1.2 Massless pointwise particle in the whole plane

(1)

Small moving rigid body into a viscous incompressible fluid

Christophe Lacave & Tak´ eo Takahashi November 4, 2016

Abstract

We consider a single disk moving under the influence of a 2D viscous fluid and we study the asymptotic as the size of the solid tends to zero.

If the density of the solid is independent ofε, the energy equality is not sufficient to obtain a uniform estimate for the solid velocity. This will be achieved thanks to the optimalL^p−L^q decay estimates of the semigroup associated to the fluid-rigid body system and to a fixed point argument. Next, we will deduce the convergence to the solution of the Navier-Stokes equations inR².

1 Introduction

We study in this paper the asymptotic of a fluid-solid system as the solid is a rigid disk which shrinks to a point. We first describe the fluid-solid system.

1.1 The fluid-solid system

We consider a rigid disk

S^ε(t) =B(h^ε(t), ε)

immersed into a viscous incompressible fluid. At timet >0, the lagrangian coordinates to the body read η(t, x) :=h^ε(t) +R_θε(t)(x−h0),

where for allt,

h^ε(t)∈R² withh^ε(0) =h0, θ^ε(t)∈R, Rθ=

cosθ −sinθ sinθ cosθ

. The domain of the fluid evolves through the formula

F^ε(t) :=R²\ S^ε(t). (1.1)

We denote byn:=n^ε(t, x) the exterior unit normal of ∂F^ε(t). The equations for the fluid-solid system read

∂u^ε

∂t + (u^ε· ∇)u^ε−divσ(u^ε, p^ε) = 0 t >0, x∈ F^ε(t), (1.2) divu^ε= 0 t >0, x∈ F^ε(t), (1.3) lim

|x|→∞u^ε(t, x) = 0 t >0, (1.4) u^ε= (h^ε)⁰(t) + (θ^ε)⁰(t)(x−h^ε(t))^⊥ t >0, x∈∂S^ε(t), (1.5) m^ε(h^ε)⁰⁰(t) =−

ˆ

∂S^ε(t)

σ(u^ε, p^ε)n dγ t >0, (1.6)

J^ε(θ^ε)⁰⁰(t) =− ˆ

∂S^ε(t)

(x−h^ε)^⊥·σ(u^ε, p^ε)n dγ t >0, (1.7)

u^ε(0,·) =u^ε₀ inF₀^ε, (1.8)

h^ε(0) =h0, (h^ε)⁰(0) =`^ε₀, θ^ε(0) = 0, (θ^ε)⁰(0) =r₀^ε. (1.9)

(2)

Here and in what follows

σ(u, p) = 2νD(u)−pI2, withν >0 is the constant viscosity and

D(u) := 1

2((∇u) + (∇u)^∗). We write for anyx∈R²,

x^⊥:=

−x2

x₁

=Rπ/2x.

It is convenient to extend the velocity field u^ε inside the rigid disk as follows:

u^ε(t, x) = (h^ε)⁰(t) + (θ^ε)⁰(t)(x−h^ε(t))^⊥ t >0, x∈ S^ε(t). (1.10) To apply the result in [12], we also need to assume that the center of the mass corresponds with the center of the disk. For simplicity, let us assume that the density ρ^ε >0 is constant in the disk. We define a global density inR² by

ρ^ε(t, x) =

1 x∈ F^ε(t),

ρ^ε x∈ S^ε(t). t>0.

For any smooth open set O, we define

• V(O) :=n

ϕ∈C₀^∞(O)| divϕ= 0 inOo

;

• H(O) the closure ofV(O) in the normL²: H(O) =n

ϕ∈L²(O)| divϕ= 0 inO, ϕ·n= 0 at∂Oo

;

• V(O) the closure ofV(O) in the normH¹: V(O) =n

ϕ∈H₀¹(O)| divϕ= 0 inOo and its dual space byV⁰(O) with respect toH(O).

We also defineVR(F^ε(t)) the subspace ofV(R²) of velocity fields that are rigid in the solid:

V_R(F^ε(t)) :=

ϕ∈H¹(R²) ; D(ϕ) = 0 inS^ε(t), divϕ= 0 . (1.11) Under the following hypotheses on the initial conditions

u^ε₀∈L²(F₀^ε), divu^ε₀= 0, u^ε₀·n=`^ε₀·non∂S₀^ε, (1.12) there exists a unique global weak solution (u^ε, h^ε, θ^ε) see [34], in the sense of the definition below.

Definition 1.1. We say that (u^ε, h^ε, θ^ε)is a global weak solution of (1.2)–(1.9)if, for any T >0, we have u^ε∈L^∞(0, T;L²(R²))∩L²(0, T;H¹(R²)), u^ε(t,·)∈ VR(F^ε(t)),

and if it satisfies the weak formulation

− ˆ T

0

ˆ

R²

ρ^εu^ε· ∂ϕ^ε

∂t + (u^ε· ∇)ϕ^ε

dxds+ 2ν ˆ T

0

ˆ

R²

D(u^ε) :D(ϕ^ε)dxds= ˆ

R²

ρ^εu^ε₀(x)·ϕ^ε(0, x)dx, (1.13) for any ϕ^ε∈C_c¹([0, T);H¹(R²))such that ϕ^ε(t,·)∈ VR(F^ε(t)).

