Non decaying solutions to the Navier-Stokes equations in the - - PowerPoint PPT Presentation

Non decaying solutions to the Navier-Stokes equations in the half-space

Yasunori Maekawa (Kyoto University) Hideyuki Miura (Tokyo Institute of Technology) Christophe Prange (CNRS & Universit´ e de Bordeaux) Mathflows, Porquerolles September 5, 2018

Navier-Stokes equations

Navier-Stokes equations

(NS)      ∂tV + V · ∇V − ∆V + ∇P = 0 , x ∈ Ω , t ∈ (0, T) , ∇ · V = 0 , x ∈ Ω , t ∈ [0, T) , V |t=0 = V0 , x ∈ Ω . Boundary condition V = 0 on ∂Ω × (0, T) Scaling Ω = R3 or R3

+, λ ∈ (0, ∞),

V0,λ(y) = λV0(λy) Vλ(y, s) = λV (λy, λ2s), ∀y ∈ Ω, s > 0 Criticality L3(R3), ˙ H

1 2 (R3) scale critical norms 2 / 27

Navier-Stokes equations

Leray-Hopf solutions

V is a Leray-Hopf or a finite energy weak solution to (NS) for initial data V0 ∈ L2

σ(Ω) if

for all T < ∞, V ∈ L∞((0, T); L2(Ω)) ∩ L2((0, T); H1(Ω)); V satisfies (NS) in the sense of distributions; V satisfies the global energy inequality for all t ∈ (0, ∞)

• Ω

|V (·, t)|2 + 2 t

• Ω

|∇V |2 ≤

• Ω

|V0|2; we have V (·, t) − V0L2(Ω) → 0, when t → 0.

Regular vs. singular point

A point (x0, t0) is a regular point if V is bounded in a parabolic cylinder Qr(x0, t0) for r > 0, otherwise it is a singular point.

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Outline of the talk

1 Motivations 2 Mild solutions and concentration near blow-up 3 Non uniqueness 4 Local energy weak solutions and blow-up of critical norms 4 / 27

Motivation 1: blow-up of critical norms

Theorem (Seregin 2012)

Let V a Leray-Hopf solution with initial data V0 ∈ C∞

c,σ(R3).

Assume that T > 0 is a blow-up time. Then V (·, t)L3(R3) → ∞, t → T −. By contraposition, assume that there exists M ∈ (0, ∞) and tn → T − such that V (·, tn)L3(R3) ≤ M. Consider Vn(y, s) := λnV (λny, T + λ2

ns),

λn :=

• T − tn

S . Then Vn(·, −S)L3 ≤ M. Remains to see: a priori bounds, convergence to a limit blow-up solution V , V = 0 by backward uniqueness and ε-regularity for smoothness.

5 / 27

Motivation 1: blow-up of critical norms

Bounds There exists S(M) ∈ (0, ∞) and A(M) ∈ (0, ∞) such that sup

s∈(−S,0)

sup

x0∈R3 Vn(·, s)2 L2(B(x0,1)) + sup x0∈R3 −S

• B(x0,1)

|∇Vn|2 dyds ≤ A. Convergence Vn converges to a V (Blow-up solution) which is a Local Energy Weak Solution (LEWS) to (NS) in R3 × (−S, 0): local energy inequality, weak solution, weak continuity in time, representation formula for the pressure. Strong convergence to initial data: sup

x0∈R3 V (·, s) − e(s+S)∆V (·, −S)2 L2(B(x0,1)) ≤ C(M)(s + S)

1 5 .

At final time V (·, T) ∈ L3, implies V (·, 0) = 0. Liouville Transfer mild decay of V (·, −S) ∈ L3 to V . Implies smoothness of V in R3 \ B(0, R) × (−S′, 0), S′ < S and V = 0 by backward uniqueness and unique continuation.

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Motivation 2: large forward self-similar solutions

Forward self-similar V (x, t) =

1 √ tV ( x √ t, 1).

Initial data −1 homogeneous: for example V0(x) = ( −x2

|x|2 , x1 |x|2 , 0) belongs to

L2

uloc(R3) :=

• v ∈ L2

loc :

sup

x0∈R3 vL2(B(x0,1)) < ∞

and vL2(B(x0,1))

x0→∞

− → 0

• .

Jia, Sverak 2014

For scale-invariant divergence-free V0 ∈ C∞(R3 \ {0}) and any scale invariant LEWS V , we have V (·, 1) ∈ C∞ and |∂α(V (x, 1) − V0(x))| ≤ C(α, V0) (1 + |x|)3+|α| , ∀|α| ≥ 0. A priori estimate coming from local in space near initial time regularity result for the LEWS V .

