Between homogeneous and inhomogeneous Navier-Stokes systems: the - - PowerPoint PPT Presentation

Between homogeneous and inhomogeneous Navier-Stokes systems: the issue of stability

Liutang Xue

joint with Prof. Piotr B. Mucha and Prof. Xiaoxin Zheng

Beijing Normal University, China

September 3 - Mathflows Confernece 2018

Liutang Xue (BNU) Between HNS and INS: Stability Issue Mathflows 2018 1 / 42

Inhomogeneous Navier-Stokes Equations

3D inhomogeneous Navier-Stokes equations (INS)                  ∂tρ + v · ∇ρ = 0, in R3 × R+, ρ∂tv + ρv · ∇v − ν∆v + ∇p = 0, in R3 × R+, div v = 0, in R3 × R+, ρ|t=0(x) = ρ0(x), v|t=0(x) = v0(x),

• n R3.

(1) describe the motion of incompressible flows of viscous Newtonian fluid with variable density (Lions [Lions96]). x = (x1, x2, x3) ∈ R3; ν > 0 the kinematic viscosity (below ν ≡ 1 for brevity); ρ the scalar density field; v = (v1, v2, v3) the incompressible velocity vector field. If ρ0 = const, (INS) system reduces to 3D homogeneous Navier-Stokes equations (Millennium Problem).

Liutang Xue (BNU) Between HNS and INS: Stability Issue Mathflows 2018 2 / 42

Research History - I

Mathematical properties of (INS) system are almost the same as those of the 3D homogeneous Navier-Stokes equations (see Kazhikhov [Kaz74], Ladyzhenskaya, Solonikov [LS73]). The main difference is found in the issue related to the density. Questions concerned with the low regularity of initial density or the possibility of vacuum states are the subjects of current studies of (INS) system (see e.g. Danchin, Mucha [DanM12,DanM13,DanM17]1, Paicu, Zhang et al [HPZ13,PZZ13]2, Li [Li17]3).

• 1R. Danchin, P

. Mucha, [DanM12] Commun. Pure Appl. Math., 65 (2012), 1458–1480; [DanM13]

• Arch. Rational Mech. Anal., 207 (2013), 991–1023;

[DanM17] ArXiv:1705.06061 [math.AP].

2[HPZ13] Huang, Paicu, Zhang, Arch. Ration. Mech. Anal., 209(2), (2013), 631–682;

[PZZ13] M. Paicu, P . Zhang, Z. Zhang, Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire, 32 (2015), no. 4, 813–832.

• 3J. Li, J. Differential Equations, 263 (2017), Issue 10, 6512–6536.

Liutang Xue (BNU) Between HNS and INS: Stability Issue Mathflows 2018 3 / 42

Stability Between (INS) and (HNS)

Our Goal: to find solutions to 3D (INS) system with large velocity field like [ChPZ14]4, [PZ15]5 (where data with slow variable ǫx3). Main Plan: to study stability issue of 3D (INS) system around the 2D homogeneous Navier-Stokes system with constant density 1. Let us emphasize that such a stability analysis has been well developed for the 3D homogeneous Navier-Stokes equations (see e.g. [CG06]6, [ChGP11]7, [Muc01]8, [KRZ14]9), but has not been pursued for (INS) system.

4J.-Y. Chemin, M. Paicu, P

. Zhang, J. Differential Equations, 256 (2014), no. 1, 223–252.

• 5M. Paicu, P

. Zhang, Ann. Inst. H. Poincar´ e Anal. Non Lin., 32 (2015), no. 4, 813–832.

6J-Y. Chemin, I. Gallagher, Ann. Sci. ´

Ecole Norm. Sup., 39 (2006), no. 4, 679–698.

7J.-Y. Chemin, I. Gallagher, M. Paicu, Ann. of Math., 173 (2011), no. 2, 983–1012. 8P

. B. Mucha, J. Differential Equations, 172 (2001), no. 2, 359–375.

• 9I. Kukavica, W. Rusin, M. Ziane, J. Math. Fluid Mech., 16 (2014), no. 2, 293–305.

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Perturbed System

Let v2d = (v2d

1 , v2d 2 , v2d 3 ) be a three component 2D vector which solves

(HNS)            ∂tv2d + v2d

h · ∇hv2d − ∆hv2d + ∇p2d = 0,

∇h · v2d

h = 0,

v2d|t=0(xh) = v2d

0 (xh),

(2) where xh = (x1, x2) ∈ R2, ∇h := (∂x1, ∂x2), ∇ := (∂x1, ∂x2, ∂x3), ∆h := ∂2

x1 + ∂2 x2.

For solution (v, ρ, p) of (INS) system and solution (v2d, p2d) of 2D (HNS), set w(t, x) := v(t, x)−v2d(t, xh), h(t, x) := ρ(t, x)−1, q(t, x) := p(t, x)−p2d(t, xh), we obtain the following perturbed system                ht + v · ∇h = 0, wt + v · ∇w − ∆w + ∇q = F, div w = 0, w|t=0 = w0, h|t=0 = h0, (3) F := −h(v2d)t − h wt − h(v · ∇w) − ρ(wh · ∇hv2d) − h(v2d

h · ∇hv2d).

(4)

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Main Result - I

Our first result concerns the flow in the whole space with regular initial density.

