Long time and small obstacle problem for the 2D NS equations - - PowerPoint PPT Presentation

Introduction Massive particle Long time behavior Massless particle

Long time and small obstacle problem for the 2D NS equations

Christophe Lacave

IMJ-PRG, University of Paris Diderot (Paris 7), France in collaboration with S. Ervedoza (Toulouse), M. Hillairet (Montpellier) and T. Takahashi (Nancy).

Porquerolles, Mathflows 2015, September

1 / 32 Long time and small obstacle problem for the 2D NS equations

Introduction Massive particle Long time behavior Massless particle

Outline

1

Introduction

2

“Massive” pointwise particle

3

Long time behavior for the unit disk in R2

4

“Massless” pointwise particle

2 / 32 Long time and small obstacle problem for the 2D NS equations

Introduction Massive particle Long time behavior Massless particle

Outline

1

Introduction

2

“Massive” pointwise particle

3

Long time behavior for the unit disk in R2

4

“Massless” pointwise particle

3 / 32 Long time and small obstacle problem for the 2D NS equations

Introduction Massive particle Long time behavior Massless particle

Navier-Stokes equations

Let uε = (uε

1, uε 2) be the velocity of an incompressible, viscous

flow in a domain Fε(t) := R2 \ Sε(t). Let pε be the pressure. The evolution of such a flow is governed by the Navier-Stokes equations :              ∂tuε − ν∆uε + uε · ∇uε = −∇pε in (0, ∞) × Fε(t) div εu = 0 in [0, ∞) × Fε(t) lim

|x|→∞ |uε| = 0

for t ∈ [0, ∞) uε(0, x) = uε

0(x)

in Fε We consider the no-slip boundary condition :

uε = 0 on (0, ∞) × ∂Sε

0 if the domain is fixed;

4 / 32 Long time and small obstacle problem for the 2D NS equations

Introduction Massive particle Long time behavior Massless particle

Navier-Stokes equations

Let uε = (uε

1, uε 2) be the velocity of an incompressible, viscous

flow in a domain Fε(t) := R2 \ Sε(t). Let pε be the pressure. The evolution of such a flow is governed by the Navier-Stokes equations :              ∂tuε − ν∆uε + uε · ∇uε = −∇pε in (0, ∞) × Fε(t) div εu = 0 in [0, ∞) × Fε(t) lim

|x|→∞ |uε| = 0

for t ∈ [0, ∞) uε(0, x) = uε

0(x)

in Fε We consider the no-slip boundary condition :

uε = 0 on (0, ∞) × ∂Sε

0 if the domain is fixed;

4 / 32 Long time and small obstacle problem for the 2D NS equations

Introduction Massive particle Long time behavior Massless particle

if the solid moves under the influence of the fluid: uε(t, x) = (hε)′(t) + ωε(t)(x − hε(t))⊥ for (t, x) ∈ [0, ∞) × ∂Sε(t),

mε(hε)′′(t) = −

• ∂Sε(t)

Σεn ds for t ∈ (0, ∞), J ε(ωε)′(t) = −

• ∂Sε(t)

(x − hε(t))⊥ · Σεn ds for t ∈ (0, ∞). Where Σε is the Cauchy stress tensor of the fluid: Σε(t, x) = −pε(t, x)Id + 2νD(uε), and D(u)k,l = 1 2 ∂uk ∂xl + ∂ul ∂xk

• 1 ≤ k, l ≤ 2 .

The solid is rigid: Sε(t) := (hε(t) − h0) + Rε(θ)Sε

0.

5 / 32 Long time and small obstacle problem for the 2D NS equations

Introduction Massive particle Long time behavior Massless particle

if the solid moves under the influence of the fluid: uε(t, x) = (hε)′(t) + ωε(t)(x − hε(t))⊥ for (t, x) ∈ [0, ∞) × ∂Sε(t),

mε(hε)′′(t) = −

• ∂Sε(t)

Σεn ds for t ∈ (0, ∞), J ε(ωε)′(t) = −

• ∂Sε(t)

(x − hε(t))⊥ · Σεn ds for t ∈ (0, ∞). Where Σε is the Cauchy stress tensor of the fluid: Σε(t, x) = −pε(t, x)Id + 2νD(uε), and D(u)k,l = 1 2 ∂uk ∂xl + ∂ul ∂xk

• 1 ≤ k, l ≤ 2 .