7 / 27

Motivation 3: singular Burgers vortex

Moffatt (2000) found blow-up solutions to (NS) which have a similar structure to the regular Burgers vortex: V sB(t, x) = µ T − t   −x1 −x2 2x3  +α

• µ − 1

T − t U G µ − 1 T − t x′ , x′ = (x1, x2)⊤, ΩsB(t, x) = ∇ × V SB(t, x) = α µ − 1 T − t G

• µ − 1

T − t x′ where U G(X′) = 1 − e− |X′|2

4

2π|X′|2   −X2 X1   , G(X′) =   

1 4πe− |X′|2

4

   and µ > 1 : magnitude of the strain α ∈ R : circulation at infinity. Blow-up in backward self-similar form, but out of Cafarelli-Kohn-Nirenberg (1982) or Necas-Ruzicka-Sverak (1997), Tsai (1998). Stability of blow-up solution: Maekawa, Miura, P. 2018.

8 / 27

Linear theory: resolvent estimates

Resolvent problem

(R)      λU − ∆U + ∇P = f , x ∈ R3

+,

∇ · U = 0 , x ∈ R3

+,

U|x3=0 = 0.

Theorem (Maekawa, Miura, P. 2017)

For all ε > 0, λ ∈ Sπ−ε, q ∈ (1, ∞), there exists C(ε, q) ∈ (0, ∞), for all f ∈ Lq

uloc,σ(R3 +), there is a unique solution to the Stokes resolvent

problem with ∇′PL1(|x′|<1,R<xd<R+1)

R→∞

− → 0 and |λ|ULq

uloc + |λ| 1 2 ∇ULq uloc ≤ CfLq uloc,

∇2ULq

uloc + ∇PLq uloc ≤ C

• 1 + e−c|λ|

1 2 log |λ|

• fLq

uloc,

q = ∞. Desch, Hieber, Pr¨ uss 2001; Abe, Giga, Hieber 2015

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Linear theory: proof of the resolvent estimates

Main source of inspiration: Desch, Hieber, Pr¨ uss 2001. Decompose U = UD.L. + Unonloc: for all ξ ∈ R2, x3 > 0,

• UD.L.(ξ, x3) =

1 2ωλ(ξ) ∞ (e−ωλ(ξ)|x3−z3| − e−ωλ(ξ)(x3+z3)) f(ξ, z3)dz3

• U ′

nonloc(ξ, x3) = −iξ

2ωλ(ξ) ∞ (e−ωλ(ξ)|x3−z3| − e−ωλ(ξ)(x3+z3))e−|ξ|z3 P0(ξ)dz3

• Unonloc,3(ξ, x3) =

|ξ| 2ωλ(ξ) ∞ (e−ωλ(ξ)|x3−z3| − e−ωλ(ξ)(x3+z3))e−|ξ|z3 P0(ξ)dz3, where ωλ(ξ) :=

• λ + |ξ|2 and for ξ = 0
• P0(ξ) = −ωλ(ξ) + |ξ|

|ξ| ∞ e−ωλ(ξ)z3 f3(ξ, z3)dz3. Integration by parts:

• Unonloc(ξ, x3) ≃ 1

λ(e−|ξ|x3 − e−ωλ(ξ)x3)ξ ⊗ ξ |ξ| ∞ e−ωλ(ξ)y3 f′(ξ, y3)dy3

10 / 27

Linear theory: proof of the resolvent estimates

Estimates singularity at 0 and decay for kernel sλ(x′, x3, z3) := 1 λ

• Rd−1 eix′·ξ

e−|ξ|x3 − e−ωλ(ξ)x3 e−ωλ(ξ)z3 ξ ⊗ ξ |ξ| dξ.

Pointwise estimate

There exist c(ε), C(ε) ∈ (0, ∞) such that for all λ ∈ Sπ−ε, x′ ∈ R2, z3, x3 > 0, |sλ| ≤ Cx3 (x3 + z3 + |x′|)2 e−c|λ|

1 2 z3

• 1 + |λ|

1 2 (x3 + z3 + |x′|)

• 1 + |λ|

1 2 (x3 + z3)

• 11 / 27

Linear theory: proof of the resolvent estimates

I(f′)(x′, x3) :=

• R2

∞ sλ(x′ − z′, x3, z3)f′(z′, z3)dz3dz′ Convolution estimates in horizontal direction: I[f′](·, y3)Lp((0,1)2) ≤