Theorem 1 (Mucha, Xue, Zheng, 2018)

Let p > 3, v2d

0 ∈ L2 ∩ ˙

B3−2/p

p,p

(R2), ρ0 − 1 ∈ L2 ∩ L∞(R3), ∇ρ0 ∈ L3(R3) and v0 − v2d

0 ∈ L2 ∩ ˙

B2−2/p

p,p

(R3). There exist c0, C′ > 0 such that if (ρ0, v0) satisfies ρ0−1L2∩L∞+v0−v2d

0 L2∩ ˙ B2−2/p

p,p

≤ c0 exp

• −C′
• v2d

0 4p L2∩ ˙ B2−2/p

p,p

+ 1

• eC′(1+v2d

0 4 L2)

• ,

(5) then we have a unique global-in-time solution (ρ, v) to the system (INS) which satisfies ρ − 1L∞(0,∞;L2∩L∞(R3)) ≤ ρ0 − 1L2∩L∞(R3), sup

t<∞

v − v2dL2∩ ˙

B2−2/p

p,p

+ (v − v2d)t, ∇2(v − v2d), ∇(p − p2d)Lp(R3×(0,∞)) ≤ C c0, sup

t∈[0,T]

∇ρ(t)L3(R3) ≤ ∇ρ0L3(R3)e(1+T)C(h0,w0,v2d

0 ),

for any T > 0. where v2d = (v2d

1 , v2d 2 , v2d 3 ) is the unique solution to 2D (HNS) system.

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Main Result - II

Our second global result removes this extra regularity condition of density.

Theorem 2 (Mucha, Xue, Zheng, 2018)

Let p > 3, v2d

0 ∈ L2 ∩ ˙

B4−2/p

p,p

(R2), ρ0 − 1 ∈ L2 ∩ L∞(R3) and v0 − v2d

0 ∈ L2 ∩ ˙

B2−2/p

p,p

(R3). There exist two generic constants c0, C′ > 0 such that if (ρ0, v0) satisfies ρ0−1L2∩L∞+v0−v2d

0 L2∩ ˙ B2−2/p

p,p

≤ c0 exp

• −C′
• v2d

0 4p L2∩ ˙ B2−2/p

p,p

+ 1

• eC′(1+v2d

0 4 L2)

• ,

(6) then we have a unique global-in-time solution (ρ, v) to the system (INS) which

• beys the uniform estimates

ρ − 1L∞(0,∞;L2∩L∞(R3)) ≤ ρ0 − 1L2∩L∞(R3), (7) sup

t<∞

v−v2dL2∩ ˙

B2−2/p

p,p

+(v−v2d)t, ∇2(v−v2d), ∇(p−p2d)Lp(R3×(0,∞)) ≤ C c0. (8)

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Remark 1

Theorems 1 and 2 say that if the initial perturbation is small (5), then the solutions exist globally in time and they are close to the ones of 2D (HNS) system. Hence the physical interpretation is the following: all solutions to 2D (HNS) are stable globally in time in the inhomogeneous Navier-Stokes system regime, provided that perturbation of density is close to a constant in the L2 ∩ L∞-norm. It means the higher norms of the density have no influence of our issue

• f stability.

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Density Patch Problem

Remark 2

Let ρ0 = 1Dc + (1 − η)1D = 1 − η1D with η a small constant, and D ⊂ R3 a bounded, simple, connected set. According to Theorem 2, if (ρ0, v0) satisfies

|η|(1 + |D|1/2) + v0 − v2d

0 L2∩ ˙ B2−2/p

p,p

≤ c0 exp

• −C′
• v2d

0 4p L2∩ ˙ B2−2/p

p,p

+ 1

• eC′(1+v2d

0 4 L2 )

• ,

then the (INS) system generates a unique global solution (ρ, v) which satisfies (7)-(8). In particular, we have for γ ∈ (0, 1 − 2/p], p > 3, vLp(0,∞;C1,γ(R3)) ≤ v2dLp(0,∞;C1,γ(R2)) + Cv − v2dLp(0,∞;W2

p (R3)) < ∞.

It guarantees that if initial boundary ∂D is C1,γ-regular, then its evolution ∂D(t) = Xv(t, ∂D) remains C1,γ-regular, with Xv(t, ·) the flow of v. For more results on density patch problem of (INS) system, e.g. considering ρ0 = ρ11D + ρ21Dc, with ρ1, ρ2 > 0 constants, one can see Danchin, Zhang [DanZ17], Gancedo, Garcia-Juarez [GanGJ18], Liao, Zhang [LZ16] etc.

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Unsolved Problem

Remark 3

It is not clear that 2D (INS) system is globally stable in the 3D (INS) regime, if perturbation of initial density is small in the norm of L2 ∩ L∞(R3). For example, for the data ρ0 = ρ11D + ρ21Dc + small perturbation in L2 ∩ L∞(R3), v0 = v2d

0 + small w0,

with ρ1, ρ2 > 0 constants, it is not clear that the 3D (INS) system will has a unique global solution.

Liutang Xue (BNU) Between HNS and INS: Stability Issue Mathflows 2018 10 / 42

Outline of The Proof of Theorem 1

For the proof of Theorem 1, we consider the perturbed system (3): Energy estimate of w.

◮ Regularity and decay estimates of smooth solution to 2D (HNS).

Estimates in the maximal regularity regime (in the Lp-spaces). Using Lemma 4. Estimate on ∇ρ in L3-norm. Note that the proof of uniqueness is based on the Eulerian coordinates approach, and that is why an information about the gradient of the density is required. Approximate system of the considered perturbed equations + L2-strong convergence and uniqueness.

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Outline of The Proof of Theorem 2

For the proof of Theorem 2, Existence is followed from the uniform estimates of approximative solutions. Uniqueness: We adopt the Lagrangian coordinates approach

• riginated in [DanM12]10,[DanM13]11.

we consider the difference system of two solutions in the Lagrangian coordinates, and we can adapt an energy type argument to show the uniqueness.

• 10R. Danchin and P

. B. Mucha, A Lagrangian approach for the incompressible Navier-Stokes equation with variable density. Commun. Pure Appl. Math., 65 (2012), 1458–1480.

• 11R. Danchin and P

. B. Mucha, Incompressible flows with piecewise constant density. Arch. Rational Mech. Anal., 207 (2013), 991–1023.