(3)

1.2 Massless pointwise particle in the whole plane

Whenε→ 0, we establish the convergence ofu^ε to the unique solution of the Navier-Stokes equations in the whole planeR².

The asymptotic behavior of the fluid motion around shrinking obstacles is already considered in several recent papers. Iftimie, Lopes Filho and Nussenzveig Lopes [21] have studied the case of one small fixed obstacle in an incompressible viscous fluid in 2D. Iftimie and Kelliher [20] have treated the same situation in 3D. In [24, 25] Lacave has considered the case of one thin obstacle shrinking to a curve in 2D and 3D.

There is also a large literature about porous medium in the homogenization framework. Since the pioneer work of Cioranescu and Murat [6] for the Laplace problem, the Navier-Stokes system was studied, in particular, by Allaire [1, 2]. We also mention [7, 8, 27, 30, 31, 35] for the fluid motion through a perforated domain.

In all the above studies, the general strategy relies on energy estimate to get a uniform estimate in H¹. It turns out that such an estimate is sufficient to pass to the limit in the weak formulation by a troncature procedure. Namely, for a test functionϕ∈ D(Ω) and for a cutoff functionχ^ε, we note thatχ^εϕis an admissible test function for the Laplace problem in the perforated domain Ω^ε. If the inclusions are far enough, for standard cutoff function, kχ^εk_W1,p remains bounded only forp62 in dimension two, which allows to pass to the limit in terms like´

∇u^ε:∇(χ^εϕ). For the Navier-Stokes equations, the cutoff procedure is more complicated and relies on Bogovski˘ı operators. Indeed, we need approximated test functions that are divergence free (see [1, 25]

and Section 4.2).

When the obstacles can move under the influence of the fluid, we also need to control uniformly the velocities of the solids. In the case of the system (1.2)–(1.9) or more generally in the case of a system with several rigid bodies moving into a viscous incompressible fluid, this control can be obtained from the energy estimate if the masses are independent ofε(see Section 5.2 for details).

If the masses tend to zero, it is no more possible to deduce estimates of the velocities of the solids independently of ε from the energy estimate. One could try to get an estimate of rigid velocities from the boundary condition. However, since the size of the solids tend to zero, this leads to look for aC⁰-estimate for the fluid velocity, and thus for H^s estimates with s > 1. It was the strategy followed in [10, 33] with a H² analysis.

Unfortunately, these articles are based on uniform elliptic estimates in the exterior of a small obstacle which fail fors >1 (see a counter-example related to these estimates in [5]).

Our strategy is different here. Our basic remark is that the small obstacle problem is related to the long-time behavior though the scaling property of the Navier-Stokes equationsu^ε(t, x) =ε⁻¹u¹(ε⁻²t, ε⁻¹x). For one disk moving in the plane, the long-time behavior has been recently studied by Ervedoza, Hillairet and Lacave in [12]. In particular, they have obtained the optimal decay estimates of the Stokes semigroup, i.e. with the rates corresponding to the heat kernel (which are invariant to the parabolic scaling). These estimates are the key to treat the massless pointwise particle.

The goal of the main theorem is to treat the case where the disk shrinks to amassless pointwise particle:

ρ^ε=ρ0 (1.14)

hence,

m^ε=ε²m¹ and J^ε=ε⁴J¹. (1.15)

We consider the massless case for small data:

Theorem 1.2. Assume (1.14). Then there exists λ0 such that the following holds.

Let (u^ε₀, `^ε₀, r₀^ε)be a family inL²(F₀^ε)×R²×R verifying (1.12)and such that

ε|`^ε₀|, ε²|r₀^ε|, ku^ε₀k_L2(F₀^ε)6λ0 (1.16) and

u^ε₀* u₀ in L²(R²). (1.17)

Then for any T >0 we have

u^ε* u^∗ in L^∞(0, T;L²(R²))∩L²(0, T;H¹(R²)) (1.18)

(4)

whereuis the weak solution of the Navier-Stokes equations inR²associated tou0: for anyϕ∈C_c¹([0, T);V(R²)),

− ˆ T

0

ˆ

R²

u· ∂ϕ

∂t + (u· ∇)ϕ

dxds+ν ˆ T

0

ˆ

R²

∇u:∇ϕ dxds= ˆ

R²

u0(x)·ϕ(0, x)dx.

Remark 1.3. For anyT >0, we will actually establish in Section 4.1 that, up to a subsequence, we have h^ε→h uniformly in [0, T],

and in Section 4.3 that, for anyObR²\ {h(t)} (for allt∈(t1, t2) with somet1, t2∈[0, T]), we have POu^ε→POustrongly in L²(t1, t2;L⁴(O)),

wherePO is the Leray projector. This strong limit will be used to pass to the limit in the non-linear term.

As we recover at the limit the weak solution of the Navier-Stokes equations in the whole plane, and as this solution is unique by the Leray theorem, we will deduce that we do not need to extract a subsequence in (1.18).

For a 2D ideal incompressible fluid governed by the Euler equations, the case of a massive pointwise particle (i.e. wherem^ε=m¹is independent ofε) in the whole plane was treated in [16], a massless pointwise particle in the whole plane in [17] and both case in a bounded domain in [18]. In these works, non-trivial limit was obtained (namely, Kutta-Joukowski lift force or vortex-wave system) when we consider non-zero initial circulations around the small solids.