The solid is rigid: Sε(t) := (hε(t) − h0) + Rε(θ)Sε

0.

5 / 32 Long time and small obstacle problem for the 2D NS equations

Introduction Massive particle Long time behavior Massless particle

Well posedness

Theorem (Takahashi and Tucsnak, 04) There exists a unique global strong solution (uε, pε, hε, ωε) if uε

0 ∈ H1(F0). If uε 0 belongs to L2(F0), there exists a unique

weak solution. In the case of moving solid, we only consider the case of the disk: hence the regularity of strong solutions is computed through the classical change of unknown: vε(t, x) = uε(t, x − hε(t)) , ˜ pε(t, x) = pε(t, x − hε(t)). which are defined in the fixed domain [0, ∞) × (R2 \ Bε

0).

What is the behavior of the solution when the solids tends to zero : Bε

0 = εB(0, 1) with ε → 0 ?

6 / 32 Long time and small obstacle problem for the 2D NS equations

Introduction Massive particle Long time behavior Massless particle

Well posedness

Theorem (Takahashi and Tucsnak, 04) There exists a unique global strong solution (uε, pε, hε, ωε) if uε

0 ∈ H1(F0). If uε 0 belongs to L2(F0), there exists a unique

weak solution. In the case of moving solid, we only consider the case of the disk: hence the regularity of strong solutions is computed through the classical change of unknown: vε(t, x) = uε(t, x − hε(t)) , ˜ pε(t, x) = pε(t, x − hε(t)). which are defined in the fixed domain [0, ∞) × (R2 \ Bε

0).

What is the behavior of the solution when the solids tends to zero : Bε

0 = εB(0, 1) with ε → 0 ?

6 / 32 Long time and small obstacle problem for the 2D NS equations

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Large literature on such problems

For Euler: Iftimie, Lopes Filho, Nussenzveig Lopes (03); Lopes Filho (07); C.L. (09,12); Glass, C.L., Sueur (14,15?); Glass, Munnier, Sueur (15?). For Navier-Stokes: Iftimie, Lopes Filho, Nussenzveig Lopes (06,09); Iftimie, Kelliher (09); C.L. (09, 15). In porous medium: Cioranescu, Murat (79); Sanchez-Palencia (80); Tartar (80); Conca (87); Allaire (90); Mikelic, Paoli (99); Lions, Masmoudi (05), Bonnaillie-Noel, C.L., Masmoudi (15); C.L., Masmoudi (15?).

7 / 32 Long time and small obstacle problem for the 2D NS equations

Introduction Massive particle Long time behavior Massless particle

Outline

1

Introduction

2

“Massive” pointwise particle

3

Long time behavior for the unit disk in R2

4

“Massless” pointwise particle

8 / 32 Long time and small obstacle problem for the 2D NS equations

Introduction Massive particle Long time behavior Massless particle

Assumptions:

mε

i = m1 i > 0

and Jε

i = ε2J1 i > 0;

uε

0 ⇀ u0

in L2(R2); |ℓε

0| ≤ C,

ε|r ε

0 | ≤ C.

Theorem For any T > 0 we have uε

∗

⇀ u in L∞(0, T; L2(R2)) ∩ L2(0, T; H1

0(R2))

where u is the weak solution of the Navier-Stokes equations associated to u0.

9 / 32 Long time and small obstacle problem for the 2D NS equations

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Scheme of the proof

1) energy estimate:

• R2 ρε |uε|2 + 4ν

t

• R2 |D(uε)|2 ≤
• R2 ρε(x) |uε

0(x)|2 .