• max |α′

i|≤2, max |α′ i+β′ i|≤2

+

• max |α′

i|≥3, max |α′ i+β′ i|≤2

• 1

+

∞

• n=1

n+1

n

• s′

λ(·, x3, z3)Ls(α′+(0,1)2)f′(·, z3)Lq(β′+(0,1)2)dz3

where sλ(·, x3, z3)Ls(R2) ≤ Ce−c|λ|

1 2 z3

|λ|

1 2 (1 + |λ| 1 2 (x3 + z3))(x3 + z3)2( 1 q − 1 p ) .

q = p = 1 excluded, also noticed in Desch, Hieber, Pr¨ uss 2001: I(f′)L1,∞

x3 ((0,1);L1((0,1)2) ≤ |λ|−1f′L1 x3((0,1);L1 uloc(R2)). 12 / 27

Linear theory: semigroup estimates

Theorem (Maekawa, Miura, P. 2017)

For q ∈ (1, ∞), let A the Stokes operator in Lq

uloc,σ(R3 +). Then −A

generates a bounded analytic semigroup in Lq

uloc,σ(R3 +).

Moreover, for 1 ≤ q < p ≤ ∞ or 1 < q = p ≤ ∞, there is C(d, p, q) ∈ (0, ∞), ∇αe−tAfLp

uloc ≤ Ct− |α| 2

• t− 3

2 ( 1 q − 1 p ) + 1

• fLq

uloc, t ∈ (0, ∞), |α| ≤ 1,

Abe, Giga 2013, 2014 (compactness method) Solonnikov 2003, Maremonti, Starita 2003 (Green tensor)

13 / 27

Mild solutions: bilinear estimates

For θ ∈ (0, 2), P∇·(U ⊗V ) = ∂α(UβVγ)+(−∆′)

2−θ 2 Gθ,≥|λ| 1 2 (U ⊗ V ) + G≤|λ| 1 2 (U ⊗ V )

For 1 ≤ q < p ≤ ∞ or 1 < q = p ≤ ∞ and 0 ≤ 1

q − 1 p < 1 3

• Gθ,≥|λ|

1 2 (U ⊗ V )

• Lp

uloc

≤ C|λ|− 1−θ

2

• 1 + |λ|

3 2 ( 1 q − 1 p )

U ⊗ V Lq

uloc,

• G≤|λ|

1 2 (U ⊗ V )

• Lq

uloc

≤ C|λ|

1 2 U ⊗ V Lq uloc.

Estimates for Oseen’s kernel

Let 1 < q ≤ p ≤ ∞ or 1 ≤ q < p ≤ ∞. Then for |α| ≤ 1 and for all t ∈ (0, ∞), ∇αe−tAP∇ · (U ⊗ V )Lp

uloc ≤ Ct− 1+α 2

t− 3

2 ( 1 q − 1 p ) + 1

• U ⊗ V Lq

uloc,

∇e−tAP∇ · (U ⊗ V )Lq

uloc ≤ Ct− 1 2

U · ∇V Lq

uloc + V · ∇ULq uloc

• .

14 / 27

Mild solutions

Proposition (Maekawa, Miura, P. 2017)

Let q ≥ 3. There exist constants γ, C > 0 such that for any V0 ∈ Lq

uloc,(ρ),σ(R3 +) satisfying V0Lq

uloc,(ρ) ≤ γρ 3 q −1 for some ρ > 0,

there exist T ≥ ρ2 and a unique mild solution V ∈ L∞(0, T; Lq

uloc,(ρ),σ(R3 +)) such that

sup

0<t<T

• V (·, t)Lq

uloc,(ρ) + t 3 2q V (·, t)L∞

≤ CV0Lq

uloc,(ρ) .

Maekawa, Terasawa 2006 (whole space) L∞

σ : Solonnikov 2003, Bae, Jin 2012 (half-space), Abe 2015

15 / 27

Localized concentration

Corollary

Let T ∈ (0, ∞). For all V ∈ C((0, T); L∞

σ (R3 +)) mild solution to (NS), if

V blows up at T, then for all t ∈ (0, T), there exists x(t) ∈ R3

+ with the

following estimate V (·, t)Lq(|·−x(t)|≤√T−t) ≥ γ (T − t)

1 2 (1− 3 q ) .

Global Leray 1934-Giga 1986 V (·, t)Lq(R3) ≥ C(T − t)− 1

2 (1− 3 q )

for q ∈ (3, ∞], t ∈ (0, T). q = 3: Escauriaza, Seregin, Sverak 2003, Seregin 2012 (whole space), Barker, Seregin 2016 (half-space) Local Li, Ozawa, Wang 2016 V (·, tk)Lq(|·−xk|≤cω(tk)−1) ≥ Cω(tk)1− 3

q ,

where ω(t) := V (·, t)L∞(R3) (T − t)− 1

2 . 16 / 27

Non uniqueness and Liouville theorem

Parasitic solutions: flows driven by the pressure V (x, t) := f(t) and P(x, t) := −f′(t) · x (in R3) V (x, t) := (V1(x3, t), V2(x3, t), 0) and P(x, t) := −f(t) · x′, (in R3

+)

with ∂tV ′ − ∂2

dV ′ = f,

V ′(0, t) = 0.