1.3 Plan of the paper

The remainder of this work is organized in four sections.

In the next section, we provide three examples where the initial convergence (1.17) holds.

We establish in Section 3 some uniform estimates on (u^ε,(h^ε)⁰). The energy estimate will give us directly a good estimate for the fluid velocityu^εbut not for the disk velocity (h^ε)⁰. Thanks to the results of [12], we will prove that someL^p−L^q estimates of the Stokes semigroup are independent ofε, and by a fixed point argument we will get a uniform estimate of the disk velocity.

Section 4 is dedicated to the passing to the limit. We introduce the cutoff procedure which follows the trajectory of the solid. A crucial point is to construct a corrected test functionϕ^η which satisfies the divergence free condition. This will be obtained by the Bogovski˘ı operator [3, 4]. Then we follow the analysis developed in [25]. Roughly, we pass first to the limitε→0 far away from the solid to get thatusatisfies the Navier-Stokes equations in this region. Next, we pass to the limitη → 0 in the cutoff function, to prove that the equations are also verified in the vicinity of the massless pointwise particle.

We will discuss in the last section the three dimensional case and we will give the extension of our main result in the case where several solids (with any shapes) tend to massive pointwise particles in bounded domains. In this case the energy estimate is sufficient to obtain a uniform estimate of the solid velocities, which allows us to reach more general geometric configurations than in the massless case.

2 Examples of initial conditions

In this paragraph, we develop three examples of family (u^ε₀, `^ε₀)ε satisfying the compatibility condition (1.12) which converges inL²(R²).

Example 2.1. The first trivial example is the case where u^ε₀ is independent of ε. Namely, let us consider (u^ε₀⁰, `^ε₀⁰, r₀^ε⁰) satisfying (1.12) inF₀^ε⁰ for some ε0 >0. The extension of u^ε₀⁰ by`^ε₀⁰+r^ε₀⁰(x−h0)^⊥ inside the disk verifies also (1.12) inF₀^εfor anyε∈(0, ε₀].

Example 2.2. In domains depending onε, a standard setting is to give an initial data in terms of an independent vorticityω₀= curlu^ε₀ and initial circulationγ₀ =¸

∂B(h0,ε)u^ε₀·τ ds(see, e.g., [16, 17, 20, 21, 24, 25]). For the 2D Euler equations, the vorticity is the natural quantity because it satisfies a transport equation, which implies some conserved properties (e.g. theL^p norm of the vorticity forp∈[1,∞] and the circulation of the velocity).

(5)

For the 2D Navier-Stokes equations in domains with fixed boundaries, the vorticity and the circulation are less relevant, because the Dirichlet boundary condition implies that the circulation is zero for t >0, and the vorticity equation does not give anymore the conservation of the L^p norm. For these equations, the standard framework is related to the energy estimate (3.12), i.e. to consider initial data belonging toL²(F₀^ε). In terms of the vorticity, we recall that u^ε₀(x) = (γ0+´

F₀^εω0)_2π|x|^x^⊥2 +O(_|x|¹2) at infinity (see for instance [16, Section 2]

or [21, Section 3]), hence

u^ε₀∈L²(F₀^ε)⇐⇒γ0+ ˆ

F₀^ε

ω0= 0.

For these reasons, the most natural condition isγ₀ =´

ω₀ = 0. In this case, it is easy to prove the following result.

Lemma 2.3. Let (`0, r0)∈R³, ω0 ∈L^∞_c (R²\ {0}) fixed such that ´

ω0 = 0. Then, for ε small enough such that suppω0∩B(0, ε) =∅, we have a unique solutionu^ε₀ in L²(F₀^ε)of

divu^ε₀= 0 inF₀^ε, curlu^ε₀=ω0 inF₀^ε, lim

|x|→∞u^ε₀(x) = 0, u^ε₀·n=`₀·non ∂B(0, ε),

˛

∂B(0,ε)

u^ε₀·τ ds= 0.

Moreover, extendingu^ε₀ by`0+r0x^⊥ inB(0, ε)we have

u^ε₀* u0 weakly in L²(R²),

whereu0=K_R2[ω0] = _2π|x|^x^⊥2 ∗ω0 is the unique vector field inL²(R²)such that divu0= 0inR², curlu0=ω0 inR², lim

|x|→∞u0(x) = 0.

Proof. The existence and uniqueness ofu^ε₀ is well-known (see e.g. [16, Section 2]):

u^ε₀(x) = 1 2π

ˆ

B(0,ε)^c

(x−y)^⊥

|x−y|² ω0(y)dy+ 1 2π

ˆ

B(0,ε)^c

x

|x|² − x−ε²y^∗

|x−ε²y^∗|² ⊥

ω0(y)dy−ε²

2

X

j=1

(`0)j∇ x_j

|x|²

with the notationy^∗=y/|y|². By a standard computation, we note that

ε²

2

X

j=1

(`0)j∇ xj

|x|²

_L₂

(F₀^ε)6Cε|`0|.

