Ancient Solutions to Navier-Stokes Equations in Half Space T.Barker (joint with G.Seregin) University of Oxford September 11, 2015 T.Barker (joint with G.Seregin) Ancient Solutions to Navier-Stokes Equations in Half Space
Bounded ancient solutions of Navier-Stokes equations in whole space Definition A u ∈ L ∞ ( Q − ) , where Q − = R 3 × ] − ∞ , 0[ , is an whole space bounded ancient solution of the Navier-Stokes equations if � � � u · ( ∂ t ϕ + ∆ ϕ ) + u ⊗ u : ∇ ϕ dz = 0 (0.1) Q − for any ϕ ∈ C ∞ 0 , 0 ( Q − ) := { ϕ ∈ C ∞ 0 ( Q − ) : div ϕ = 0 } and � u · ∇ qdz = 0 (0.2) Q − for any q ∈ C ∞ 0 ( Q − ) . T.Barker (joint with G.Seregin) Ancient Solutions to Navier-Stokes Equations in Half Space
Mild bounded ancient solution of Navier-Stokes equations Definition Bounded solenoidal u is a whole space mbas if for any A < 0 and for ( x , t ) ∈ Q A := R 3 × ] A , 0[ , � u i ( x , t ) = Γ( x − y , t − A ) u i ( y , A ) dy + R 3 t � � + K ijm ( x − y , t − τ ) u j ( y , τ ) u m ( y , τ ) dyd τ (0.3) R 3 A Here K is obtained from Φ : ∆Φ( x , t ) = Γ( x , t ) . (0.4) ∂ 3 Φ ∂ 3 Φ K mjs ( x , y , t ) = δ mj ( x , y , t ) − ( x , y , t ) . ∂ y i ∂ y i ∂ y s ∂ y m ∂ y j ∂ y s T.Barker (joint with G.Seregin) Ancient Solutions to Navier-Stokes Equations in Half Space
Motivation for mbas :Navier Stokes finite time blow up in whole space Consider the following initial value problem ∂ t v + v · ∇ v − ∆ v = −∇ q , div v = 0 in Q ∞ = R 3 × ]0 , ∞ [, 0 , 0 ( R 3 ) = { v ∈ C ∞ 0 ( R 3 ) : div v = 0 } v ( · , 0) = u 0 ( · ) ∈ C ∞ Suppose that there is a finite timeblowup at t = T , i.e., � v ( · , t ) � ∞ , R 3 → ∞ as t → T − . Then there exists a sequence z n = ( x ( n ) , t n ) such that t n > 0, t n increasing, t n → T − , and M n = | v ( z ( n ) ) | = sup x ∈ R 3 | v ( x , t ) | → ∞ . sup 0 < t ≤ t n T.Barker (joint with G.Seregin) Ancient Solutions to Navier-Stokes Equations in Half Space
Rescaling about singularities Key is that Navier Stokes equations invariant under scaling v λ ( x , t ) → λ v ( λ x + x 0 , λ 2 t + t 0 ) , p λ ( x , t ) → λ 2 q ( λ x + x 0 , λ 2 t + t 0 ) . Scaling for initial value problem (Seregin, Sverak ’09) 1 v ( y + x k , s u k ( y , s ) := + t k ) , M 2 M k M k k 1 q ( y + x k , s p k ( x , t ) := + t k ) . M 2 M 2 M k k k Notice that | u k (0 , 0 | = 1 , | u k | � 1 in R 3 × ] − M 2 k t k , 0[ . T.Barker (joint with G.Seregin) Ancient Solutions to Navier-Stokes Equations in Half Space
Limit of rescaled solutions, Liouville Conjecture It was shown by Seregin, Sverak ’09 that in C ( ¯ B ( a ) × [ − a 2 , 0]): u k → u for any a > 0. Where, u is whole space mbas with | u (0 , 0) | = 1. The whole space Liouville conjecture made by Koch, Nadirashvili, Seregin and Sverak ’09 is Conjecture Any whole space mbas is a constant. Notice: Liouville theorem + scale invariant estimate → no finite time blow up! T.Barker (joint with G.Seregin) Ancient Solutions to Navier-Stokes Equations in Half Space
