Vortex filaments in the 3D Navier-Stokes equations Jacob Bedrossian - PowerPoint PPT Presentation

Vortex filaments in the 3D Navier-Stokes equations Vortex filaments in the 3D Navier-Stokes equations Jacob Bedrossian joint work with Pierre Germain and Ben Harrop-Griffiths Partially supported by the NSF University of Maryland, College Park Department of Mathematics Center for Scientific Computation and Mathematical Modeling July 12, 2018

Vortex filaments in the 3D Navier-Stokes equations Vortex filaments Vortex filaments Vortex filaments are one the most common coherent structures in 3D incompressible fluids 1 2 Models and analysis for their motion and and behavior have been studied, going back at least to Kelvin in his 1880 work. However, the mathematically rigorous derivation of dimension-reduced models, such as the local induction approximation, is not yet developed. 1 AirTeamImages/Daily Mail UK 2 Robert Kozloff/University of Chicago

Vortex filaments in the 3D Navier-Stokes equations Vortex filaments as (extra-)critical initial data 3D Navier-Stokes In momentum form ∂ t u + u · ∇ u + ∇ p = ∆ u ∇ · u = 0; and in vorticity form for ω = ∇ × u ∂ t ω + u · ∇ ω − ω · ∇ u = ∆ ω u = ∇ × ( − ∆) − 1 ω. The scaling symmetry is (hence, L d is critical for u , L d / 2 for ω ): � t � t u ( t , y ) �→ 1 λ 2 , y ω ( t , y ) �→ 1 λ 2 , y � � λ u , λ 2 ω . (1) λ λ Vortex filaments are regions of vorticity highly concentrated along thin tubular neighborhoods:

Vortex filaments in the 3D Navier-Stokes equations Vortex filaments as (extra-)critical initial data Mild solutions We will be interested only in mild solutions satisfying ω ∈ C ∞ ((0 , T ) × R d ): � t ω ( t ) = e t ∆ µ − e ( t − s )∆ ∇ · ( u ⊗ ω − ω ⊗ u ) ds . (2) 0 Generally, well-posedness of mild solutions is closely tied to the scaling symmetry. In momentum form, one of largest critical spaces for which one has local well-posedness for all data is u 0 ∈ L 3 ; in vorticity it is ω 0 ∈ L 3 / 2 .

Vortex filaments in the 3D Navier-Stokes equations Vortex filaments as (extra-)critical initial data Vortex Filaments as (extra-)critical initial data We model vortex filament initial data via measure-valued vorticity directed along a smooth curve γ with constant circulation α ∈ R . 3 They also prove something stronger: if the “scaling-critical” piece of the initial data is small, one gets local existence. E.g. if one has a vortex filament with | α | ≪ 1 and a smooth (but large) background vorticity.

Vortex filaments in the 3D Navier-Stokes equations Vortex filaments as (extra-)critical initial data Vortex Filaments as (extra-)critical initial data We model vortex filament initial data via measure-valued vorticity directed along a smooth curve γ with constant circulation α ∈ R . As observed by Giga-Miyakawa ‘89, measures of this type are in the scaling-critical Morrey space � µ � M 3 / 2 = sup x , R R − 1 | µ ( B ( x , R )) | < ∞ . They proved global well-posedness for small data in this space 3 . The associated velocity field is in the Koch-Tataru space BMO − 1 , but not in L 2 loc , so one cannot associate Leray-Hopf weak solutions to this data. These two larger critical spaces contain self-similar solutions: local well-posedness of mild solutions is known only for small data . 3 They also prove something stronger: if the “scaling-critical” piece of the initial data is small, one gets local existence. E.g. if one has a vortex filament with | α | ≪ 1 and a smooth (but large) background vorticity.

Vortex filaments in the 3D Navier-Stokes equations Vortex filaments as (extra-)critical initial data 2D NSE and 3D axisymmetric flows The Oseen vortex column:   0 0 ω ( t , x , z ) = (3)     4 π t e − | x | 2 α 4 t is a self-similar solution to both 2D and 3D Navier-Stokes. In 3D, it is the canonical infinite, straight vortex filament. It is known to be unique in the class of 2D measure valued initial data [Gallagher-Gallay-Lions ‘05, Gallagher/Gallay ‘05] (in fact the 2D NSE in vorticity form is globally well-posed with measure valued vorticity). Gallay-ˇ Sver´ ak ‘15 later considered vortex ring initial data and obtained existence and uniqueness of mild solutions in the axisymmetric class for such initial data (see also Feng/ˇ Sver´ ak ‘15).