It is also rather classical to prove that the second integral in the right hand side tends to zero asε→0. For instance, the authors establish in [21, Lemmas 7 and 10] the uniform estimate inL²(R²) and the convergence to zero inD⁰(R²), which implies the weak convergence inL²(R²).

As _2π¹ ´

B(0,ε)^c (x−y)^⊥

|x−y|²ω0(y)dy=K_R2[ω] =u0, this ends the proof.

Remark 2.4. Even if the zero circulation condition is mandatory for strong solutions to the Navier-Stokes equations in fixed domain, for the fluid-solid system the no-slip boundary condition would imply

˛

∂B(h₀,ε)

u^ε₀·τ ds= 2πε²r₀^ε.

Hence, an interesting extension could be to study the case of non zero initial circulationγ₀^ε. If we assume that γ₀^εis independent ofε, some singular terms appear at the limit of the formγ0 (x−h0)^⊥

2π|x−h0|², which does not belong to L²_loc(R²), but only toL^p_loc(R²) for allp∈[1,2). Another difficulty in this case is to ensure thatγ₀^ε+´

F₀^εω0= 0

(6)

for anyε, in order to state that the initial velocity is square integrable at infinity. A possibility could be to chose γ₀^ε=´

B(h₀,ε)ω0, i.e. r^ε₀=ffl

B(h₀,ε)ω0. Without this condition,u^ε₀ belongs only toL^p(F₀^ε) for allp∈(2,∞].

Therefore, a circulation independent of εand a vorticity with non zero mean value would require to work in the Marcinkiewicz space L^2,∞(F₀^ε) (weakL² space). Even if this space is less classical that L², there is a large literature for the well-posedness of the Navier-Stokes equations in fixed domain, because it corresponds to relevant initial data for the Euler equations, and also because the self-similar solutions (as the Lamb-Oseen vortex) in the whole plane belongs toL^2,∞. In this case, the Cauchy theory is well-known, and Iftimie, Lopes Filho and Nussenzveig Lopes managed in [21] to consider the small obstacle problem with non-zero initial circulation and initial vorticity with non-zero mean value. For the fluid-solid problem, such a Cauchy theory is not yet established. As the optimal decay estimates for the Stokes semigroup are now known inL^p−L^q [12], we guess that it would be possible to extend it by interpolation to Marcinkiewicz spaces, and then to prove a well-posedness result for the full fluid-solid system. Such an extension would require more work and could be interesting, but the main goal of this article is to stay in the standard framework for the fluid solid problem and to treat the same question as [10, 33].

Example 2.5. Another example of initial conditions satisfying (1.12) can be obtained by truncating a stream function associated to a vector fieldu₀ defined onR².

Let us consider u₀=∇^⊥ψ₀∈L²(R²) such that

divu₀= 0 inR², curlu₀∈L¹∩L^q(R²) withq >1, lim

|x|→∞u₀(x) = 0.

We denote byχa smooth cutoff function such thatχ(x)≡0 inB(0,3/2) andχ(x)≡1 inB(0,2)^c. We consider (`0, r0)∈R³given, then we define

u^ε₀:=∇^⊥ ψ₀(x)χ

x−h₀ ε

+

1−χ

x−h₀ ε

(`₀·(x−h₀)^⊥+r₀^|x−h₂⁰^|²)

!

which is divergence free, tending to 0 at infinity, equal tou₀ far away the solid, and equal to`₀+r₀(x−h₀)^⊥ in the vicinity of S₀^ε. By local elliptic regularity, we can show that ψ₀ ∈ L^∞(B(h₀,2)) and as ¹_ε∇χ ^·−h_ε⁰ converges weakly to 0 inL², one can check thatu^ε₀* u₀ inL²(R²).

Actually, as we consider only one solid, we can choseψ0such thatψ0(h0) = 0 and in that case, we can prove that the convergence holds strongly inL²(R²).

A last cutoff example comes from the porous medium analysis or the thin obstacle problem (see [1, 25]).

Instead of truncating the steam function, we truncate directly the vector fieldu0 and we add a correction to restore the divergence free condition:

u^ε₀:=χ

x−h0

ε

u0(x) + 1−χ

x−h0

ε

(`0+r0(x−h0)^⊥) +g^ε(x) whereg^ε∈H₀¹(B(h0,2ε)∩ F₀^ε) satisfies

divg^ε=−1 ε∇χ

x−h0

ε

·u₀(x) +1 ε∇χ

x−h0

ε

·(`₀+r₀(x−h₀)^⊥).

Such a function can be constructed thanks to the Bogovski˘ı operator, and we can prove that kg^εk_L2 6Cε

ku₀k_L2+ε|`₀|+ε²|r₀| . See later Proposition 4.1 for details. This implies thatu^ε₀→u0 in L²(R²).

3 Uniform estimates

Let (u^ε₀, `^ε₀, r^ε₀) be a family inL²(F₀^ε)×R²×Rverifying (1.12) and such that, up to the extension (1.10),u^ε₀ is bounded inL²(R²) (see (1.17)).

(7)

3.1 Change of variables and energy estimate

As in [12, 34], we make the change of variables

v^ε(t, x) =u^ε(t, x−h^ε(t)), q^ε(t, x) =p^ε(t, x−h^ε(t)), and we define

`^ε(t) = (h^ε)⁰(t), r^ε(t) = (θ^ε)⁰(t).

