UG

2 − ∂2UG 1

)(t, x) = G(t, x) = 1 4πt e− |x|2

4t . Yasunori Maekawa (Tohoku University) Two-dimensional scale-critical flows

1-3. Kambe-Lundgren (similarity) transform and Burgers vortex t

1 2 u(t, x) = v(τ, ξ) ,

τ = log(1 + t) , ξ = x t

1 2

. . The following two are equivalent. (i) Asymptotic stability of the Lamb-Oseen vortex αUG for the two- dimensional Navier-Stokes equations (ii) Two-dimensional stability of the Burgers vortex with circulation α, which is a stationary solution to the three-dimensional Navier-Stokes equations Giga-Kambe (1988); Carpio (1994); Gallay-Wayne (2005); Gallagher-Gallay-Lions (2005) Two-dimensional vorticity equations: ω = rot u

∂tω + u · ∇ω = ∆ω , t > 0 , x ∈ R2 .

Giga-Miyakawa-Osada (1988), Kato (1994), Gallagher-Gallay-Lions (2005), Gallagher-Gallay (2005)

Yasunori Maekawa (Tohoku University) Two-dimensional scale-critical flows

1-4. Some properties of circular swirling flows

Let V(x) = x⊥ f(|x|) , x⊥ = (−x2, x1) for a scalar function f in R2. (i) div V(x) = x⊥ · x

|x| f ′(|x|) = 0

(ii) V · ∇V = 1

2∇|V|2 + V⊥rot V = 1 2∇|V|2 + ∇P, where P(x) = ∫ ∞

|x|

f ′(r)w(r) dr , w(|x|) = rot V(x) .

Therefore, we have P(V · ∇V) = 0, where P is the Helmholtz projection. (iii) For any radial function ρ we have

∫

Ω

ρg V · ∇g dx = −1 2 ∫

Ω

|g|2V · ∇ρ dx = 0 , g ∈ C∞

0 (Ω) .

Yasunori Maekawa (Tohoku University) Two-dimensional scale-critical flows

2-1. Stability of the Lamb-Oseen vortex in exterior problem

Problem: Large time behavior of solutions to (NS) for the initial velocity

u0(x) = αUG(1, x) + v0(x) , |x| ≫ 1 , v0 ∈ L2

σ(Ω) .

. Difficulty when Ω in an exterior domain (even for 0 < |α| ≪ 1) . . (i) The vorticity equation is not useful to obtain the uniform bound in the scale-critical norms. (ii) The Hardy inequality is not available in the two-dimensional case:

∥ 1 |x| f∥L2(Ω) ≤ C∥∇f∥L2(Ω) , f ∈ ˙ W1,2

0 (Ω).

In particular, one can not expect the coercive (positive) estimate such as

⟨ −∆v + α(UG · ∇v + v · ∇UG) + v · ∇v , v ⟩L2(Ω) ≥ c∥∇v∥2

L2(Ω)

for v ∈ C∞

0,σ(Ω), even when 0 < |α| ≪ 1.

Yasunori Maekawa (Tohoku University) Two-dimensional scale-critical flows

2-2. Asymptotic behavior of two-dimensional exterior flows

. Theorem 1 (Gallay-M. (2013), M. (2015)). . . There is a constant δ > 0 such that for any u0 ∈ L2,∞

σ (Ω) of the form

u0 = αUG|t=1 + v0 , |α| ≤ δ , v0 ∈ L2(Ω)2

there exists a unique solution u to (NS) with initial data u0 satisfying

lim

t→∞ t

k 2 ∥∇k(u(t) − αUG(t))∥L2(Ω) = 0 ,

k = 0, 1 . · The local L2 stability is proved by Iftimie-Karch-Lacave (2011). · The similar result holds even when αUG is replaced by the strong

solution U obtained in Kozono-Yamazaki (1995) which satisfies

sup

t>0

∥U(t)∥L2,∞(Ω) + sup

t>0

t

1 4 ∥U(t)∥L4(Ω) ≤ δ ≪ 1 . Yasunori Maekawa (Tohoku University) Two-dimensional scale-critical flows

2-3. Key ingredients of the proof

(i) The logarithmic growth energy estimate for the perturbation

v(t) = u(t) − αUG(t): ∥v(t)∥2

L2(Ω) +

∫ t

1

∥∇v(s)∥2

L2(Ω) ds

≤ C(∥v0∥L2(Ω)) + C0α2 log(1 + t) , t > 1 .

(ii) The analysis of the low frequency part of v by using the argument in Borchers-Miyakawa (1992) and Kozono-Ogawa (1993).

• Note. The argument essentially uses the scale-critical temporal decay of

UG such as ∥UG(t)∥L4(Ω) ≤ Ct−1/4.

Yasunori Maekawa (Tohoku University) Two-dimensional scale-critical flows

3-1. Steady circular flows with a scale-critical decay

We set

U(x) = lim

t→0 UG(t, x) =

x⊥ 2π|x|2 , x ∈ R2 \ {0} . · For each α ∈ R the velocity αU is a stationary solution to the following

Navier-Stokes system in Ω = {x ∈ R2 | |x| > 1}. . Navier-Stokes flows around a rotating disk . .