Situations where Liouville theorem holds A whole space mbas is constant when the original velocity: is 2-D is axisymmetric with no swirl i.e u ( ρ, x 3 ) = ( u ρ ( ρ, x 3 , t ) , 0 , u 3 ( ρ, x 3 , t )). has a finite Ladyzhenskaya-Serrin-Prodi quantity i.e 0 � l � � � s dt , | u ( x , t ) | s dx R 3 −∞ with 3 s + 2 l = 1 and l < ∞ . See Seregin, Sverak ’09 and Koch, Nadirashvili, Seregin and Sverak ’09. T.Barker (joint with G.Seregin) Ancient Solutions to Navier-Stokes Equations in Half Space
Equivalent definition of wholespace mbas and smoothness Proposition A whole space bounded ancient solution u is a whole space mbas if and only if for the corresponding pressure we have p ∈ L ∞ ( −∞ , 0; BMO ) Theorem If u is whole space mbas in Q − . Then u is of class C ∞ and moreover t ∇ l +1 p ( x , t ) | )+ ( | ∂ k t ∇ l u ( x , t ) | + | ∂ k sup ( x , t ) ∈ Q + − + � ∂ k t p � L ∞ ( −∞ , o ; BMO ( R 3 ) � C ( k , l , � u � L ∞ ( Q − ) ) < ∞ for any k, l = 0 , 1 . . . . T.Barker (joint with G.Seregin) Ancient Solutions to Navier-Stokes Equations in Half Space
Main ideas of whole space mbas proofs Smoothness Use p ∈ L ∞ ( −∞ , 0; BMO ) and velocity bounded to apply CKN and (interior) local theory to get spatial smoothness for u and ∇ p . Get spatial smoothness for higher time derivatives from the NSE and estimate pressure through singular integral representation. Equivalence Main ingredients are to apply Tychonoff’s uniqueness theorem with rhs f = − div u × u − ∇ p along with kernel decay (Solonnikov ’64): c ( k ) |∇ k Φ( x , t ) | � . 1+ k ( t + | x | 2 ) 2 Remark Pressure condition rules out parasitic solutions of form u ( x , t ) = c ( t ) , p ( x , t ) = − c ′ ( t ) · x . T.Barker (joint with G.Seregin) Ancient Solutions to Navier-Stokes Equations in Half Space
Bounded ancient solutions of Navier-Stokes equations in half space Definition u ∈ L ∞ ( Q + − ) , where Q + − = R 3 + × ] − ∞ , 0[ , is a half space bounded ancient solution of the Navier- Stokes equations if � � u · ( ∂ t ϕ + ∆ ϕ ) + u ⊗ u : ∇ ϕ ) dxdt = 0 (0.5) Q + − for any ϕ ∈ C ∞ 0 , 0 ( Q − := R 3 × ] − ∞ , 0[) with ϕ ( x ′ , 0 , t ) = 0 � u · ∇ qdz = 0 (0.6) Q + − for any q ∈ C ∞ 0 ( Q − ) . T.Barker (joint with G.Seregin) Ancient Solutions to Navier-Stokes Equations in Half Space
Half space Kernels G ij ( x , y , t ) is Green’s function for Stokes system in half space with no-slip boundary condition Φ = (Φ ij ) are solutions to the following boundary value problems in half space: ∆ y Φ mn ( x , y , t ) = G mn ( x , y , t ) (0.7) with Neumann b.c if n < 3 and Dirichlet b.c Φ mn ( x , y , t ) = 0 if n = 3 K = ( K mjs ) is obtained from Φ as follows: ∂ 3 Φ mj ∂ 3 Φ mn K mjs ( x , y , t ) = ( x , y , t ) − ( x , y , t ) ∂ y i ∂ y i ∂ y s ∂ y n ∂ y j ∂ y s T.Barker (joint with G.Seregin) Ancient Solutions to Navier-Stokes Equations in Half Space