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A priori estimates for 2D (HNS) on R2

Lemma 3

Let v2d

0 = (v2d 1,0, v2d 2,0, v2d 3,0) ∈ L2(R2), then there exists a unique strong solution

v2d to 2D (HNS) system on (0, ∞) which is smoothly regular for all t > 0. (1) v2d satisfies (cf. [CheG06]) v2d(t)2

L2(R2) + 2∇v2d2 L2(R2×(0,t)) ≤ v2d 0 2 L2(R2),

∀t ≥ 0. (9) v2d2

L2(R+;L∞(R2)) ≤ C0v2d 0 2 L2(R2)

• 1 + v2d

0 2 L2 log2(e + v2d 0 L2)

• .

(10) (2) v2d also satisfies the following energy type estimates that for every t ≥ 0, t∇hv2d(t)2

L2 +

t τ

• ∂τv2d2

L2 + ∇2 hv2d2 L2

• dτ ≤ Cv2d

0 2 L2eCv2d

0 4 L2,

(11) t2∂tv2d(t)2

L2 + t2∇2 hv2d2 L2 +

t τ2∇h∂τv2d2

L2 ≤ Cv2d 0 2 L2eC(1+v2d

0 4 L2), (12)

t3∇h∂tv2d(t)2

L2(R2) + t4∇2 h∂tv2d(t)2 L2(R2) ≤ Cv2d 0 2 L2(R2)eC(1+v2d

0 4 L2).

(13)

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Remark on Decay Estimates of 2D (HNS)

By interpolation and energy type estimates (11)-(13), we have the following.

Corollary 1

Under the assumptions of Lemma 3, v2d(t)Lp(R2) ≤ Cv2d

0 L2(R2)eC(1+v2d

0 4 L2) t 1 p − 1 2 ,

p ∈ [2, ∞], (14) ∇hv2d(t)Lp(R2) ≤ Cv2d

0 L2(R2)eC(1+v2d

0 4 L2) t 1 p −1,

p ∈ [2, ∞], (15) ∂tv2d(t)L∞(R2) ≤ Cv2d

0 L2(R2)eC(1+v2d

0 4 L2) t− 3 2 .

(16) The proof of (11)-(13) uses the method of [PZZ13]12 or [Hoff02]13 (Integration By Parts). REMARK: From (11)-(16), we can say the decay estimates of solution v2d to 2D (HNS) system are (almost) the same with those of solution et∆hv2d

0 to free

heat equation.

Compared with [KozO93], (14)-(16) at p = ∞ removes the additional logarithmic term

• log(1 + t) on r.h.s.
• 12M. Paicu, P

. Zhang, Z. Zhang, Comm. Part. Diff. Equa., 38 (2013), 1208–1234.

• 13D. Hoff, Comm. Pure Appl. Math., 55 (2002), 1365–1407.

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Linear Estimate of Stokes System

We will extensively use the following a priori estimate of the Stokes system (e.g. cf. Theorem 5 of [DanM13]).

Lemma 4

Let 1 < p < ∞, u0 ∈ ˙ B2−2/p

p,p

(Rn), f ∈ Lp(Rn × (0, T)). Then the Stokes system            ∂tu − ∆u + ∇Q = f, in Rn × (0, T), div u = 0, in Rn × (0, T), u|t=0 = u0,

• n Rn,

(17) has a unique solution (u, ∇Q) to (17) with u ∈ C(0, T; ˙ B2−2/p

p,p

(Rn)) and ut, ∇2u, ∇Q ∈ Lp(Rn × (0, T)), and the following estimate holds true sup

0≤t≤T

u(t) ˙

B2−2/p

p,p

(Rn) + ut, ∇2u, ∇QLp(Rn×(0,T)) ≤ C

• fLp(Rn×(0,T)) + u0 ˙

B2−2/p

p,p

• .

(18)

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A priori estimates for 2D (HNS) on R2-continued

If v2d

0 is more regular, we also have the following refined a priori estimates.

Lemma 5

Let v2d

0 ∈ L2 ∩ ˙

B3−2/p

p,p

(R2) with p > 3, then the unique global smooth solution v2d = (v2d

h , v2d 3 ) of the 2D (HNS) system satisfies that

sup

t≥0

v2d ˙

B

3− 2 p p,p

+ ∂tv2d, ∇2

hv2dLp(R+;W1

p ) ≤ C

• v2d

0 3 L2∩ ˙ B3−2/p

p,p

+ 1

• v2d

0 L2∩ ˙ B3−2/p

p,p

eC(1+v2d

0 4 L2 ).

Estimate ∇hv2d

h (t)Lp and ∇hv2d 3 (t)Lp.

From ∂tv2d − ∆hv2d + ∇p2d = −v2d

h · ∇hv2d, div v2d h = 0, use Lemma 4.

From ∂t(∇hv2d) − ∆h(∇hv2d) + ∇(∇hp2d) = −∇h(v2d

h · ∇hv2d), div v2d h = 0,

use Lemma 4 again.

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A priori estimates for 2D (HNS) on R2-continued

We show the refined regularity estimate for the solution v2d other than (5).

Lemma 6

Let v2d

0 ∈ L2 ∩ ˙

B4−2/p

p,p

(R2) with p > 3, then the solution v2d = (v2d

h , v2d 3 ) of the

2D (HNS) system satisfies sup

t≥0

v2d ˙

B4−2/p

p,p

+ ∂tv2d, ∇2

hv2dLp(R+;W2

p ) ≤ C

• v2d

0 6 L2∩ ˙ B4−2/p

p,p

+ 1

• eC(1+v2d

0 4 L2).

From ∂t(∇2

hv2d) − ∆h(∇2 hv2d) + ∇(∇2 hp2d) = −∇2 h(v2d h · ∇hv2d), div v2d h = 0,

use Lemma 4. Such a result is only used in the uniqueness part of Theorem 2, since we need ∇2

hv2d(t) is Lipschitzian there.