Then up to a subsequence, uε

∗

⇀ u in L∞(0, T; L2(R2)) ∩ L2(0, T; H1(R2)) hε → h uniformly in [0, T], with h ∈ W 1,∞(0, T). We need a strong compactness on the support of the test

• function. Problem of compatibility of test function

10 / 32 Long time and small obstacle problem for the 2D NS equations

Introduction Massive particle Long time behavior Massless particle

Scheme of the proof

1) energy estimate:

• R2 ρε |uε|2 + 4ν

t

• R2 |D(uε)|2 ≤
• R2 ρε(x) |uε

0(x)|2 .

Then up to a subsequence, uε

∗

⇀ u in L∞(0, T; L2(R2)) ∩ L2(0, T; H1(R2)) hε → h uniformly in [0, T], with h ∈ W 1,∞(0, T). We need a strong compactness on the support of the test

• function. Problem of compatibility of test function

10 / 32 Long time and small obstacle problem for the 2D NS equations

Introduction Massive particle Long time behavior Massless particle

Scheme of the proof

2) modified test function: Proposition Let T > 0 and ϕ ∈ C∞

c ([0, T) × R2) with div ϕ = 0. For any

η > 0, there exists ϕη ∈ W 1,∞

c

([0, T); H1

0(R2)) satisfying

div ϕη = 0 in [0, T) × Ω, ϕη ≡ 0 t ∈ (0, T), x ∈ B

• h(t), η

2

• ,

ϕη

∗

⇀ ϕ L∞(0, T; H1(Ω)), ∂tϕη

∗

⇀ ∂tϕ L∞(0, T; L2(Ω)).

11 / 32 Long time and small obstacle problem for the 2D NS equations

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Scheme of the proof

3) passing to the limit far away: (i.e. η fixed and ε → 0) There exist open relatively compact sets Oj and ˜ εη > 0 such that for all ε < ˜ εη we have Sε(t) ∩ Oj = ∅ for all t ∈ (tj, tj+1) and supp ϕη ⊂ M

j=0(tj, tj+1) × Oj.

One decompose as follows: uε = POjuε + ∇qε. We have the estimate of POjuε and ∇qε in L∞L2 ∩ L2H1. We use the Aubin Lions lemma for POjuε with H1 ∩ H(Oj) ֒ → L4 ∩ H(Oj) ֒ → V′(Oj) Then we can pass to the limit in the non linear term because:

• Oj

(∇˜ q ⊗ ∇˜ q) : ∇ϕη = −

• Oj

div (∇˜ q ⊗ ∇˜ q) · ϕη = −

• Oj

1 2∇|∇˜ q|2 · ϕη + ∆˜ q∇˜ q · ϕη

• = 0.

12 / 32 Long time and small obstacle problem for the 2D NS equations

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Scheme of the proof

3) passing to the limit far away: (i.e. η fixed and ε → 0) There exist open relatively compact sets Oj and ˜ εη > 0 such that for all ε < ˜ εη we have Sε(t) ∩ Oj = ∅ for all t ∈ (tj, tj+1) and supp ϕη ⊂ M

j=0(tj, tj+1) × Oj.

One decompose as follows: uε = POjuε + ∇qε. We have the estimate of POjuε and ∇qε in L∞L2 ∩ L2H1. We use the Aubin Lions lemma for POjuε with H1 ∩ H(Oj) ֒ → L4 ∩ H(Oj) ֒ → V′(Oj) Then we can pass to the limit in the non linear term because:

• Oj

(∇˜ q ⊗ ∇˜ q) : ∇ϕη = −

• Oj

div (∇˜ q ⊗ ∇˜ q) · ϕη = −

• Oj

1 2∇|∇˜ q|2 · ϕη + ∆˜ q∇˜ q · ϕη

• = 0.

12 / 32 Long time and small obstacle problem for the 2D NS equations

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Scheme of the proof

3) passing to the limit far away: (i.e. η fixed and ε → 0) There exist open relatively compact sets Oj and ˜ εη > 0 such that for all ε < ˜ εη we have Sε(t) ∩ Oj = ∅ for all t ∈ (tj, tj+1) and supp ϕη ⊂ M

j=0(tj, tj+1) × Oj.

