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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− = R3×] − ∞, 0[, is an whole space bounded ancient solution of the Navier-Stokes equations if

• Q−
• u · (∂tϕ + ∆ϕ) + u ⊗ u : ∇ϕ
• dz = 0

(0.1) for any ϕ ∈ C ∞

0,0(Q−) := {ϕ ∈ C ∞ 0 (Q−) : div ϕ = 0} and

• Q−

u · ∇qdz = 0 (0.2) 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) ∈ QA := R3×]A, 0[, ui(x, t) =

• R3

Γ(x − y, t − A)ui(y, A)dy+ +

t

• A
• R3

Kijm(x − y, t − τ)uj(y, τ)um(y, τ)dydτ (0.3) Here K is obtained from Φ: ∆Φ(x, t) = Γ(x, t). (0.4) Kmjs(x, y, t) = δmj ∂3Φ ∂yi∂yi∂ys (x, y, t) − ∂3Φ ∂ym∂yj∂ys (x, y, t).

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 ∂tv + v · ∇v − ∆v = −∇q, div v = 0 in Q∞ = R3×]0, ∞[, v(·, 0) = u0(·) ∈ C ∞

0,0(R3) = {v ∈ C ∞ 0 (R3) : div v = 0}

Suppose that there is a finite timeblowup at t = T, i.e., v(·, t)∞,R3 → ∞ as t → T−. Then there exists a sequence zn = (x(n), tn) such that tn > 0, tn increasing, tn → T−, and Mn = |v(z(n))| = sup

0<t≤tn

sup

x∈R3 |v(x, t)| → ∞.

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 vλ(x, t) → λv(λx +x0, λ2t +t0), pλ(x, t) → λ2q(λx +x0, λ2t +t0). Scaling for initial value problem (Seregin, Sverak ’09) uk(y, s) := 1 Mk v( y Mk + xk, s M2

k

+ tk), pk(x, t) := 1 M2

k

q( y Mk + xk, s M2

k

+ tk). Notice that |uk(0, 0| = 1, |uk| 1 in R3×] − M2

ktk, 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) × [−a2, 0]): uk → 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, x3) = (uρ(ρ, x3, t), 0, u3(ρ, x3, t)). has a finite Ladyzhenskaya-Serrin-Prodi quantity i.e

• −∞

R3

|u(x, t)|sdx l

s dt,

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 sup

(x,t)∈Q+

−

(|∂k

t ∇lu(x, t)| + |∂k t ∇l+1p(x, t)|)+

+∂k

t pL∞(−∞,o;BMO(R3) C(k, l, uL∞(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): |∇kΦ(x, t)| c(k) (t + |x|2)

1+k 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+ − = R3 +×] − ∞, 0[, is a half space

bounded ancient solution of the Navier- Stokes equations if

• Q+

−

• u · (∂tϕ + ∆ϕ) + u ⊗ u : ∇ϕ)dxdt = 0

(0.5) for any ϕ ∈ C ∞

0,0(Q− := R3×] − ∞, 0[) with ϕ(x′, 0, t) = 0

• Q+

−

u · ∇qdz = 0 (0.6) for any q ∈ C ∞

0 (Q−).

T.Barker (joint with G.Seregin) Ancient Solutions to Navier-Stokes Equations in Half Space

Half space Kernels

Gij(x, y, t) is Green’s function for Stokes system in half space with no-slip boundary condition Φ = (Φij) are solutions to the following boundary value problems in half space: ∆yΦmn(x, y, t) = Gmn(x, y, t) (0.7) with Neumann b.c if n < 3 and Dirichlet b.c Φmn(x, y, t) = 0 if n = 3 K = (Kmjs) is obtained from Φ as follows: Kmjs(x, y, t) = ∂3Φmj ∂yi∂yi∂ys (x, y, t) − ∂3Φmn ∂yn∂yj∂ys (x, y, t)

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 := R3 +×]A, 0[,

ui(x, t) =

• R3

+

Gij(x, y, t − A)ui(y, A)dy+ +

t

• A
• R3

+

Kijm(x, y, t − τ)uj(y, τ)um(y, τ)dydτ. (0.8)

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+

∞ := R3 +×]0, ∞[. Assume that there is a finite time blow up at

t = T, also Mn = |v(x(n), tn)| → ∞. In the case x(n)

3 Mn → 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) is 2D with sup−∞<t<0(−t)

1 2 u(·, t)∞ < ∞ along with

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 = (Hij) ∈ L∞(R3

+), there exists unique function p1 H with the

following properties: p1

H,even belongs to the space BMO

[p1

H]B+ = 0

• R3

+

p1

H∆ϕdx = −

• R3

+

H : ∇2ϕdx for any ϕ ∈ C ∞

0 (R3 +) with ϕ,3(x′, 0) = 0

p1

H,evenBMO ≤ AH∞,R3

+ 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 = p1

u⊗u + p2, where

∆p2(·, t) = 0 (0.9) |∇p2(x, t)| ≤ c ln(2 + 1/x3) (0.10) for all (x, t) ∈ Q+

−.

sup

x′∈R2 |∇p2(x, t)| → 0

(0.11) as x3 → ∞ 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+

− := R3 +×] − ∞, 0[ with

non-slip boundary condition is only achieved through shear flows of form w(x3, t) = (w1(x3, t), w2(x3, 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

Smoothness up to boundary for half space mbas (Barker,Seregin ’15)

Theorem Suppose u ∈ L∞(Q+

−) is an arbitrary mild bounded ancient

solution in Q+

−. Then u is of class C ∞ and moreover

sup

(x,t)∈Q+

−

(|∂k

t ∇lu(x, t)| + |∂k t ∇l+1p(x, t)|)+

+∂k

t p1L∞(−∞,o;BMO(R3) C(k, l, uL∞(Q+

−)) < ∞

for any k, l = 0, 1. . . . Here, p1 = p1

u⊗u.

T.Barker (joint with G.Seregin) Ancient Solutions to Navier-Stokes Equations in Half Space

Comparison with whole space proof/ ingredients of proof

Comparison with whole space Cannot get boundedness of higher spatial derivatives near boundary via CKN/bootstrapping (see e.g counterexample by Kang ’05, Seregin-Sverak ’10) Ingredients of proof Split Green’s function into heat part (G 1) and the perturbed part (G 2).Split u = u1 + u2. Global arguments involving G 2 give ∇u2Ls,unif < ∞. Estimating ∇u1Ls,unif is standard Pressure equation (and u solenoidal) → ∇p1

u⊗u bounded

→ ∇u1 bounded

T.Barker (joint with G.Seregin) Ancient Solutions to Navier-Stokes Equations in Half Space

Ingredients of proof continued

Use global estimates for G 2 to get ∇u bounded. Also get ∂tu ∈ Ls,unif and bounded away from boundary. Split singular integral ∂tp1

u⊗u close to and far from boundary.

Use this to show ∂tu bounded Bootstrap to get ∇∂k

t u, ∇∂k t p1 u⊗u bounded

Conclusion reached by regularity of stationary problem involving higher time derivatives and bootstrapping

T.Barker (joint with G.Seregin) Ancient Solutions to Navier-Stokes Equations in Half Space