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A Priori Estimates For Perturbed System

First we show the a priori energy estimate of (h, w).

Proposition 1

Let h0 = ρ0 − 1 ∈ L2 ∩ L∞(R3), v2d

0 = (v2d h,0, v2d 3,0) ∈ L2 ∩ ˙

B3−2/p

p,p

(R2) (p > 3) and w0 = v0 − v2d

0 ∈ L2(R3), and let (h, w) be a smooth solution to perturbed

system (3)-(4) over R3 × [0, T]. Then under the condition that h0L∞(R3) ≤ 1

2,

we have for all t ∈ [0, T], h(t)Lp(R3) = h0Lp(R3) for each p ∈ [2, ∞], sup

0≤t<T

w(t)L2(R3) + T ∇w(τ)2

L2(R3) dτ

1/2 ≤ C(w0, h0)L2(R3)eB(v2d

0 ), (19)

where B(v2d

0 ) := C

• v2d

0 4 L2∩ ˙ B3−2/p

p,p

(R2) + 1

• eC(1+v2d

0 4 L2).

(20) In above, the statement still holds replacing time interval [0, T] by [0, ∞).

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Proof of Proposition 1

We start from 1 2 d dt w(t)2

L2 + ∇w(t)2 L2

= −

• R3 h(v2d)t · w dx −
• R3 hwt · w dx −
• R3 h(v · ∇)w · w dx

−

• R3 ρ(wh · ∇h)v2d · w dx −
• R3 h(v2d

h · ∇h)v2d · w dx.

Integration by parts + H¨

• lder, interpolation etc.

From

• R3 h(v2d)t · w dx
• , estimate ∂tv2dL1(0,∞;L∞(R2));

from

• R3 h(v2d

h · ∇h)v2d · w dx

• , estimate ∇hv2dL2(0,∞;L∞(R2)).

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A Priori Estimates For Perturbed System - Continued

Next result is of the a priori Lp-based estimate of w.

Proposition 2

Let h0 = ρ0 − 1 ∈ L2 ∩ L∞(R3), v2d

0 ∈ L2 ∩ ˙

B3−2/p

p,p

(R2), p > 3 and w0 = v0 − v2d

0 ∈ L2 ∩ ˙

B2−2/p

p,p

(R3), and let (h, w) be a smooth solution to perturbed system (3)-(4) over R3 × [0, T]. There exists a small absolute constant ¯ c∗ > 0 such that if (h0, w0, v2d

0 ) satisfies

• h0L2∩L∞ + w0L2∩ ˙

B2−2/p

p,p

• exp
• C′
• v2d

0 4p L2∩ ˙ B3−2/p

p,p

+ 1

• eC′(1+v2d

0 4 L2)

• ≤ ¯

c∗, (21) with C′ some absolute constant, then we have sup

t≤T

w(t) ˙

B

2− 2 p p,p (R3) + wt, ∇2w, ∇qLp(R3×(0,T)) ≤ C¯

c∗. (22) In the above, the time interval [0, T] can be replaced by [0, ∞).

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Proof of Proposition 2

Applying Lemma 4 to wt − ∆w + ∇q = F, div w = 0, we have

sup

τ≤t

wp

˙ B

2− 2 p p,p

+ wτ, ∇2w, ∇qp

Lp(R3×(0,t)) ≤ C

      

6

• i=1

Fip

Lp(R3×(0,t)) + w0p B2−2/p

p,p

       , F1 := ρ(v2d · ∇w), F2 := ρ(w · ∇w), F3 := h (v2d)τ, F4 := h wτ, F5 := ρ(wh · ∇hv2d), F6 := h(v2d

h · ∇hv2d).

Denoting by Xw(t) := sup

τ≤t

w(τ)p

˙ B

2− 2 p p,p (R3)

, Yw(t) := wτ, ∇2w, ∇qp

Lp(R3×(0,t)),

(23) and assuming h0L∞(R3) small enough, we get for every p > 3,

Xw(t) + Yw(t) ≤ C t Xw(τ)

• v2d(τ)p

L∞(R2) + ∇hv2d(τ)p L∞(R2)

• dτ+

+ C1

• Xw(t) + Yw(t)

8p−10

7p−10

(w0, h0)L2eB(v2d

0 ) p(6p−10) 7p−10 +

+ Cw0p

˙ B2−2/p

p,p

(R3) + C

• w0p

L2(R3) + h0p L2∩L∞(R3)

• e2pB(v2d

0 ).

Continuity method.

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A Priori Estimates For Perturbed System - Continued

If h0 has some gradient regularity, we have regularity preservation estimate.

Proposition 3

Under the assumptions of Proposition 2 and additionally assume that ∇h0 ∈ L3(R3), then we have sup

t∈[0,T]

∇h(t)L3(R3) ≤ ∇h0L3(R3)e(1+T)C(h0,w0,v2d

0 ),

(24) where C(h0, w0, v2d

0 ) is depending only on the initial data h0, w0, v2d 0 .

Equation of ∇h: ∂t(∇h) + v · ∇(∇h) = −(∇v) · ∇h. ∇vL1(0,T;L∞(R3)) ≤ ∇hv2dL1(0,T;L∞(R2)) + wL1(0,T;L∞(R3)) from previous results.

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Proof of Theorem 1: Approximate System

Let v2d be the unique strong solution to 2D (HNS) system associated with v2d

0 ∈ L2 ∩ ˙

B3−2/p

p,p

(R2), then it is smooth for all t > 0 and satisfies the regularity and decay estimates stated above. We construct (wn+1, hn+1) (n ∈ N) as solutions to the approximate system                      hn+1

t

+ vn · ∇hn+1 = 0, wn+1

t

+ vn · ∇wn+1 − ∆wn+1 + ∇pn+1 = −hn(wn+1

t

+ vn−1 · ∇wn+1) −(1 + hn)(wn+1 · ∇v2d) + f(hn, v2d), div wn+1 = 0, hn+1|t=0 = h0, wn+1|t=0 = w0, with vn = v2d + wn and f(hn, v2d) = −hn(v2d)t − hn(v2d · ∇v2d). Set v−1(t, x) ≡ 0; w0(t, x) ≡ w0(x), h0(t, x) ≡ h0(x), v0(t, x) = v0(x) = w0 + v2d

0 .