The vector fieldsv^εis the weak solution of a system similar to (1.2)–(1.9):

∂v^ε

∂t + ([v^ε−`^ε]· ∇)v^ε−divσ(v^ε, q^ε) = 0 t >0, x∈ F₀^ε, (3.1) divv^ε= 0 t >0, x∈ F₀^ε, (3.2) lim

|x|→∞v^ε(x) = 0 t >0, (3.3) v^ε(t, x) =`^ε(t) +r^ε(t)x^⊥ t >0, x∈∂S₀^ε, (3.4) m^ε(`^ε)⁰(t) =−

ˆ

∂S₀^ε

σ(v^ε, q^ε)n dγ t >0, (3.5)

J^ε(r^ε)⁰(t) =− ˆ

∂S₀^ε(t)

x^⊥·σ(v^ε, q^ε)n dγ t >0, (3.6)

v^ε(0,·) =v^ε₀ inF₀^ε, (3.7)

`^ε(0) =`^ε₀, r^ε(0) =r^ε₀. (3.8)

We set the global density inR²:

ρ^ε(x) =

1 x∈ F₀^ε, ρ x∈ S₀^ε. We can define a weak solution

Definition 3.1. We say that (v^ε, `^ε, r^ε)is a global weak solution of (3.1)–(3.8)if, for anyT >0, we have v^ε∈L^∞(0, T;L²(R²))∩L²(0, T;H¹(R²)), v^ε(t, x) =`^ε(t) +r^ε(t)x^⊥ inS₀^ε,

if it satisfies the weak formulation

− ˆ T

0

ˆ

R²

ρ^εv^ε· ∂ϕ^ε

∂t + ([v^ε−`^ε]· ∇)ϕ^ε

dxds+ 2ν ˆ T

0

ˆ

R²

D(v^ε) :D(ϕ^ε)dxds= ˆ

R²

ρ^εv₀^ε(x)·ϕ^ε(0, x)dx, for any ϕ^ε∈C_c¹([0, T);H¹(R²))such that ϕ^ε(t,·)∈ VR(F₀^ε).

One can check thatu^εis a weak solution in the sense of Definition 1.1 if and only ifv^εis a weak solution in the sense of the above definition.

We also define the following functional spaces forp∈[1,∞], L^p_ε=

v∈L^p(R²) ; divv= 0 inR², D(v) = 0 inS₀^ε , with the norm (forp6=∞)

kvk_L^p_ε = ˆ

R²

ρ^ε|v|^pdx ^1/p

.

Forp=∞, the norm L^∞ is the classicalL^∞ norm. We recall that for anyv ∈ L^p_ε, there exists (`_v, r_v)∈R³ such that

v(y) =`v+rvy^⊥ (y∈ S₀^ε).

(8)

Moreover, one can deduce`v fromv by

`v:= 1

|S₀^ε| ˆ

S₀^ε

v dy. (3.9)

One can write the system (3.1)–(3.8) in the following abstract form:

∂_tv^ε+A^εv^ε=P^εdivF^ε(v^ε), v^ε(0) =v^ε₀, where

D(A^ε) :=

v∈H²(R²) ; divv= 0 inR², D(v) = 0 inS₀^ε ,

A^εv:=











−ν∆v inF₀^ε,

2ν m^ε

ˆ

∂S₀^ε

D(v)n ds+2ν J^ε

ˆ

∂S₀^ε

y^⊥·D(v)n dy

!

x^⊥ inS₀^ε.

(v∈ D(A^ε)),

A^ε:=P^εA^ε, F^ε(v^ε) =

v^ε⊗(`v^ε−v^ε) onF₀^ε

0 onS₀^ε, (3.10)

and where P^ε denotes the projector from L^p(R²) to L^p_ε, and `v^ε is defined through (3.9). Note that in the definition ofA^ε,D(v)ncorresponds to the trace of the restriction ofD(v) to the fluid domain.

The operator−A^εis the infinitesimal generator of a semigroup of (S^ε(t))_t_>₀in L^p_ε forp∈(1,∞) (see [12]).

Then, Duhamel’s formula gives the following integral formulation of the above equations:

v^ε(t) =S^ε(t)v₀^ε+ ˆ t

0

S^ε(t−s)P^εdivF^ε(v^ε(s))ds. (3.11) By Sobolev embedding, it is classical to deduce from the weak formulation (see Definition 3.1) that ∂tv^ε belongs toL²(0, T;V_R⁰(F₀^ε)) and that the relation

ˆ T 0

ˆ

R²

ρ^ε(∂_tv^ε·ϕ^ε−v^ε·([v^ε−`^ε]· ∇ϕ^ε))dxds+ 2ν ˆ T

0

ˆ

R²

D(v^ε) :D(ϕ^ε)dxds= 0 is satisfied for anyϕ^ε∈L²(0, T;VR(F₀^ε)). In particular, we can takeϕ^ε=v^ε1[0,t], and we remark that

ˆ t 0

ˆ

R²

ρ^εv^ε·([v^ε−`^ε]· ∇v^ε)dxds=1 2

ˆ t 0

ˆ

R²

ρ^ε(v^ε−`^ε)· ∇|v^ε|²dxds= 0

because divv^ε = 0 and that [v^ε−`^ε]·n|∂S^ε₀ = 0. Next, we observe that ∂tv^ε ∈ L²(0, T;V_R⁰ (F₀^ε)) and v^ε ∈ L²(0, T;VR(F₀^ε)) implies thatv^ε is equal for a.e. time to a functionC([0, T];L²_ε) and that

1 2

ˆ

R²

ρ^ε(x)|v^ε(t, x)|²dx−1 2 ˆ

R²

ρ^ε(x)|v₀^ε(x)|²dx= ˆ t

0

ˆ

R²

ρ^ε(x)v^ε(t, x)·∂tv^ε(t, x)dx for a.e. t∈(0, T).

