. 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 . ∂ t u + u · ∇ u = ∆ u − ∇ p , t > 0 , x ∈ Ω , div u = 0 , t ≥ 0 , x ∈ Ω , (NS) u = 0 , t > 0 , x ∈ ∂ Ω , u | t = 0 = u 0 , x ∈ Ω . . u = ( u 1 ( t , x ) , u 2 ( t , x )) : velocity field p = p ( t , x ) : pressure Ω = R 2 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 + × R 2 then ( u λ , p λ ) also solves (NS) in R + × R 2 ; u λ ( t , x ) = λ u ( λ 2 t , λ x ) , p λ ( t , x ) = λ 2 p ( λ 2 t , λ x ) , λ > 0 . Lamb-Oseen vortex (forward self-similar solution, circular swirling flow) . x ⊥ 2 π | x | 2 (1 − e − | x | 2 x ⊥ = ( − x 2 , x 1 ) U G ( t , x ) = 4 t ) , (i) For each α ∈ R the velocity α U G is a forward self-similar solution to (NS) (with Ω = R 2 ): U G λ ( t , x ) = U G ( t , x ) , λ > 0 . | U G ( t , x ) | ≤ C min { | x | − 1 , t − 1 2 } ; infinite energy flow (ii) The vorticity field is the two-dimensional Gaussian: 4 π t e − | x | 2 1 (rot U G )( t , x ) = ( ∂ 1 U G 2 − ∂ 2 U G 4 t . ) ( t , x ) = G ( t , x ) = 1 . Yasunori Maekawa (Tohoku University) Two-dimensional scale-critical flows
1-3. Kambe-Lundgren (similarity) transform and Burgers vortex ξ = x 1 2 u ( t , x ) = v ( τ, ξ ) , t τ = log(1 + t ) , 1 t 2 . The following two are equivalent. (i) Asymptotic stability of the Lamb-Oseen vortex α U G 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 x ∈ R 2 . ∂ t ω + u · ∇ ω = ∆ ω , t > 0 , 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 ⊥ = ( − x 2 , x 1 ) for a scalar function f in R 2 . (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 ∫ ∞ f ′ ( r ) w ( r ) d r , P ( x ) = w ( | x | ) = rot V ( x ) . | x | Therefore, we have P ( V · ∇ V ) = 0 , where P is the Helmholtz projection. (iii) For any radial function ρ we have ∫ ∫ ρ g V · ∇ g d x = − 1 | g | 2 V · ∇ ρ d x = 0 , g ∈ C ∞ 0 ( Ω ) . 2 Ω 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 u 0 ( x ) = α U G (1 , x ) + v 0 ( x ) , v 0 ∈ L 2 | x | ≫ 1 , 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 W 1 , 2 f ∈ ˙ | x | f ∥ L 2 ( Ω ) ≤ C ∥∇ f ∥ L 2 ( Ω ) , 0 ( Ω ) . In particular, one can not expect the coercive (positive) estimate such as ⟨ − ∆ v + α ( U G · ∇ v + v · ∇ U G ) + v · ∇ v , v ⟩ L 2 ( Ω ) ≥ c ∥∇ v ∥ 2 L 2 ( 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 u 0 ∈ L 2 , ∞ σ ( Ω ) of the form u 0 = α U G | t = 1 + v 0 , v 0 ∈ L 2 ( Ω ) 2 | α | ≤ δ , there exists a unique solution u to (NS) with initial data u 0 satisfying k 2 ∥∇ k ( u ( t ) − α U G ( t ) ) ∥ L 2 ( Ω ) = 0 , t →∞ t lim k = 0 , 1 . . · The local L 2 stability is proved by Iftimie-Karch-Lacave (2011). · The similar result holds even when α U G is replaced by the strong solution U obtained in Kozono-Yamazaki (1995) which satisfies 1 4 ∥ U ( t ) ∥ L 4 ( Ω ) ≤ δ ≪ 1 . sup ∥ U ( t ) ∥ L 2 , ∞ ( Ω ) + sup t t > 0 t > 0 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 ) − α U G ( t ) : ∫ t ∥ v ( t ) ∥ 2 ∥∇ v ( s ) ∥ 2 L 2 ( Ω ) + L 2 ( Ω ) d s 1 ≤ C ( ∥ v 0 ∥ L 2 ( Ω ) ) + C 0 α 2 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 U G such as ∥ U G ( t ) ∥ L 4 ( 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 x ⊥ x ∈ R 2 \ { 0 } . t → 0 U G ( t , x ) = U ( x ) = lim 2 π | x | 2 , · For each α ∈ R the velocity α U is a stationary solution to the following Navier-Stokes system in Ω = { x ∈ R 2 | | x | > 1 } . . Navier-Stokes flows around a rotating disk . ∂ t u + u · ∇ u = ∆ u − ∇ p , t > 0 , | x | > 1 , div u = 0 , t ≥ 0 , | x | > 1 , (NS α ) u = α 2 π x ⊥ , t > 0 , | x | = 1 , u | t = 0 = u 0 , | x | > 1 . . Yasunori Maekawa (Tohoku University) Two-dimensional scale-critical flows
3-2. Two important aspects . Steady circular flow with a scale-critical decay . α U ( x ) = α x ⊥ α 2 x ⊥ = ( − x 2 , x 1 ) α 2 P ( x ) = − 2 π | x | 2 8 π 2 | x | 2 . (I) Simple model of the flow around a rotating obstacle · The existence of two-dimensional periodic flows around a rotating obstacle 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 . α U ( x ) = α x ⊥ α 2 x ⊥ = ( − x 2 , x 1 ) α 2 P ( x ) = − 2 π | x | 2 8 π 2 | x | 2 . (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 on both domains and given data. · Hillairet and Wittwer (2013) proved the existence of the steady exterior flows in Ω = { x ∈ R 2 | | x | > 1 } near α U 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 L 2 stability of α U for small | α | . Theorem 2 (M.). . For any sufficiently small | α | there is a constant ϵ = ϵ ( α ) > 0 such that if ∥ u 0 − α U ∥ L 2 ( Ω ) ≤ ϵ then there exists a unique solution u to (NS α ) satisfying k 2 ∥∇ k ( u ( t ) − α U ) ∥ L 2 ( Ω ) = 0 , lim t →∞ t k = 0 , 1 . . The Helmholtz projection P : L 2 ( Ω ) 2 → L 2 σ ( Ω ) satisfies P ∇ p = 0 . . The perturbed Stokes operator . D ( A α ) = W 2 , 2 ( Ω ) 2 ∩ W 1 , 2 0 ( Ω ) 2 ∩ L 2 σ ( Ω ) A α v = − P ∆ v + α P ( U · ∇ v + v · ∇ U ) , v ∈ D ( A α ) . 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 A α by using the polar coordinates and the streamfunction-vorticity formulation . We set ∫ ∞ s 1 −| n | K µ n ( α ) ( sz ) d s , F n ( z ; α ) = Re( z ) > 0 , n ∈ Z \ { 0 } , 1 where K µ is the modified Bessel function of second kind of order µ ∫ ∞ K µ ( z ) = 1 e − z 2 ( t + 1 t ) t − µ − 1 d t , Re( z ) > 0 2 0 and µ n ( α ) = ( n 2 + i α 2 π n ) 1 2 , n ∈ Z \ { 0 } Yasunori Maekawa (Tohoku University) Two-dimensional scale-critical flows
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