One decompose as follows: uε = POjuε + ∇qε. We have the estimate of POjuε and ∇qε in L∞L2 ∩ L2H1. We use the Aubin Lions lemma for POjuε with H1 ∩ H(Oj) ֒ → L4 ∩ H(Oj) ֒ → V′(Oj) Then we can pass to the limit in the non linear term because:

• Oj

(∇˜ q ⊗ ∇˜ q) : ∇ϕη = −

• Oj

div (∇˜ q ⊗ ∇˜ q) · ϕη = −

• Oj

1 2∇|∇˜ q|2 · ϕη + ∆˜ q∇˜ q · ϕη

• = 0.

12 / 32 Long time and small obstacle problem for the 2D NS equations

Introduction Massive particle Long time behavior Massless particle

Scheme of the proof

4) equation in the vicinity of the solid: We have for any η > 0 − T

• Ω

u · ∂ϕη ∂t + (u · ∇)ϕη

• dx

+ ν T

• Ω

∇u : ∇ϕη dx =

• Ω

u0(x) · ϕη(0, x) dx. then passing to the limit η → 0 we have that u satisfies the Navier-Stokes equations in R2, for any test function ϕ. By uniqueness, the weak convergence uεn

∗

⇀ u in L∞(0, T; L2(Ω)) ∩ L2(0, T; H1(Ω)) holds for all sequence (εn) converging to 0 (without extracting a subsequence).

13 / 32 Long time and small obstacle problem for the 2D NS equations

Introduction Massive particle Long time behavior Massless particle

Scheme of the proof

4) equation in the vicinity of the solid: We have for any η > 0 − T

• Ω

u · ∂ϕη ∂t + (u · ∇)ϕη

• dx

+ ν T

• Ω

∇u : ∇ϕη dx =

• Ω

u0(x) · ϕη(0, x) dx. then passing to the limit η → 0 we have that u satisfies the Navier-Stokes equations in R2, for any test function ϕ. By uniqueness, the weak convergence uεn

∗

⇀ u in L∞(0, T; L2(Ω)) ∩ L2(0, T; H1(Ω)) holds for all sequence (εn) converging to 0 (without extracting a subsequence).

13 / 32 Long time and small obstacle problem for the 2D NS equations

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Comments

Remark

For the massive case, everything can be applied in bounded domain with several obstacles (with any shapes). If the mass tends to zero, we loose the uniform estimate of (hε)′ (step 1), and everything falls down... We need an other argument. Be careful with H2 analysis... Our idea is to relate the small obstacle problem with the long time behavior, though the scaling property of the Navier-Stokes equations: uε(t, x) = ε−1u1(ε−2t, ε−1x).

14 / 32 Long time and small obstacle problem for the 2D NS equations

Introduction Massive particle Long time behavior Massless particle

Comments

Remark

For the massive case, everything can be applied in bounded domain with several obstacles (with any shapes). If the mass tends to zero, we loose the uniform estimate of (hε)′ (step 1), and everything falls down... We need an other argument. Be careful with H2 analysis... Our idea is to relate the small obstacle problem with the long time behavior, though the scaling property of the Navier-Stokes equations: uε(t, x) = ε−1u1(ε−2t, ε−1x).

14 / 32 Long time and small obstacle problem for the 2D NS equations

Introduction Massive particle Long time behavior Massless particle

Outline

1

Introduction

2

“Massive” pointwise particle

3

Long time behavior for the unit disk in R2

4

“Massless” pointwise particle

15 / 32 Long time and small obstacle problem for the 2D NS equations

Introduction Massive particle Long time behavior Massless particle

Convection-diffusion equations (no pressure)

Escobedo and Zuazua (91): ∂tu − ∆u = a · ∇(|u|q−1u). V` azquez (03): the porous medium equation ∂tu = ∆(um). V` azquez and Zuazua (03, 06): 1D fluid-solid model (viscous Burger). Munnier and Zuazua (04): N-dimensional heat equation where a rigid ball B(t) moves under the force −ν

• ∂B(t)

∂u ∂n ds.