Liutang Xue (BNU) Between HNS and INS: Stability Issue Mathflows 2018 23 / 42

Proof of Theorem 1: Uniform Estimates

First consider the estimate of (h1, w1). By assuming h0L∞(R3) sufficiently small and w0L∞(R3) ≤ 1, we obtain that for any t > 0, h1(t)L2∩L∞(R3) = h0L2∩L∞(R3), and w1(t)L2(R3) + t ∇w1(τ)2

L2(R3) dτ

1/2 ≤ C(w0, h0)L2(R3)eB(v2d

0 ), (25)

and sup

t<∞

w1(t)p

˙ B2−2/p

p,p

(R3) + w1 t , ∇2w1p Lp(R3×(0,∞))

≤ 2C

• w0p

L2∩ ˙ B2−2/p

p,p

+ h0p

L2∩L∞

• exp
• C′
• v2d

0 4p L2∩ ˙ B3−2/p

p,p

+ 1

• eC′(1+v2d

0 4 L2)

• .

(26)

Liutang Xue (BNU) Between HNS and INS: Stability Issue Mathflows 2018 24 / 42

We use the induction method. Under the condition (21), that is, there is an absolute constant c∗ > 0 s.t.

• w0L2∩ ˙

B2−2/p

p,p

+ h0L2∩L∞

• exp
• C′
• v2d

0 4p L2∩ ˙ B3−2/p

p,p

+ 1

• eC′(1+v2d

0 4 L2)

• ≤ c∗,

(27) we suppose that for each n ∈ N+ and k ≤ n we have wk(t)L2 + t ∇wk(τ)2

L2 dτ

1/2 ≤ C(w0, h0)L2eB(v2d

0 ), ∀t > 0,

(28) and sup

t<∞

wk(t)p

˙ B

2− 2 p p,p (R3)

+ wk

t , ∇2wkp Lp(R3×(0,∞))

≤ 2C

• w0p

L2∩ ˙ B

2− 2 p p,p (R3)

+ h0p

L2∩L∞

• exp
• C′t′3p
• v2d

0 7p L2∩ ˙ B3−2/p

p,p

+ 1

• ,

(29) which in terms of the notation (23) means that Xwk (t) + Ywk (t) ≤ 2Ccp

∗ ,

∀t > 0. We indeed can prove the similar uniform estimates for wn+1.

Liutang Xue (BNU) Between HNS and INS: Stability Issue Mathflows 2018 25 / 42

Proof of Theorem 1: Uniform Estimates and L2-Contraction

For any T > 0, by Proposition 3 and the above uniform estimates, sup

t∈[0,T]

∇hn+1(t)L3(R3) ≤ C∇h0L3(R3)e(1+T)C(h0,w0,v2d

0 ),

∀n ∈ N. (30) {(hn, wn)}n∈N is Cauchy sequence in L2(R3) on a small interval [0, T0]. Denoting by δhn = hn − hn−1, δwn = wn − wn−1, δpn = pn − pn−1, n ∈ N with the convention h−1 = p−1 = 0 and w−1 = v−1 = 0, we write                (δhn+1)t + vn · ∇δhn+1 = −δwn · ∇hn, (δwn+1)t + vn · ∇δwn+1 − ∆δwn+1 + ∇δpn+1 = H, div δwn+1 = 0, δhn+1|t=0 = 0, δwn+1|t=0 = 0, (31) where H = 10

i=1 Hi with

H1 := −δwn · ∇wn, H2 := −δhn wn+1

t

, H3 := −hn−1 (δwn+1)t, etc. Based on the uniform estimates, we can show sup

τ∈[0,T0]

δhn(τ)2

L2 + sup τ∈[0,T0]

δwn(τ)2

L2 +

T0 ∇δwn(τ)2

L2dτ ≤ C0

2n .

Liutang Xue (BNU) Between HNS and INS: Stability Issue Mathflows 2018 26 / 42

Proof of Theorem 1

Strong Convergence: hn → h, wn → w strongly in various topology. Uniqueness. The maximal time T∗ can equal ∞.

Liutang Xue (BNU) Between HNS and INS: Stability Issue Mathflows 2018 27 / 42

Proof of Theorem 2: Existence

Consider (INS) system (1) for (ρǫ, vǫ) with initial data vǫ|t=0 = v0 and ρǫ|t=0 = ρǫ

0 := πǫ ∗ ρ0, where πǫ is the standard mollifier.

Theorem 1 implies that under the uniform-in-ǫ condition (5), the solution (ρǫ, vǫ) satisfies uniform-in-ǫ estimates (7) and (8). Up to a subsequence, ρǫ − 1 ⇀∗ ρ − 1 in L∞(0, ∞; L2 ∩ L∞(R3)) vǫ−v2d ⇀∗ v−v2d in L∞(0, ∞; L2∩ ˙ B2−2/p

p,p

)∩ ˙ W1

p (0, ∞; Lp)∩Lp(0, ∞; ˙

W2

p ).

By Rellich type theorems, vǫ → v a.e. pointwisely in R3 × (0, ∞). (32) For the problematic term ρǫ∂tvǫ, one has ρǫ∂tvǫ + ρǫvǫ · ∇vǫ = ∂t(ρǫvǫ) + div(ρǫvǫ ⊗ vǫ). We can pass to the limit using its distributional form and (32). The convergences of other terms are direct.

