Ancient Solutions to the Navier-Stokes Equations

Gregory A. Seregin

December 19, 2012

DefinitionA divergence free field u ∈ L∞(Q−;Rn) is called a weak boundedancient solution (or simply bounded ancient solution) to theNavier-Stokes equations if∫

Q−

(u · ∂tw + u ⊗ u : ∇w + u ·∆w)dz = 0

for any w ∈ C∞0,0(Q−).

Let u be an arbitrary bounded ancient solution. For any numberm > 1,

|∇u|+ |∇2u|+ |∇pu⊗u| ∈ Lm(Q−).

Moreover, for each t0 ≤ 0, there exists a functionbt0 ∈ L∞(t0 − 1, t0) with the following property

supt0≤0‖bt0‖L∞(t0−1,t0) ≤ c(n) < +∞.

If we let ut0(x , t) = u(x , t) + bt0(t) in Qt0 = Rn×]t0 − 1, t0[, then,for any number m > 1 and for any point x0 ∈ Rn, the uniformestimate

‖ut0‖W 2,1

m (Q(z0,1))≤ c(m, n) < +∞, z0 = (x0, t0),

is valid and, for a.a. z = (z , t) ∈ Qt0 , functions u and ut0 obey thesystem of equations

∂tut0 + divu ⊗ u −∆u = −∇pu⊗u, divu = 0.

The first equation of the latter system can be rewritten in thefollowing way

∂tu + divu ⊗ u −∆u = −∇pu⊗u − b′t0 , b′t0(t) = dbt0(t)/dt,

in Qt0 in the sense of distributions. So, the real pressure field inQt0 is the following distribution pu⊗u + b′t0 · x

We can find a measurable vector-valued function b defined on]−∞, 0[ and having the following property. For any t0 ≤ 0, thereexists a constant vector ct0 such that

supt0≤0‖b − ct0‖L∞(t0−1,t0) < +∞.

Moreover, the Navier-Stokes system takes the form

∂tu + divu ⊗ u −∆u = −∇(pu⊗u + b′ · x), divu = 0

in Q− in the sense of distributions.

A vector field u is called a mild bounded ancient solution if u is abounded ancient solution and there exists a pressure fieldp ∈ L∞(−∞, 0;BMO(R3)) such that∫

Q−

(u · (∂tϕ+ ∆ϕ) + u ⊗ u : ∇ϕ

)dz = −

∫Q−

p divϕ (1)

for any ϕ ∈ C∞0 (Q−). Without loss of generality, we may assumethat p = pu⊗u.

LemmaLet u a mild bounded ancient solution. Then u is of class C∞ andmoreover

sup(x ,t)∈Q−

(|∂kt ∇lu(x , t)|+ |∂kt ∇l+1p(x , t)|)+

+‖∂kt p‖L∞(BMO) ≤ C (k , l , ‖p‖L∞(BMO)) <∞

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

Any mild bounded ancient solution has the following property: forany A < 0, they can be presented in the form

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, (2)

where Γ is a standard heat kernel and K is obtained from theOseen tensor

T (x , t) = Γ(x , t)I−∇2Φ(x , t),

where∆Φ(x , t) = Γ(x , t),

in the following way

Kijm(x , t) = Tij ,m(x , t).

A vector-valued function u is a mild bounded ancient solution ifand only if u is a bounded and, for any A < 0, (2) holds.

Any mild bounded ancient solution is a constant.

TheoremLet u be a mild bounded ancient solution to the Navier-Stokesequations.Assume that

sup0<r<∞

Ms,l(u; r) =

0∫−∞

(∫R3

|u(x , t)|sdx) l

s<∞

with 3/s + 2/l = 1 and l <∞. Then u ≡ 0 in Q−.

TheoremAssume that n = 2 and u is an arbitrary bounded ancient solution.Then u(x , t) = b(t) for any x ∈ R2.

LemmaLet functions

ω ∈ W2,1m (Q−) = {u ∈W 2,1

m,loc(Q−) : supz0∈Q−

‖u‖W 2,1

m (Q(z0,1))<∞},

with m > 3, and u ∈ L∞(Q−) satisfy the equation

∂tω + u · ∇ω −∆ω = 0 in Q−

and the inequality|u| ≤ 1 in Q−.

Then, for any positive numbers ε and R, there exists a pointz0 = (x0, t0), x0 ∈ R2 and t0 ≤ 0, such that

ω(z) ≥ M − ε, z ∈ Q(z0,R),

where M = supz∈Q−

ω(z).

TheoremStrong maximum principle Let functionsw ∈W 2,1

m (Q(z0,R)) with m > n + 1 and a ∈ L∞(Q(z0,R);Rn)satisfy the equation

∂tw + a · ∇w −∆w = 0 in Q(z0,R).

Let, in addition,w(z0) = sup

z∈Q(z0,R)w(z).