17 / 27

Non uniqueness and Liouville theorem

Parasitic solutions: flows driven by the pressure V (x, t) := f(t) and P(x, t) := −f′(t) · x (in R3) V (x, t) := (V1(x3, t), V2(x3, t), 0) and P(x, t) := −f(t) · x′, (in R3

+)

with ∂tV ′ − ∂2

dV ′ = f,

V ′(0, t) = 0. (V, P) is a weak solution to the Stokes system with zero initial data and zero source term if V ∈ L∞((0, T); L1

uloc,σ(R3 +)), P, ∇P ∈ L1 loc(R3 + × (0, T)),

(V, P) satisfies Stokes in the sense of distributions, V is weakly continuous in time, for all δ ∈ (0, T), supx0∈R3

+

∞

δ

∇P(·, t)L1(B(x0,1)) < ∞.

Theorem (Maekawa, Miura, P. 2017)

Any weak solution (V, P) to the Stokes system in R3

+ with zero initial

data and zero source term is a parasitic solution. Jia, Seregin, Sverak 2012, 2013

18 / 27

Local energy weak solutions in R3

Definition

The pair (V, P) is a Local Energy Weak Solution (LEWS) to Navier-Stokes in R3 × (0, T) with initial data V0 ∈ L2

uloc,σ(R3) if:

V ∈ L∞((0, T); L2

uloc,σ(R3)), ∇V ∈ L2((0, T); L2 uloc,σ(R3)),

P ∈ L

3 2

loc(R3 × (0, T)),

(V, P) is a solution in the sense of distributions, t →

• R3 V (·, t)ϕ is C0([0, T)) for all ϕ ∈ C∞

c (R3) and for all

K ⋐ R3, limt→0 V (·, t) − V0L2(K) = 0, for all χ ∈ C∞

c (R3 × (0, T))

• R3 |χV (·, t)|2dx + 2

t

• R3 |χ∇V |2dxds

≤ t

• R3 |V |2(∂sχ2 + ∆χ2) + V · ∇χ2(|V |2 + 2P)dxds.

(LEI)

19 / 27

Pressure formula in R3

Two approaches: Kikuchi, Seregin 2007 include the pressure formula in the definition of LEWS: for all x0 ∈ R3, there is cx0(t) ∈ L

3 2 (0, T), for all

(x, t) ∈ R3 × (0, T), with K = ∇2( 1

|x|)

Px0(x, t) = P(x, t) − cx0(t) = P x0

loc(x, t) + P x0 nonloc(x, t)

Ploc(x, t) = − 1 3|V (x, t)|2 + 1 4π

• B(x0,2)

K(x − y) · V ⊗ V (y, t)dy Pnonloc(x, t) = 1 4π

• R3\B(x0,2)

(K(x − y) − K(x0 − y)) · V ⊗ V (y, t)dy. Lemari´ e-Rieusset 2002, Jia, Sverak 2013 assume mild decay of V0, e.g. V0 ∈ L2

uloc,σ(R3) and recover the pressure formula via a Liouville theorem

for stationary Stokes.

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Helmholtz-Leray pressure and harmonic pressure in the half-space

Pressure solves

• − ∆P = ∇ · (∇ · (V ⊗ V ))

in R3

+,

∇P · e3 = γ|x3=0∆V3 − γ|x3=0∇ · (V ⊗ V ) · e3 in ∂R3

+,

Representation formula more complicated: P := P (x0) = P V0

loc + P V0 nonloc

• =Pli

+ P V ⊗V

loc,H + P V ⊗V loc,harm

• =P V ⊗V

loc

+ P V ⊗V

nonloc,H + P V ⊗V nonloc,harm

• =P V ⊗V

nonloc 21 / 27

Helmholtz-Leray pressure and harmonic pressure in the half-space

Pressure solves

• − ∆P = ∇ · (∇ · (V ⊗ V ))