Liutang Xue (BNU) Between HNS and INS: Stability Issue Mathflows 2018 28 / 42

Lagrangian Coordinates

Let Xv(t, y) solve the following ODE (treating y as a parameter)

dXv(t, y) dt = v(t, Xv(t, y)), Xv(t, y)|t=0 = y, (33)

which leads to Xv(t, y) = y + t

0 v(τ, Xv(τ, y))dτ.

Denote ¯ v(t, y) := v(t, Xv(t, y)), then Xv(t, y) = y + t

0 ¯

v(τ, y)dτ.

Lemma 7

Assume that v ∈ L1(0, T; ˙ W1

∞(Rn)). Then the system (33) has a unique

solution Xv(t, y) on [0, T] such that ∇yXv(t)L∞(Rn) ≤ exp t

0 ∇xv(τ)L∞(Rn)dτ

• .

Let Y(t, ·) be inverse diffeomorphism of X(t, ·), then ∇xYv(t, x) =

• ∇yXv(t, y)

−1 with x = Xv(t, y), and if

t ∇y¯ v(τ)L∞(Rn)dτ ≤ 1 2, (34)

we have |∇xYv(t, x) − Id| ≤ 2 t

0 |∇y¯

v(τ, y)|dτ.

Liutang Xue (BNU) Between HNS and INS: Stability Issue Mathflows 2018 29 / 42

Lagrangian Coordinates - Continued

Set ¯ h(t, y) := h(t, Xv(t, y)), ¯ w(t, y) := w(t, Xv(t, y)), q(t, y) := q(t, Xv(t, y)), ¯ F(t, y) := F(t, Xv(t, y)), then according to [DanM12] or [DanM13], the perturbed system (3) recasts in                  ¯ ht = 0, ¯ wt − div

• AvA T

v ∇y ¯

w

• + A T

v ∇yq = ¯

F, divy (Av ¯ w) = 0, ¯ h|t=0 = h0, ¯ w|t=0 = w0, (35) where Av(t, y) := (∇yXv(t, y))−1. (36) As pointed out by [DanM12,DanM13], under the condition (34), the system (35) in the Lagrangian coordinates is equivalent to the system (3) in the Eulerian coordinates.

Liutang Xue (BNU) Between HNS and INS: Stability Issue Mathflows 2018 30 / 42

Lagrangian Coordinates - Continued

Also denote v2d(t, y) := v2d(t, Xv,h(t, y)), and ¯ v(t, y) := v(t, Xv(t, y)), with Xv,h(t, y) = (Xv,1(t, y), Xv,2(t, y)), then ¯ v(t, y) = v2d(t, y) + ¯ w(t, y). (37) The first equation of (35) implies ¯ h(t, y) ≡ h0(y), ∀t ∈ [0, T]. (38) From wt(t, Xv(t, y)) = ¯ wt(t, y) − (v · ∇w)(t, Xv(t, y)), we have ¯ F(t, y) = − h0(y) (v2d)t (t, Xv,h) − h0(y) ¯ wt (t, y) − h0(y) (v2d

h · ∇hv2d)(t, Xv,h) − ρ0(y) wh(t, y) · (∇hv2d) (t, Xv,h)

:= ¯ F1(t, y) + ¯ F2(t, y) + ¯ F3(t, y) + ¯ F4(t, y). (39)

Liutang Xue (BNU) Between HNS and INS: Stability Issue Mathflows 2018 31 / 42

Proof of Theorem 2: Uniqueness

Consider (h1, w1, q1) and (h2, w2, q2) to perturbed system (3) starting from the same initial data (h0, v2d

0 , w0) with

h0 ∈ L2 ∩ L∞(R3), v2d

0 ∈ L2 ∩ ˙

B4−2/p

p,p

(R2) and w0 ∈ L2 ∩ ˙ B2−2/p

p,p

(R3). We have sup

t<∞

wi(t)L2∩ ˙

B2−2/p

p,p

(R3) + (wi)t, ∇2wi, ∇qiLp(R3×(0,∞)) ≤ C c0,

which implies viL∞(R3×(0,T)) ≤ wiL∞(0,T;L∞(R3)) + v2dL∞(0,T;L∞(R2)) ≤ C, and T ∇vi(t)L∞(R3)dt ≤ C0T 1− 1

p wiLp(0,T;L2∩ ˙

W2

p (R3)) + C0Tv2dL∞(0,T;L2∩ ˙

B3−2/p

p,p

)

≤ C(T 1− 1

p + T).

(40)

Liutang Xue (BNU) Between HNS and INS: Stability Issue Mathflows 2018 32 / 42

Proof of Theorem 2: Uniqueness

The system of (hi, wi) (i = 1, 2) in the Lagrangian coordinates writes

                   ∂t ¯ hi = 0, ∂t ¯ wi − div

• AviA T

vi∇y ¯

wi

• + A T

vi∇yqi = ¯

Fi, divy (Avi ¯ wi) = 0, ¯ hi|t=0 = h0, ¯ wi|t=0 = w0. (41)

We have ¯ h1(t, y) = ¯ h2(t, y) ≡ h0(y). The difference equation of ¯ w1 − ¯ w2 =: δ¯ w reads as

             δ¯ wt − div

• Av1A T

v1∇yδ¯

w

• + A T

v1∇yδq = δ¯

F + div

• (Av1A T

v1 − Av2A T v2)∇y ¯

w2

• − (A T

v1 − A T v2)∇yq2,

divy (Av1δ¯ w) = div

• (Av1 − Av2)¯

w2

• ,

δ¯ w|t=0 = 0,

where δ¯ q := ¯ q1 − ¯ q2, and δ¯ F := ¯ F1 − ¯ F2 is decomposed as 5

i=1 δ¯

Fi with

δ¯ F1 := −h0(y)

• ∂tv2d(t, Xv1,h) − ∂tv2d(t, Xv2,h)
• ,

δ¯ F2 := h0(y)δ¯ wt, δ¯ F3 := −h0(y)

• (v2d

h · ∇hv2d)(t, Xv1,h) − (v2d h · ∇hv2d)(t, Xv2,h)

• ,

δ¯ F4 := −ρ0(y) δwh(t, y) · ∇hv2d (t, Xv1,h) , δ¯ F5 := −ρ0(y) w2,h(t, y) ·

• ∇hv2d(t, Xv1,h) − ∇hv2d(t, Xv2,h)
• .