(NSα)                      ∂tu + u · ∇u = ∆u − ∇p, t > 0 , |x| > 1 , div u = 0 , t ≥ 0 , |x| > 1 , u = α 2π x⊥ , t > 0 , |x| = 1 , u|t=0 = u0 , |x| > 1 .

Yasunori Maekawa (Tohoku University) Two-dimensional scale-critical flows

3-2. Two important aspects

. Steady circular flow with a scale-critical decay . .

αU(x) = αx⊥ 2π|x|2 α2P(x) = − α2 8π2|x|2 x⊥ = (−x2, x1)

(I) Simple model of the flow around a rotating obstacle

· The existence of two-dimensional periodic flows around a rotating

• bstacle is still open in general.

· Hishida (2015): the asymptotic estimates for the steady Stokes flows

around a rotating obstacle.

· The unique existence and the stability for the three-dimensional problem:

Borchers (1992), Galdi (2003), Silvestre (2004), Farwig-Hishida (2007), Hishida -Shibata (2009), ...

Yasunori Maekawa (Tohoku University) Two-dimensional scale-critical flows

3-2. Two important aspects

. Steady circular flow with a scale-critical decay . .

αU(x) = αx⊥ 2π|x|2 α2P(x) = − α2 8π2|x|2 x⊥ = (−x2, x1)

(II) Stability of scale-critical flows

· The unique existence of stationary solutions having the spatial decay O(|x|−1) is proved by Yamazaki (2011) under some symmetry conditions

• n both domains and given data.

· Hillairet and Wittwer (2013) proved the existence of the steady exterior

flows in Ω = {x ∈ R2 | |x| > 1} near αU for large |α|. The stability of these stationary solutions decaying in the order O(|x|−1) is widely open even when the initial perturbation is small.

Yasunori Maekawa (Tohoku University) Two-dimensional scale-critical flows

3-3. Local L2 stability of αU for small |α|

. Theorem 2 (M.). . . For any sufficiently small |α| there is a constant ϵ = ϵ(α) > 0 such that if

∥u0 − αU∥L2(Ω) ≤ ϵ then there exists a unique solution u to (NSα) satisfying lim

t→∞ t

k 2 ∥∇k(u(t) − αU)∥L2(Ω) = 0 ,

k = 0, 1 .

The Helmholtz projection P : L2(Ω)2 → L2

σ(Ω) satisfies P∇p = 0.

. The perturbed Stokes operator . .

D(Aα) = W2,2(Ω)2 ∩ W1,2

0 (Ω)2 ∩ L2 σ(Ω)

Aαv = −P∆v + αP(U · ∇v + v · ∇U) , v ∈ D(Aα)

Yasunori Maekawa (Tohoku University) Two-dimensional scale-critical flows

3-4. Spectral analysis for the linearized operator

. Key ingredient . . Spectral analysis of the perturbed Stokes operator Aα by using the polar coordinates and the streamfunction-vorticity formulation We set

Fn(z; α) = ∫ ∞

1

s1−|n|Kµn(α)(sz) ds , Re(z) > 0 , n ∈ Z \ {0} ,

where Kµ is the modified Bessel function of second kind of order µ

Kµ(z) = 1 2 ∫ ∞ e− z

2 (t+ 1 t )t−µ−1 dt ,

Re(z) > 0

and

µn(α) = (n2 + iα 2π n) 1

2 ,

n ∈ Z \ {0}

Yasunori Maekawa (Tohoku University) Two-dimensional scale-critical flows

3-4. Spectral analysis for the linearized operator Aαv = −P∆v + αP(U · ∇v + v · ∇U) , U(x) = x⊥ 2π|x|2

. Structure of the spectrum of Aα (M.) . . Let α ∈ R. Then the following statements hold.

(1) σ(−Aα) = R− ∪ σdisc(−Aα) and σdisc(−Aα)= {λ ∈ C \ R−

• Fn(

√ λ; α) = 0 for some n ∈ Z \ {0} } . (2) For any ϵ ∈ (0, π

2) there is a constant δϵ > 0 such that if |α| ≤ δϵ then

the sector Σπ−ϵ is included in the resolvent set ρ(−Aα).

σdisc(−Aα): isolated eigenvalues of −Aα with finite algebraic multiplicities Σϕ = {z ∈ C \ {0} | |arg z| < ϕ}

Yasunori Maekawa (Tohoku University) Two-dimensional scale-critical flows

3-5. Linear stability

. Estimates for the perturbed Stokes semigroup (M.) . . There is a constant δ > 0 such that if |α| ≤ δ then the following statement

• holds. Let 1 < q ≤ 2 ≤ p < ∞. Then it follows that

∥e−t Aα f∥Lp(Ω) ≤ Ct− 1

q+ 1 p∥f∥Lq(Ω) ,

t > 0 ,

(1)

∥∇e−t Aα f∥L2(Ω) ≤ Ct− 1

q ∥f∥Lq(Ω) ,

t > 0 ,

(2) for f ∈ L2

σ(Ω) ∩ Lq(Ω)2. Here the constant C depends only on α, p, and q.

Yasunori Maekawa (Tohoku University) Two-dimensional scale-critical flows

• Special thanks -

. . Dear Professor Yoshikazu Giga, I am deeply grateful for your guidance, I wish you good health for many years to come.

Yasunori Maekawa (Tohoku University) Two-dimensional scale-critical flows