in R3

+,

∇P · e3 = γ|x3=0∆V3 − γ|x3=0∇ · (V ⊗ V ) · e3 in ∂R3

+,

Representation formula more complicated: P := P (x0) = P V0

loc + P V0 nonloc

• =Pli

+ P V ⊗V

loc,H + P V ⊗V loc,harm

• =P V ⊗V

loc

+ P V ⊗V

nonloc,H + P V ⊗V nonloc,harm

• =P V ⊗V

nonloc

Estimates for linear harmonic pressure t

3 4 P V0

loc(t)L2(B(x0,1)) ≤ CV0L2(B(x0,5)),

t

1 2 P V0

nonloc(t)L∞(B(x0,1))) ≤ CV0L2

uloc(R3 +). 22 / 27

Helmholtz-Leray pressure and harmonic pressure in the half-space

Pressure solves

• − ∆P = ∇ · (∇ · (V ⊗ V ))

in R3

+,

∇P · e3 = γ|x3=0∆V3 − γ|x3=0∇ · (V ⊗ V ) · e3 in ∂R3

+,

Representation formula more complicated: P := P (x0) = P V0

loc + P V0 nonloc

• =Pli

+ P V ⊗V

loc,H + P V ⊗V loc,harm

• =P V ⊗V

loc

+ P V ⊗V

nonloc,H + P V ⊗V nonloc,harm

• =P V ⊗V

nonloc

Estimates for nonlocal pressure terms P V ⊗V

nonloc,H(·, t)L∞(B(x0,1)) ≤ CV (·, t)2 L2

uloc(R3 +),

P V ⊗V

nonloc,harm(·, t)L∞(B(x0,1)) ≤ CV 2 L∞(0,t;L2

uloc(R3 +)). 23 / 27

Global in time local energy weak solutions in the half-space

LEWS in half-space

The pair (V, P) is a LEWS in R3

+ × (0, ∞) with initial data

V0 ∈ L2

uloc,σ(R3 +) if addition to continuity, weak formulation and (LEI) we

have V ∈ L∞

loc([0, ∞); L2 uloc,σ(R3 +)), P ∈ L

3 2

loc((0, ∞) × R3 +), and

sup

x∈R3

+

T ′ ∇V 2

L2(B(x,1))dt + sup x∈R3

+

T ′

δ

∇PL1(B(x,1))dt < ∞ for all T ′ ∈ (0, ∞) and δ ∈ (0, T ′). Condition on pressure enables to apply the Liouville theorem. Condition V0 ∈ L2

uloc,σ(R3 +) rules out parasitic solutions.

Theorem (Maekawa, Miura, P. (2017))

For all V0 ∈ L2

uloc,σ(R3 +), there exists a LEWS on R3 + × (0, ∞).

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Global in time local energy weak solutions in the half-space

Decay at ∞

For all T ∈ (0, ∞), for all V0 ∈ L2

uloc,σ(R3 +), there exists

C(T, V0L2

uloc) < ∞ such that any LEWS with initial data V0 bounded

in the local energy norm on R3

+ × (0, T) satisfies for all R > 1, for all

t ∈ (0, T), αR(t) + βR(t) ≤C t α21

R (s)ds

1

21 + R−1(log R) + ϑRV0L2 uloc(R3 +)

• ,

with αR(t) := sup

0<t′<t

sup

x0∈R3

+

• B(x0,1)

|ϑRV (·, t′)|2, βR(t) := sup

x0∈R3

+

t

• B(x0,1)

|ϑR∇V |2 and ϑR cutoff supported outside B(0, R).

25 / 27

Blow-up of the L3 norm in the half-space

Theorem (Barker, Seregin 2016, Maekawa, Miura, P. 2017)

Let V a Leray-Hopf solution with initial data V0 ∈ C∞

c,σ(R3 +).

Assume that T > 0 is a blow-up time. Then V (·, t)L3(R3

+) → ∞,

t → T −. By contraposition, assume that there exists M ∈ (0, ∞) and tn → T − such that V (·, tn)L3(R3

+) ≤ M. Consider

Vn(y, s) := λnV (λny, T + λ2

ns),

λn :=

• T − tn

S . Then Vn(·, −S)L3 ≤ M.

Goal

Vn converges to a blow-up solution V such that V (·, 0) = 0 and V is smooth in (R3

+ \ B(0, R)) × (−S′, 0) for some S′ < S.

26 / 27

Blow-up of the L3 norm in the half-space

Two approaches: Barker, Seregin 2016 Stability result of weakly converging solutions. Seregin 2012, Maekawa, Miura, P. 2017 V converges to a LEWS V ; V (·, −S) ∈ L3 ⊂ L2

uloc so we can transfer the mild decay to

V and get V (·, t) ∈ L2

uloc.

End of the proof identical: through backward uniqueness uniqueness and unique continuation get V = 0 in R3

+ × (−S′, 0),

rescaling and ε-regularity then imply that T is not a blow-up time.

27 / 27