(42)

Liutang Xue (BNU) Between HNS and INS: Stability Issue Mathflows 2018 33 / 42

Proof of Theorem 2: Uniqueness

Goal: δ¯ w(t)L2 is zero. By energy type estimates. Basic problem is related to the nonhomogeneous rhs of the second equation of (41). Because of it, we are not allowed to test the equation (33) by δ¯ w. We instead split it into two parts δ¯ w = z1 + z2, (43) where z1 is given as a solution to the following divergence equation div(Av1z1) = div((Av1 − Av2)¯ w2) = (Av1 − Av2) : ∇y ¯ w2. (44) For our purpose, we introduce Np(T) :=

• f
• f = a + b, a ∈ L

2p 2p−n (0, T; L 2p p+2 (Rn)), b ∈ L2(0, T; L2(Rn))

• with the norm fNp(T) =

inf

• aL

2p 2p−n

(0,T;L 2p

p+2

) + bL2(0,T;L2)

• f = a + b, a ∈ L

2p 2p−n (0, T; L 2p p+2 ), b ∈ L2(0, T; L2)

• .

Liutang Xue (BNU) Between HNS and INS: Stability Issue Mathflows 2018 34 / 42

Lemma 8

Let p > n, and A be a matrix-valued function on Rn × [0, T] satisfying det A ≡ 1. Let g : Rn × [0, T] → R be given by g = div R which satisfies

g ∈ L2(Rn × [0, T]), R ∈ L∞(0, T; L2(Rn)), Rt ∈ L

2p 2p−n (0, T; L 2p p+2 (Rn)).

(45)

There exists a constant c > 0 depending only on n, such that if Id − AL∞(Rn×(0,T)) + AtL2(0,T;L∞(Rn)) ≤ c, (46) then the twisted divergence equation div(A z) = g in Rn × [0, T] admits a solution z in the space XT :=

• f| f ∈ L∞(0, T; L2(Rn)), ∇f ∈ L2(0, T; L2(Rn)), ft ∈ Np(T)
• ,

(47) which satisfies the following estimates for some constant C = C(n): zL∞(0,T;L2(Rn)) ≤ CRL∞(0,T;L2), ∇zL2(0,T;L2(Rn)) ≤ CgL2(0,T;L2), ztNp(T) ≤ CRL∞(0,T;L2(Rn)) + CRtL

2p 2p−n

(0,T;L 2p

p+2 (Rn)).

(48)

Liutang Xue (BNU) Between HNS and INS: Stability Issue Mathflows 2018 35 / 42

We need to verify the conditions (45) and (46) in Lemma 8. Verifying (46). By letting T be small enough, T ∇y¯ v1(t)L∞(R3)dt ≤ T ∇xv1(t)L∞(R3)∇yXv1(t)L∞(R3)dt ≤ CT 1−1/peCT1−1/p ≤ min c 2, 1 4

• ,

(49) with c > 0 the constant in (46), thus Id − Av1L∞(0,T;L∞(R3)) ≤ 2 T ∇y¯ v1(t)L∞(R3)dt ≤ min

• c, 1

2

• ,

We also see that (Av1)t =

∞

• k=1

(−1)kk t ∇y¯ v1(τ, y)dτ k−1 ∇y¯ v1(t, y)

• ,

thus by letting T > 0 be small enough we get (Av1)tL2(0,T;L∞(R3)) ≤ C∇y¯ v1L2(0,T;L∞(R3)) ≤ CT

1 2 − 1 p eCT1−1/p ≤ c.

(50)

Liutang Xue (BNU) Between HNS and INS: Stability Issue Mathflows 2018 36 / 42

Verifying (45). We have to justify that

(Av1−Av2)¯ w2 ∈ L∞(0, T; L2), (Av1−Av2) : ∇y ¯ w2 ∈ L2(0, T; L2),

• (Av1−Av2)¯

w2

• t ∈ Np(T).

(51)

It reduces to control Av1 − Av2L∞(0,T;L2(R3)) and (Av1 − Av2)tL2(0,T;L2(R3)). After a tedious computation, we manage to get for sufficiently small T, Av1 − Av2L∞(0,T;L2(R3)) ≤ CT 1/2(δ¯ wL∞(0,T;L2) + ∇δ¯ wL2(0,T;L2(R3))), (52) (Av1 −Av2)tL2(0,T;L2(R3)) ≤ C(δ¯ wL∞(0,T;L2(R3)) +∇δ¯ wL2(0,T;L2(R3))). (53) Hence, Lemma 8 ensures that z1L∞(0,T;L2(R3))+∇z1L2(R3×(0,T)) ≤ CT 1/2 δ¯ wL∞(0,T;L2)+∇δ¯ wL2(0,T;L2)

• ,

(54) and ∂tz1N(T) ≤ CT

p−3 2p

δ¯ wL∞(0,T;L2(R3)) + ∇δ¯ wL2(0,T;L2(R3))

• .