Thenw(z) = w(z0) in Q(z0,R).

TheoremLet u be an arbitrary axially symmetric bounded ancient solutionwith zero swirl. Then u(x , t) = b(t) for any x ∈ R3 and for anyt ≤ 0. Moreover, u1(x , t) = 0 and u2(x , t) = 0 for the same x andt or, equivalently, u%(%, x3, t) = 0 for any % > 0, for any x3 ∈ R,and for any t ≤ 0.

TheoremLet u be an arbitrary axially symmetric bounded ancient solutionsatisfying assumption

|u(x , t)| ≤ A

|x ′|, x = (x ′, x3) ∈ R3, −∞ < t ≤ 0, (1)

where A is a positive constant independent of x and t. Then u ≡ 0in Q−.

∂tv + v · ∇v −4v = −∇q, div v = 0 (2)

in R3×]0,∞[,v(x , 0) = v0(x) ∈ C∞0,0(R3) (3)

for any x ∈ R3. Here, C∞0,0(R3) = {v ∈ C∞0 (R3) : div v = 0}.

g(t) = sup0<τ≤t

M(τ)→∞ as t → T − 0, (4)

whereM(t) = sup

x∈R3

|v(x , t)|.

Let ω be a open set in R3. We say that a pair u and p is a suitableweak solution to the Navier-Stokes equations in ω×]T1,T [ if uand p satisfy the conditions:

u ∈ L2,∞(ω×]T1,T [) ∩ L2(T1,T ;W 12 (ω)); (5)

p ∈ L 32(ω×]T1,T [); (6)

∂tu + u · ∇u −∆u = −∇p, div u = 0 (7)

in the sense of distributions;the local energy inequality∫

ωϕ(x , t)|u(x , t)|2 dx + 2

∫ω×]T1,t[

ϕ|∇u|2 dxdt ′

≤∫

ω×]T1,t[

(|u|2(∆ϕ+ ∂tϕ) + u · ∇ϕ(|u|2 + 2q)) dxdt ′

(8)

holds for a.a. t ∈]T1,T [ and all nonnegative functionsϕ ∈ C∞0 (ω×]T1,∞[).

Now, we are in a position to explain ε-regularity theory. Quantitiesthat are invariant with respect to the Navier-Stokes scaling

vλ(y , s) = λv(x0 + λy , t0 + λ2s),

qλ(y , s) = λ2q(x0 + λy , t0 + λ2s). (9)

play the crucial role in this theory. By the definition, suchquantities are defined on parabolic balls Q(r) and have theproperty

F (v , q; r) = F (vλ, qλ; r/λ).

Suppose that v and q are a suitable weak solution to theNavier-Stokes equations in Q. There exist universal positiveconstants ε and {ck |}∞k=1 such that if F (v , q; 1) < ε then|∇kv(0)| < ck , k = 0, 1, 2, .... Moreover, the function z 7→ ∇kv(z)is Holder continuous (relative to the usual parabolic metric) withany exponent less 1/3 in the closure of Q(1/2).

F (v , q; r) =1

r2

∫Q(r)

(|v |3 + |q|

32

)dz .

Let v and q be a suitable weak solution in Q. There exists auniversal positive constant ε with the property: ifsup0<r<1 F (v ; r) < ε then z = 0 is a regular point. Moreover, forany k = 0, 1, 2, ..., the function z 7→ ∇kv(z) is Holder continuouswith any exponent less 1/3 in the closure of Q(r) for some positiver .

F (v ; r) = Ms,l(v ; r) = ‖v‖ls,l ,Q(r) =

0∫−r2

( ∫B(r)

|v |sdx) l

sdt

provided3

s+

2

l= 1

A(v ; z0, r) ≡ supt0−r2≤t≤t0

1

r

∫B(x0,r)

|v(x , t)|2 dx ,

E (v ; z0, r) ≡ 1

r

∫Q(z0,r)

|∇ v |2 dz ,

C (v ; z0, r) ≡ 1

r2

∫Q(z0,r)

|v |3 dz ,

D0(q; z0, r) ≡ 1

r2

∫Q(z0,r)

|q − [q]x0,r |32 dz .

Proposition

Let v and q be a suitable weak solution to the Navier-Stokesequations in Q. Given M > 0, there exists a positive number ε(M)having the property: if two inequalities lim supr→0 E (r) < M and

lim infr→0

E (r) < ε(M)

hold, then z = 0 is a regular point of v .

G1(v ; r) = supz=(x ,t)∈Q(r)

|x ||v(z)|,

G2(v ; r) = supz=(x ,t)∈Q(r)

√−t|v(z)|.