This means thatv^εsatisfies the energy equality 1

2 ˆ

R²

ρ^ε(x)|v^ε(t, x)|²dx+ 2ν ˆ t

0

ˆ

R²

|D(v^ε)|²dxds= 1 2

ˆ

R²

ρ^ε(x)|v^ε₀(x)|² dx for a.e. t∈(0, T), hence, we have for anyε

1 2

ˆ

R²

ρ^ε(t, x)|u^ε(t, x)|² dx+ 2ν ˆ t

0

ˆ

R²

|D(u^ε)|²dxds= 1 2

ˆ

R²

ρ^ε₀(x)|u^ε₀(x)|² dx for a.e. t∈(0, T). (3.12)

(9)

This energy inequality and the hypotheses (1.14) and (1.17) imply that

(u^ε)ε is bounded in L^∞(0, T;L²(R²))∩L²(0, T;H¹(R²)). (3.13) Let us remark that the energy estimate only implies

ε(h^ε)⁰(t), ε²(θ^ε)⁰(t) bounded inL^∞(0, T).

These estimates do not allow to localize the rigid body and then to use the method developed in Section 4. The goal of the sequel of this section is to obtain an estimate of (h^ε)⁰ independently ofε.

3.2 Semigroup estimates

The key for the uniform estimate of`^εis the following theorem concerning the Stokes-rigid body semigroup.

Theorem 3.2. For each q∈(1,∞), the semigroupS^ε(t)on L^q_ε satisfies the following decay estimates:

• Forp∈[q,∞], there existsK1=K1(p, q)>0 such that for everyv^ε₀∈ L^q_ε:

kS^ε(t)v₀^εk_L^p_ε 6K₁t¹^p⁻¹^qkv^ε₀k_L^q_ε for all t >0. (3.14)

• For26q6p <∞, there existsK₂=K₂(p, q)>0 such that for every F^ε∈L^q(R²;M_2×2(R)) satisfying F^ε= 0 inS₀^ε:

kS^ε(t)P^εdivF^εk_L^p_ε 6K2t⁻¹²⁺¹^p⁻¹^qkF^εk_Lq(R²) for all t >0. (3.15)

• For26q <∞, there existsK`=K`(q)>0 such that for everyF^ε∈L^q(R²;M_2×2(R))satisfyingF^ε= 0 onS₀^ε:

|`_Sε(t)P^εdivF^ε|6K`t⁻(¹₂⁺¹_q)kF^εk_Lq(R²) for all t >0. (3.16) For ε fixed, estimates like (3.14)-(3.15) were only established for the Stokes system with the Dirichlet boundary condition [9, 26]. For the fluid solid problem with one rigid disk in R², this result was recently obtained by Ervedoza, Hillairet and Lacave in [12]. The only point to check here is that the constantsK1, K2, K`

are independent of ε, which will be easily obtained by a scaling argument. Indeed, as the above estimates are optimal i.e. correspond to the decay of the heat solution, they are invariant to the parabolic scaling of the Navier-Stokes equations.

Proof. Forε= 1, the statements of the theorem were proved in [12], see therein Theorem 1.1, Corollaries 3.10 and 3.11.

We note that v^ε(t) :=S^ε(t)v^ε₀ satisfies

∂v^ε

∂t −divσ(v^ε, q^ε) = 0 t >0, x∈ F₀^ε, divv^ε= 0 t >0, x∈ F₀^ε, lim

|x|→∞v^ε(x) = 0 t >0,

v^ε(t, x) =`^ε(t) +r^ε(t)x^⊥ t >0, x∈∂S₀^ε, m^ε(`^ε)⁰(t) =−

ˆ

∂S₀^ε

σ(v^ε, q^ε)n dγ t >0, J^ε(r^ε)⁰(t) =−

ˆ

∂S₀^ε(t)

x^⊥·σ(v^ε, q^ε)n dγ t >0, v^ε(0,·) =v₀^ε inF₀^ε,

`^ε(0) =`^ε₀, r^ε(0) =r^ε₀.