(55)

Liutang Xue (BNU) Between HNS and INS: Stability Issue Mathflows 2018 37 / 42

Now the equation on z2 satisfies            ∂tz2 − div

• Av1A T

v1∇yz2

+ A T

v1∇yδq = δ¯

F + 4

i=1 ¯

Li divy

• Av1z2

= 0, z2|t=0 = 0, (56) with δ¯ F := 5

i=1 δ¯

Fi given by (42), and ¯ L1 = div

• (Av1A T

v1 − Av2A T v2)∇y ¯

w2

• ,

¯ L2 = −(A T

v1 − A T v2)∇yq2,

¯ L3 = −∂tz1, ¯ L4 = div

• Av1A T

v1∇yz1

. (57) Test (56) by z2, and noticing

• R3 A T

v1∇yδq(t, y) z2(t, y)dy = −

• R3 δq(t, y) div(Av1z2)(t, y)dy = 0,

we derive

1 2 d dt z22

L2 + A T v1∇yz22 L2 ≤ 5

• i=1
• R3 δ¯

Fi(t, y)z2(t, y)dy +

4

• i=1
• R3

¯ Li(t, y) z2(t, y)dy.

Liutang Xue (BNU) Between HNS and INS: Stability Issue Mathflows 2018 38 / 42

For ¯ L3 = −∂tz1, since ∂tz1 ∈ Np(T), for any ǫ > 0 we take ∂tz1 = aǫ + bǫ, with aǫ ∈ L

2p 2p−3 (0, T; L 2p p+2 (R3)), bǫ ∈ L2(0, T; L2(R3)),

so that aǫL

2p 2p−3

(0,T;L 2p

p+2

(R3)) + bǫL2(0,T;L2(R3)) ≤ ∂tz1Np(T) + ǫ,

thus by H¨

• lder and interpolation, we infer that for p ∈ (3, ∞),
• R3

¯ L3(t, y) z2(t, y)dy

• ≤ aǫ(t)L 2p

p+2

(R3)z2(t)L 2p

p−2

(R3) + bǫ(t)L2(R3)z2(t)L2(R3)

≤ aǫ(t)L 2p

p+2 z2(t) p−3 p

L2 ∇z2(t)

3 p

L2 + bǫ(t)L2z2(t)L2(R3)

≤ 1 32∇z2(t)2

L2 + Caǫ

2p 2p−3

L 2p

p+2

z2(t)

2p−6 2p−3

L2

+ bǫ(t)L2z2(t)L2.

Liutang Xue (BNU) Between HNS and INS: Stability Issue Mathflows 2018 39 / 42

Set h0L∞(R3) small enough, we finally deduce z22

L∞(0,T;L2(R3)) + ∇z22 L2(R3×(0,T))

≤ CT

p−2 p Av1 − Av22

L∞(0,T;L2) + CT

2p−5 2p−3 Av1 − Av2 2p 2p−3

L∞(0,T;L2)z2

2p−6 2p−3

L∞(0,T;L2)

+ Caǫ

2p 2p−3

L

2p 2p−3

(0,T;L 2p

p+2

)z2

2p−6 2p−3

L∞(0,T;L2(R3)) + CT 1/2bǫL2(0,T;L2)z2L∞(0,T;L2)

+ C∇z12

L2(R3×(0,T)) + CTδ¯

wL∞(0,T;L2(R3))z2L∞(0,T;L2(R3)). We combine it with (54)-(55) to get z12

L∞(0,T;L2(R3)) + ∇z12 L2(R3×(0,T)) + z22 L∞(0,T;L2(R3)) + ∇z22 L2(R3×(0,T))

≤ CT

p−3 2p−3

δ¯ w2

L∞(0,T;L2(R3)) + ∇δ¯

w2

L2(R3×(0,T)) + z22 L∞(0,T;L2(R3)) + ǫ2

. Passing ǫ to 0 and letting T > 0 be small enough, we conclude that z1 = z2 ≡ 0 and δ¯ w = z1 + z2 ≡ 0 on R3 × [0, T]. We also get (v2d)1 ≡ (v2d)2 and Xv1(t, y) ≡ Xv2(t, y) on R3 × [0, T]. Hence, by (49), coming back to Eulerian coordinates, we infer (h1, w1) ≡ (h2, w2) on R3 × [0, T].

Liutang Xue (BNU) Between HNS and INS: Stability Issue Mathflows 2018 40 / 42

Now suppose the solution to perturbed system (3) is unique on [0, T ′], with T ′ > 0 fixed. Let (hi, wi, qi) (i = 1, 2) be two solutions to (3) starting from same initial data, and also (h1, w1, q1) ≡ (h2, w2, q2) on [0, T ′]. We introduce another Lagrangian coordinate Xvi(t, y) (i = 1, 2) satisfying d Xvi(t, y) dt = vi(t, Xvi(t, y)),

• Xvi(T ′, y) = y.

By arguing as above, and from h(T ′)L∞ ≤ h0L∞ ≪ 1, we find

˜ z12

L∞(T′,T′+T;L2) + ∇˜

z12

L2(R3×(T′,T′+T)) + ˜

z22

L∞(T′,T′+T;L2) + ∇˜

z22

L2(R3×(T′,T′+T))

≤ CT

p−3 2p−3

δ ˜ w2

L∞(T′,T′+T;L2(R3)) + ∇δ ˜

w2

L2(R3×(T′,T′+T)) + ˜

z22

L∞(T′,T′+T;L2(R3))

• .

For any T∗ > T ′, there exists a small T > 0 depending only on the initial data and T∗ so that ˜ z1 = ˜ z2 = δ˜ w ≡ 0 on R3 × [T ′, T ′ + T], and also

• Xv1(t, y) ≡

Xv2(t, y), which implies (h1, w1) ≡ (h2, w2) on R3 × [T ′, T ′ + T]. By connectivity, we get δw ≡ 0 on R3 × [0, T∗] and moreover conclude uniqueness on R3 × [0, ∞).

Liutang Xue (BNU) Between HNS and INS: Stability Issue Mathflows 2018 41 / 42

Thanks for your attention!

Liutang Xue (BNU) Between HNS and INS: Stability Issue Mathflows 2018 42 / 42