Without pressure term, one can multiply by |u|p−2u and get: d dt

• yp

Lp

• + 4(p − 1)

p

• F0

|∇(|y|p/2)|2 = 0.

16 / 32 Long time and small obstacle problem for the 2D NS equations

Introduction Massive particle Long time behavior Massless particle

Convection-diffusion equations (no pressure)

Escobedo and Zuazua (91): ∂tu − ∆u = a · ∇(|u|q−1u). V` azquez (03): the porous medium equation ∂tu = ∆(um). V` azquez and Zuazua (03, 06): 1D fluid-solid model (viscous Burger). Munnier and Zuazua (04): N-dimensional heat equation where a rigid ball B(t) moves under the force −ν

• ∂B(t)

∂u ∂n ds.

Without pressure term, one can multiply by |u|p−2u and get: d dt

• yp

Lp

• + 4(p − 1)

p

• F0

|∇(|y|p/2)|2 = 0. Thanks to Sobolev embeddings, we retrieve Lp − Lq decays similar to the ones of the heat kernel: t

N 2 ( 1 q − 1 p )z(t)Lp ≤ Cz0Lq,

t > 0, p ≥ q.

16 / 32 Long time and small obstacle problem for the 2D NS equations

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Convection-diffusion equations (no pressure)

Escobedo and Zuazua (91): ∂tu − ∆u = a · ∇(|u|q−1u). V` azquez (03): the porous medium equation ∂tu = ∆(um). V` azquez and Zuazua (03, 06): 1D fluid-solid model (viscous Burger). Munnier and Zuazua (04): N-dimensional heat equation where a rigid ball B(t) moves under the force −ν

• ∂B(t)

∂u ∂n ds.

Without pressure term, one can multiply by |u|p−2u and get: d dt

• yp

Lp

• + 4(p − 1)

p

• F0

|∇(|y|p/2)|2 = 0. Using Duhamel formula, or some scaling invariance: we prove that the solution behaves like the heat kernel

• u0

(4πt)N/2 e− |x|2

4t if

u0 ∈ L1.

16 / 32 Long time and small obstacle problem for the 2D NS equations

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Navier-Stokes in the full plane

One approach consists of removing the pressure by taking the curl of the momentum equation: ∂tω + u · ∇ω − ν∆ω = 0. Theorem (Gallay and Wayne, 05) For ω0 ∈ L1 such that

• ω0 = 0, the vorticity behaves like the

heat kernel

• ω0

4πt e− |x|2

4t , and the velocity behaves like the

Lamb-Ossen vector field:

• ω0

2π x⊥ |x|2

• 1 − e− |x|2

4t

• .

17 / 32 Long time and small obstacle problem for the 2D NS equations

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Navier-Stokes in the full plane

Another approach consists to solve p = ∆−1(div (u · ∇u)) and to estimate it by the Fourier decomposition. Theorem (Carpio, 94) For u0 ∈ L1, the velocity behaves like the heat kernel

• u0

4πt e− |x|2

4t . 18 / 32 Long time and small obstacle problem for the 2D NS equations

Introduction Massive particle Long time behavior Massless particle

Navier-Stokes in exterior domains

Theorem (Maremonti-Solonnikov 97, Dan-Shibata 99) For each q ∈ (1, ∞), the Stokes operator of the linear problem generates a semigroup S(t) on Lq which satisfies :

• For p ∈ [q, ∞] :

S(t)v0Lp ≤ K1(νt)

1 p − 1 q v0Lq.

for all t > 0.

• If q ≤ 2, for p ∈ [q, 2] :

∇S(t)v0Lp ≤ K2(νt)− 1

2 + 1 p − 1 q v0Lq

for all t > 0.

• For p ∈ [max{2, q}, ∞) :

∇S(t)v0Lp ≤

• K3(νt)− 1

2+ 1 p − 1 q v0Lq

for all 0 < t < 1

ν ,

K3(νt)− 1

q v0Lq

for all t ≥ 1

ν .