Mild bounded ancient solution in half space Definition A bounded divergence free function u in Q + − is called a half space mbas if, for any A < 0 and any ( x , t ) ∈ Q + A := R 3 + × ] A , 0[ , � u i ( x , t ) = G ij ( x , y , t − A ) u i ( y , A ) dy + R 3 + t � � + K ijm ( x , y , t − τ ) u j ( y , τ ) u m ( y , τ ) dyd τ. (0.8) A R 3 + T.Barker (joint with G.Seregin) Ancient Solutions to Navier-Stokes Equations in Half Space
Motivation for half space mbas: Navier Stokes finite time blow up in half space Consider the Navier-Stokes initial boundary value problem in Q + ∞ := R 3 + × ]0 , ∞ [. Assume that there is a finite time blow up at t = T , also M n = | v ( x ( n ) , t n ) | → ∞ . In the case x ( n ) 3 M n → a < ∞ , it is shown in Seregin-Sverak ’13 that a non-trivial half space mbas is obtained as a locally uniform limit of rescaled solutions. They also showed that the case a = ∞ produces a whole space mbas . T.Barker (joint with G.Seregin) Ancient Solutions to Navier-Stokes Equations in Half Space
Half space Liouville conjecture. Situations where half space Liouville theorem holds. The half space Liouville conjecture made in Seregin, Sverak ’13 is Conjecture Any half space mbas is a constant. Notice: Liouville theorem whole space+ Liouville theorem half space + scale invariant estimate → no finite time blow up! A half space mbas is constant when the original velocity is 2D with bounded kinetic energy (Seregin ’14) 1 2 � u ( · , t ) � ∞ < ∞ along with is 2D with sup −∞ < t < 0 ( − t ) positive vorticity (Giga et al ’14) T.Barker (joint with G.Seregin) Ancient Solutions to Navier-Stokes Equations in Half Space
Defining a half space pressure operator Let H = ( H ij ) ∈ L ∞ ( R 3 + ), there exists unique function p 1 H with the following properties: p 1 H , even belongs to the space BMO [ p 1 H ] B + = 0 � � p 1 H : ∇ 2 ϕ dx H ∆ ϕ dx = − R 3 R 3 + + 0 ( R 3 for any ϕ ∈ C ∞ + ) with ϕ , 3 ( x ′ , 0) = 0 � p 1 H , even � BMO ≤ A � H � ∞ , R 3 + T.Barker (joint with G.Seregin) Ancient Solutions to Navier-Stokes Equations in Half Space
Equivalence theorem for half space mbas (Barker, Seregin ’15) Theorem A half space bounded ancient solution to Navier-Stokes equations is a half space mbas if and only if the pressure p is such that p = p 1 u ⊗ u + p 2 , where ∆ p 2 ( · , t ) = 0 (0.9) |∇ p 2 ( x , t ) | ≤ c ln(2 + 1 / x 3 ) (0.10) for all ( x , t ) ∈ Q + − . x ′ ∈ R 2 |∇ p 2 ( x , t ) | → 0 sup (0.11) as x 3 → ∞ and for any t < 0 T.Barker (joint with G.Seregin) Ancient Solutions to Navier-Stokes Equations in Half Space
Idea behind equivalence proof Jia, Seregin, Sverak ’12 showed non-uniqueness for bounded solutions of Stokes system in Q + − := R 3 + × ] − ∞ , 0[ with non-slip boundary condition is only achieved through shear flows of form w ( x 3 , t ) = ( w 1 ( x 3 , t ) , w 2 ( x 3 , t ) , 0) . Gradient of pressure is given as a function of time. We show that the shear flow (which is the difference of the two velocities) must be zero using the decay of gradient of harmonic part for both the velocities. T.Barker (joint with G.Seregin) Ancient Solutions to Navier-Stokes Equations in Half Space
Recommend
More recommend