(10)

Setting

v(t, x) :=v^ε(ε²t, εx), q(t, x) :=εq^ε(ε²t, εx), `(t) :=`^ε(ε²t), r(t) :=εr^ε(ε²t), (3.17) standard calculation gives that

∂v

∂t −divσ(v, q) = 0 t >0, x∈ F₀¹, divv= 0 t >0, x∈ F₀¹, lim

|x|→∞v(x) = 0 t >0,

v(t, x) =`(t) +r(t)x^⊥ t >0, x∈∂S₀¹, m¹`⁰(t) =−

ˆ

∂S₀¹

σ(v, q)n dγ t >0, J¹r⁰(t) =−

ˆ

∂S₀¹(t)

x^⊥·σ(v, q)n dγ t >0, v(0,·) =v0 inF₀¹,

`(0) =`₀, r(0) =r₀, where

v0(x) :=v₀^ε(εx), `0:=`^ε₀, r0:=εr₀^ε. (3.18) This means thatv(t) =v¹(t) =S¹(t)v0 and thus that

kv(t)k_L^p

1 6K1t^p¹⁻¹^qkv0k_L^q

1 for all t >0.

Using (3.17)-(3.18), this estimate is equivalent to

kv^ε(t)k_L^p_ε 6K₁t¹^p⁻¹^qkv₀^εk_L^q_ε for all t >0.

Relations (3.15) and (3.16) can be done similarly. In that case, we also set F(x) := 1

εF^ε(εx)

and we show that ifv^ε(t) =S^ε(t)P^εdivF^ε, thenv defined by (3.17) satisfies v(t) =S¹(t)P¹divF.

3.3 Uniform estimate on the solid velocity

We first show that there existsλ0>0 such that ifkv^ε₀k_L2

ε6λ0, then there exists a unique

v^ε∈ C⁰([0, T];L²_ε)∩ C_3/8⁰ ([0, T];L⁸_ε) with `^ε:=`v^ε ∈ C⁰_1/2([0, T];R²) (3.19) satisfying (3.11) (that is a mildsolution). Here we have denoted for any Banach spaceX byC_α⁰([0, T];X) the Banach space of functionsf such thatt7→t^αf(t) are continuous from [0, T] inX. The norm associated is

kfk_C0

α([0,T];X):= sup

t∈[0,T]

t^αkf(t)kX.

(11)

Proposition 3.3. There exist λ0, µ0 > 0 independent of ε such that the following holds for any T > 0. If v₀^ε∈ L²_ε satisfies

kv₀^εk_L2

ε6λ0 (3.20)

then there exists a unique v^ε satisfying (3.11)and such that kv^εk_C0([0,T];L²_ε), kv^εk_C0

3/8([0,T];L⁸_ε), k`^εk_C0

1/2([0,T];R²)are bounded by µ₀.

Moreover there exists a constant C >0 independent ofε such that, if v^ε_0a, v^ε_0b ∈ L²_ε are two initial conditions satisfying (3.20), then

kv_a^ε−v_b^εk_C⁰_([0,T];L²_ε)6Ckv_0a^ε −v^ε_0bk_L²_ε. (3.21) Proof. As we have proved in the previous theorem that the constant in the semigroup estimates are independent of ε, it is then enough to follow the fixed point argument in [12, pp. 364-371]. For completeness, let us write here the details.

Let us introduce the space X^ε:=n

v^ε∈ C⁰([0, T];L²_ε)∩ C_3/8⁰ ([0, T];L⁸_ε) with `_vε ∈ C_1/2⁰ ([0, T];R²)o endowed with the norm

kv^εkX^ε :=kv^εk_C⁰_([0,T];L²_ε)+kv^εk_C⁰

3/8([0,T];L⁸_ε)+k`v^εk_C⁰

1/2([0,T];R²). Let us also define the map

Z^ε:X^ε→ X^ε, defined by

Z^ε(v^ε)(t) =S^ε(t)v₀^ε+ ˆ t

0

S^ε(t−s)P^εdivF^ε(v^ε)(s)ds, whereF^εis defined by (3.10). One can define

Φ(v^ε, w^ε)(t) = ˆ t

0

S^ε(t−s)P^εdivG^ε(v^ε, w^ε)(s)ds, where

G^ε(v^ε, w^ε) =

v^ε⊗(`_wε−w^ε) onF₀^ε 0 onS₀^ε. We deduce from (3.15) that

t³⁸kΦ(v^ε, w^ε)(t)k_L8

ε 6t³⁸K₂(8,4) ˆ t

0

(t−s)^−5/8 kv^ε(s)⊗w^ε(s)k_L4

ε+|`_wε(s)|kv^ε(s)k_L4 ε

ds.

Using H¨older’s inequalities, we obtain from the above inequality that t³⁸kΦ(v^ε, w^ε)(t)k_L8

ε 6t³⁸K2(8,4) ˆ t

0

(t−s)^−5/8

s⁻³⁸kv^εk_C0

3/8L⁸_εs⁻³⁸kw^εk_C0 3/8L⁸_ε

+s⁻¹²|`w^ε|_C0

1/2kv^εk^1/3_L∞L²_ε(s⁻³⁸)^2/3kv^εk^2/3_C0 3/8L⁸_ε

ds 62K2(8,4)B

5 8,3

4

kv^εk_Xεkw^εk_Xε, (3.22)

whereB(·,·) is the Beta function:

B(α, β) :=

ˆ 1 0

(1−τ)^−ατ^−βdτ.