Proposition

Let v and q be a suitable weak solution to the Navier-Stokesequations in Q and z = 0 be a singular point of v . There exist twofunctions v and q having the following properties:(i) v ∈ L3(Q) and q ∈ L 3

2(Q) obey the Navier-Stokes equations in

Q in the sense of distributions;(ii) v ∈ L∞(B×]− 1,−a2[) for all a ∈]0, 1[;(iii) there exists a number 0 < r1 < 1 such thatv ∈ L∞({(x , t) : r1 < |x | < 1, −1 < t < 0}).Moreover, functions v and q are obtained from v and q with thehelp of the space-times shift and the Navier-Stokes scaling and theorigin remains to be a singular point of v .

u(k)(y , s) = λkv(x , t), p(k)(y , s) = λ2kq(x , t)

withx = x (k) + λky , x = tk + λ2ks,

where x (k) ∈ R3, −1 < tk ≤ 0, and λk > 0 are parameters of thescaling and λk → 0 as k → +∞.

λk = 1/Mk , where a sequence Mk is defined as

Mk = ‖v(, tk)‖∞,B(r) = |v(x (k), tk)|

with x (k) ∈ B(r1) for sufficiently large k .

Proposition

There exist a subsequence of u(k) (still denoted by u(k)) and amild bounded ancient solution u such that, for any a > 0, thesequence u(k) converges uniformly to u on the closure of the setQ(a) = B(a)×]− a2, 0[. The function u has the additionalproperties: |u| ≤ 1 in R3×]−∞, 0[ and |u(0)| = 1.

what happens if we drop the condition on smallness ofscale-invariant quantities, assuming their uniform boundednessonly, i.e, sup0<r<1 F (v , r) < +∞.

Boundedness of

sup0<r<1

G2(v ; r) = G2(v , 1) = G20 < +∞

can be rewritten in the form

|v(z)| ≤ G20√−t

for all z = (x , t) ∈ Q. If v satisfies the above inequality and z = 0is still a singular point of v , we say that a singularity of Type I orType I blowup takes place at t = 0.

Proposition

Let functions v and q be a suitable weak solution to theNavier-Stokes equations in Q.(i) If min{G1(v ; 1),G2(v ; 1)} < +∞, then

g = sup0<r<1

{A(v ; r) + C (v ; r) + D(q; r) + E (v ; r)} < +∞.

(ii) If

g ′ = min{ sup0<r<1

A(v ; r), sup0<r<1

C (v ; r), sup0<r<1

E (v ; r)} < +∞,

then g < +∞.

validity of the conjecture would rule out Type I blowups

∫Q+−

(u · (∂tϕ+ ∆ϕ) + u ⊗ u : ∇ϕ

)dz = 0 (10)

for any ϕ ∈ C∞0,0(Q−) with ϕ(x ′, 0, t) = 0 for any x ′ ∈ R2 and forany −∞ < t < 0; ∫

Q+−

u · ∇qdz = 0 (11)

for any q ∈ C∞0 (Q−).

u(x , t) = (u1(x3, t), u2(x3, t), 0). (12)

A bounded function u is a mild bounded ancient solution if andonly if there exists a pressure p such that p = p1 + p2, where

4p1 = −divdiv u ⊗ u (13)

in Q+− with p1,3(x ′, 0, t) = 0 and p2(·, t) is a harmonic function in

R3+ whose gradient satisfies the estimate

|∇p2(x , t)| ≤ c ln(2 + 1/x3) (14)

for all (x , t) ∈ Q+− and has the property

supx ′∈R2

|∇p2(x , t)| → 0 (15)

as x3 →∞; u and p satisfy (11) and∫Q+−

(u · (∂tϕ+ ∆ϕ) + u ⊗ u : ∇ϕ+ pdivϕ

)dxdt = 0 (16)

for any ϕ ∈ C∞0 (Q−) with ϕ(x ′, 0, t) = 0 for any x ′ ∈ R2 and forany t < 0.

If u is a mild bounded ancient solution, then ∇u ∈ L∞(Q+− ). The

function u is infinitely smooth in spatial variables in upper halfspace x3 > 0.

Conjecture There is no non-trivial mild bounded ancient solutionto the Navier-Stokes equations in the half space.

∂tv + v · ∇v −4v = −∇q, div v = 0 (17)

in R3+×]0,∞[,

v(x ′, 0, t) = 0 (18)

for any x ′ ∈ R2 and t > 0,

v(x , 0) = v0(x) ∈ C∞0,0(R3+) (19)

for any x ∈ R3+. Here, C∞0,0(R3

+) = {v ∈ C∞0 (R3+) : div v = 0}.

Assume that the initial boundary value problem has a solution thatblows up at time T . There exists at least one non-trivial(non-zero) mild bounded ancient solution either in the whole spaceor in the half space.

Assume that for some positive constant A

|v(x , t)| ≤ A

x3(20)

for all x ∈ R3+ and t ∈]0,T [. There exists a positive constant ε

such that if the initial boundary value problem has a solution thatblows up at time T . Then there must be

A ≥ ε. (21)