19 / 32 Long time and small obstacle problem for the 2D NS equations

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Navier-Stokes in exterior domains

Theorem (Iftimie-Karch-C.L. 11, 14; Gallay-Maekawa 13) There exists ε > 0 such that if v0L2,∞ ≤ ε, then the Navier-Stokes solutions behaves like the self-similar solution (in Lp for p > 2). Problem: the Lp − Lq estimates for the Stokes problem is available only with the Dirichlet boundary condition.

20 / 32 Long time and small obstacle problem for the 2D NS equations

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Navier-Stokes in exterior domains

Theorem (Iftimie-Karch-C.L. 11, 14; Gallay-Maekawa 13) There exists ε > 0 such that if v0L2,∞ ≤ ε, then the Navier-Stokes solutions behaves like the self-similar solution (in Lp for p > 2). Problem: the Lp − Lq estimates for the Stokes problem is available only with the Dirichlet boundary condition.

20 / 32 Long time and small obstacle problem for the 2D NS equations

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Decomposition for the Stokes problem

As v is divergence-free, there exists ψ such that v = ∇⊥ψ : ψ(x) = 1 2ω|x|2 + ℓ · x⊥ , in ¯ B0 . Introducing radial coordinates (r, θ) and expanding ψ in Fourier series, ψ(r, θ) =

∞

• k=0

ψk(r) cos(θ)+

∞

• k=1

φk(r) sin(θ), ∀ (r, θ) ∈ (0, ∞)×(−π, π) Actually, the following will be sufficient: v = weθ + ∇⊥[ψ cos(θ)] + ∇⊥[φ sin(θ)] + vR .

21 / 32 Long time and small obstacle problem for the 2D NS equations

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wLp((0,∞),rdr) + vRLp(R2) + ∂rψLp((0,∞),rdr) + ψ/rLp((0,∞),rdr) + ∂rφLp((0,∞),rdr) + φ/rLp((0,∞),rdr) ≤ C(p)vLp( Conversely: vLp(R2) ≤ C(p)(wLp((0,∞),rdr) + vRLp(R2) + ∂rψLp((0,∞),rdr) + +∂rφLp((0,∞),rdr) + φ/rLp((0,∞),rdr)) , and ∇vLp(F0) ≤ C(p)(∂rwLp((1,∞),rdr) + w/rLp((1,∞),rdr) + ∇vRLp +∂rrψLp((1,∞),rdr) + ∂rψ/rLp((1,∞),rdr) + ψ/r 2Lp(( +∂rrφLp((1,∞),rdr) + ∂rφ/rLp((1,∞),rdr) + φ/r 2Lp((1

22 / 32 Long time and small obstacle problem for the 2D NS equations

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∂tvR − ∆vR + ∇pR = 0 t ≥ 0, x ∈ F0 ; div vR = 0 t ≥ 0, x ∈ F0 ; vR(t, x) = 0 t ≥ 0, x ∈ ∂B0 . We can use directly the results of Maremonti-Solonnikov & Dan-Shibata !!

23 / 32 Long time and small obstacle problem for the 2D NS equations

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∂tw + ν

• −1

r ∂r(r∂rw) + 1 r 2 w

• = 0

t ≥ 0, r ≥ 1 ; ∂tw(t, 1) = 2νπ J (∂rw(t, 1) − w(t, 1)) t ≥ 0 . We can use adapt the idea used for the convection-diffusion (without pressure) equations.