Vortex filaments in the 3D Navier-Stokes equations Large filaments with large (smoother) backgrounds Perturbation of the Oseen vortex column Perturbation of the infinite straight filament Define the space (here ˆ 1 � f ( x , z ) e − iz ζ dz ), f ( x , ζ ) = √ 2 π � � ˆ � f � B z L p = f ( · , ζ ) � L p d ζ. (4) Theorem (JB/Germain/Harrop-Griffiths ‘18) For all α and ω 0 such that for some r ∈ (1 , 2) , x + � x · ω x � ω 0 � B z L 1 0 � r − 1 < ∞ , (5) r B z L r ∩ B z L there exists a time T = T ( � ω 0 � , α ) and a mild solution ω ∈ C w ([0 , T ); B z L 1 ) ∩ C ∞ ((0 , T ) × R 3 ) such that   0  + 1 � log t , x � 0 ω ( t , x , z ) = t Ω c √ t , z + ω b ( t , x , z ) , (6)    4 π t e − | x | 2 α 4 t satisfying (where lim T ց 0 ǫ 0 = 0 ), t 1 / 4 � ω b ( t ) � B z L 4 / 3 �� ξ � m Ω c ( τ ) � B z L 2 sup + sup ξ ≤ ǫ 0 ( T ) . (7) x 0 < t < T −∞ <τ< log T

Vortex filaments in the 3D Navier-Stokes equations Large filaments with large (smoother) backgrounds Perturbation of the Oseen vortex column Comments Small ω 0 implies global existence (‘small’ depends on α ). The proof is a fixed point, so the solutions are automatically unique and stable in the class of solutions whose decomposition admits similar estimates (e.g. filaments with a Gaussian core). Rules out the kind of non-uniqueness 4 discussed in Jia/ˇ Sver´ ak ‘13-‘14 for self-similar solutions in L 3 , ∞ : indeed, the linearization around the filament is stable at all α . 4 Unfortunately, this does not imply uniqueness in the general class of mild solutions satisfying suitable a priori estimates. For example, imagine there is a second, fully 3D self-similar solution that looks like e.g. a helical telephone cord twisting at a scale like O ( √ t ).

Vortex filaments in the 3D Navier-Stokes equations Large filaments with large (smoother) backgrounds Perturbation of the Oseen vortex column Comments Small ω 0 implies global existence (‘small’ depends on α ). The proof is a fixed point, so the solutions are automatically unique and stable in the class of solutions whose decomposition admits similar estimates (e.g. filaments with a Gaussian core). Rules out the kind of non-uniqueness 4 discussed in Jia/ˇ Sver´ ak ‘13-‘14 for self-similar solutions in L 3 , ∞ : indeed, the linearization around the filament is stable at all α . The key structure: in self-similar coordinates ξ = √ t (note, only in x ) the x z dependence is almost entirely subcritical at the linearized level. This turns the intractable looking 3D stability problem into a perturbation of tractable 2D linearized problems. 4 Unfortunately, this does not imply uniqueness in the general class of mild solutions satisfying suitable a priori estimates. For example, imagine there is a second, fully 3D self-similar solution that looks like e.g. a helical telephone cord twisting at a scale like O ( √ t ).

Vortex filaments in the 3D Navier-Stokes equations Large filaments with large (smoother) backgrounds Perturbation of the Oseen vortex column One of the two key linear problems The linearization in self-similar variables becomes: ∂ τ Ω ξ + α g · ∇ ξ Ω ξ − α Ω ξ · ∇ ξ g − α e 2 τ G ∂ z U ξ = 1 � � L + e τ ∂ 2 Ω ξ z ∂ τ Ω z + α g · ∇ ξ Ω z + α U ξ · ∇ ξ G − α e 2 τ G ∂ z U z = 1 � � L + e τ ∂ 2 Ω z , z where G = e −| ξ | 2 , g is the corresponding velocity, L f = ∆ f + 1 2 ∇ · ( ξ f ). After Fourier transforming in z , we can treat this perturbatively as � ∂ τ + e τ | ζ | 2 − L + α Γ � w ξ = α F ξ ∂ τ + e τ | ζ | 2 − L + α Λ w z = α F z , � � where Λ = g · ∇ ξ − ∇ ξ G · ∇ ⊥ ξ ( − ∆ ξ ) − 1 . Γ = g · ∇ ξ − ∇ ξ g , The propagator e t ( L− α Λ) was studied by Gallay/Wayne ‘02 and e t ( L− α Γ) by Gallay/Maekawa ‘11 in their study on 3D stability of the Burgers vortex.

Vortex filaments in the 3D Navier-Stokes equations Large filaments with large (smoother) backgrounds Perturbation of the Oseen vortex column One of the two key linear problems The linearization in self-similar variables becomes: ∂ τ Ω ξ + α g · ∇ ξ Ω ξ − α Ω ξ · ∇ ξ g − α e 2 τ G ∂ z U ξ = 1 � � L + e τ ∂ 2 Ω ξ z ∂ τ Ω z + α g · ∇ ξ Ω z + α U ξ · ∇ ξ G − α e 2 τ G ∂ z U z = 1 � � L + e τ ∂ 2 Ω z , z where G = e −| ξ | 2 , g is the corresponding velocity, L f = ∆ f + 1 2 ∇ · ( ξ f ). After Fourier transforming in z , we can treat this perturbatively as � ∂ τ + e τ | ζ | 2 − L + α Γ � w ξ = α F ξ ∂ τ + e τ | ζ | 2 − L + α Λ w z = α F z , � � where Λ = g · ∇ ξ − ∇ ξ G · ∇ ⊥ ξ ( − ∆ ξ ) − 1 . Γ = g · ∇ ξ − ∇ ξ g , The propagator e t ( L− α Λ) was studied by Gallay/Wayne ‘02 and e t ( L− α Γ) by Gallay/Maekawa ‘11 in their study on 3D stability of the Burgers vortex. The other linear problem we need is the vector transport-diffusion: ∂ t ω + u g · ∇ ω − ω · ∇ u g = ∆ ω, (8) 1 where u g = √ t g ( x √ t ).