24 / 32 Long time and small obstacle problem for the 2D NS equations

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∂tψ + ν

• −1

r ∂r(r∂rψ) + 1 r 2 ψ

• = −r∂rq1

t ≥ 0, r ≥ 1 ; ∂t∂rψ + ν∂r

• −1

r ∂r(r∂rψ) + 1 r 2 ψ

• = −q1

r t ≥ 0, r ≥ 1 ; ψ(t, 1) = ∂rψ(t, 1) = h′

2(t)

t ≥ 0; m π h′′

2(t) = −q1(t, 1) − ν(−1

r ∂r(r∂rψ) + 1 r 2 ψ)(t, 1) t ≥ 0

25 / 32 Long time and small obstacle problem for the 2D NS equations

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The new unknown: z(r) := ∂rψ(r) + ψ(r) r = 1 r ∂r[rψ(r)] , ∀ r ∈ (1, ∞), which solves ∂tz − ν

• ∂rr + 1

r ∂r

• z = 0

t ≥ 0, r ≥ 1; ∂tz(t, 1) =

4π π+mν∂rz(t, 1)

t > 0.

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Lp − Lq estimates

Theorem (Ervedoza-Hillairet-C.L., 14) For each q ∈ (1, ∞), the Stokes operator of the linear problem generates a semigroup S(t) on Lq which satisfies:

• For p ∈ [q, ∞] :

S(t)v0Lp ≤ K1(νt)

1 p − 1 q v0Lq.

for all t > 0.

• If q ≤ 2, for p ∈ [q, 2] :

∇S(t)v0Lp(F0) ≤ K2(νt)− 1

2 + 1 p − 1 q v0Lq

for all t > 0.

• For p ∈ [max{2, q}, ∞) :

∇S(t)v0Lp(F0) ≤

• K3(νt)− 1

2 + 1 p − 1 q v0Lq

for all 0 < t < 1

ν ,

K3(νt)− 1

q v0Lq

for all t ≥ 1

ν .

27 / 32 Long time and small obstacle problem for the 2D NS equations

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Long time behavior of the Stokes solution

Theorem (Ervedoza-Hillairet-C.L., 14) For all p ≥ 2, and for any v0 ∈ L1 ∩ L2(exp(|x|2/2); F0), we have lim

t→∞ t1−1/pS(t)v0 − (m − π)Uv0(t, ·)Lp(F0)

= 0, lim

t→∞ t

• ℓS(t)v0 − (m − π)ℓ0

8πνt

• =

0, where Uv0(t, x) = ∇⊥  1 − e− |x|2

4νt

2π|x|2 ℓ0 · x⊥   . We have also a partial result for the long time behavior for the Navier-Stokes solutions.

28 / 32 Long time and small obstacle problem for the 2D NS equations

Introduction Massive particle Long time behavior Massless particle

Outline

1

Introduction

2

“Massive” pointwise particle

3

Long time behavior for the unit disk in R2

4

“Massless” pointwise particle

29 / 32 Long time and small obstacle problem for the 2D NS equations

Introduction Massive particle Long time behavior Massless particle

Assumptions:

mε

i = ε2m1 i > 0

and Jε

i = ε4J1 i > 0;

uε

0 ⇀ u0

in L2(R2);

Theorem There exists λ0 such that if ε|ℓε

0|, ε2|r ε 0|, uε 0L2(R2\B(0,1)) ≤ λ0

For any T > 0 we have uε

∗

⇀ u in L∞(0, T; L2(R2)) ∩ L2(0, T; H1

0(R2))

where u is the weak solution of the Navier-Stokes equations associated to u0.

30 / 32 Long time and small obstacle problem for the 2D NS equations

Introduction Massive particle Long time behavior Massless particle

Scheme of the proof

1) Uniform estimates:

the energy inequality gives that uε

∗

⇀ u in L∞(0, T; L2(R2)) ∩ L2(0, T; H1(R2)). the Lp − Lq decay estimate of the Stokes problem is independent

• f ε.

we write the Duhamel formula, and by a fixed point argument in vε ∈ C0([0, T]; L2

ε)∩C0 3/8([0, T]; L8 ε)

with ℓε := ℓvε ∈ C0

1/2([0, T]; R2)

we construct a solution such that |(hε)′(t)| ≤ µ0 √ t (t > 0).

Step 2, 3 and 4: same than the massive case.

31 / 32 Long time and small obstacle problem for the 2D NS equations

Introduction Massive particle Long time behavior Massless particle

Thank you for your attention

32 / 32 Long time and small obstacle problem for the 2D NS equations