(12)

Similarly,

kΦ(v^ε, w^ε)(t)k_L2

ε62K₂(2,2)B 1

2,1 2

kv^εk_Xεkw^εk_Xε. (3.23) Finally,

`_Φ(vε,w^ε)(t)= ˆ t

0

`_Sε(t−s)P^εdivG^ε(v^ε,w^ε)(s)ds, and from (3.16), we deduce

t¹²|`_Φ(vε,w^ε)(t)|6t¹² ˆ t

0

K`(4)(t−s)⁻(¹₂⁺¹₄)kG^ε(v^ε, w^ε)(s)k_L4(R²)ds.

With the same estimates as in (3.22), we obtain

t¹²|`Φ(v^ε,w^ε)(t)|62K_`(4)B 3

4,3 4

kv^εkX^εkw^εkX^ε. (3.24) Gathering (3.22), (3.23) and (3.24) yields

kΦ(v^ε, w^ε)kX^ε 6C0kv^εkX^εkw^εkX^ε, (3.25) where

C0= 2

K2(8,4)B 5

8,3 4

+K2(2,2)B 1

2,1 2

+K`(4)B 3

4,3 4

. We assume (3.20) for λ₀that we fix below and we apply (3.14) in order to obtain

kS^εv^ε₀kX^ε 6C1λ0, (3.26)

where

C1=K1(8,2) +K1(2,2) +K1(∞,2).

Note that we have used in (3.26) the relation

|`S^ε(t)v₀^ε|6kS^ε(t)v₀^εkL^∞_ε

which is a consequence of (3.9).

Relations (3.25) and (3.26) imply that the mappingZ^ε is well-defined and that kZ^ε(v^ε)kX^ε 6C1λ0+C0kv^εk²_Xε.

Let us set

R:= 1 4C0

and λ0= min R 2C1

, R

. (3.27)

Then the closed ballB_Xε(0, R) ofX^εis invariant byZ^εand ifv^ε, w^ε∈B_Xε(0, R),

kZ^ε(v^ε)− Z^ε(w^ε)kX^ε =kΦ(v^ε, v^ε−w^ε) + Φ(v^ε−w^ε, w^ε)kX^ε62C0Rkv^ε−w^εkX^ε = 1

2kv^ε−w^εkX^ε. Using the Banach fixed point, we deduce the existence and uniqueness results. Moreover, we haveµ0=R= 1/(4C0), which is independent ofε. This strategy comes from [22] and [23]. It is originally done through an iterative method, but it was adapted as a fixed point argument in [28] (see also [32]).

Sensitivity of v^εto the initial data. Assume v^ε_0a, v_0b^ε ∈ L²_εsatisfy (3.20) withλ0 defined in (3.27). Then, kv^ε_a−v_b^εk_X^ε6kS^ε(v_0a^ε −v^ε_0b)k_X^ε+kΦ(v_a^ε, v_a^ε−v_b^ε)k_Xε+kΦ(v^ε_a−v_b^ε, v_b^ε)k_Xε

6C₁kv_0a^ε −v_0b^εk_L²_ε+C₀kv^ε_a−v_b^εkX^ε(kv_a^εkX^ε+kv^ε_bkX^ε) 6C1kv_0a^ε −v_0b^εk_L²_ε+ 2C0µ0kv^ε_a−v^ε_bkX^ε.

We conclude that

kv_a^ε−v^ε_bkX^ε 62C1kv_0a^ε −v_0b^εk_L2 ε.

(13)

One can show that a mild solution in the above sense is also a weak solution (see [12]). For sake of completeness, we give the proof of this result here.

Lemma 3.4. Assume v^ε satisfies (3.19)-(3.20) and (3.11). Then v^ε is the weak solution of (3.1)-(3.8) in the sense of Definition 3.1.

Proof. Let us consider a sequence (v_0n^ε )n with values inD((A^ε)^1/2) such that v_0n^ε →v^ε₀ inL²_ε

and such thatv_0n^ε satisfies (3.20). For alln, it is proved in [34] that there exists a unique strong solution v_n^ε ∈H¹(0, T;L²_ε)∩C([0, T];D((A^ε)^1/2))∩L²(0, T;D(A^ε))

of (3.1)-(3.8) and it satisfies (3.11) since in this case

divF^ε(v^ε_n)∈L²(0, T;L²(R²)).

It is also proved in [34] that (v_n^ε) converges towards the weak solution of (3.1)-(3.8) in the sense of Definition 3.1.

From (3.21), we also have that (v_n^ε) converges towards the mild solution of Proposition 3.3 inL^∞(0, T;L²_ε).

Consequently, the mild solutionv^ε, that satisfies (3.19)-(3.20), is the weak solution associated tov^ε₀.

4 Proof of Theorem 1.2

This section is dedicated to the proof of Theorem 1.2. As recalled in the introduction, [34] established that for anyε >0, there exists a unique weak solution (u^ε, h^ε, θ^ε) to problem (1.2)–(1.9) in the sense of Definition 1.1.

Let us fixT >0.

4.1 First convergences

Thanks to the energy estimate (3.13), we can extract a subsequence such that

u^ε* u^∗ in L^∞(0, T;L²(R²))∩L²(0, T;H¹(R²)). (4.1) By abuse of notation, we continue to writeu^εthe subsequence.

If the initial data satisfies the smallness condition (1.16) withλ0 given in Proposition 3.3, we deduce from Section 3.3 that

|(h^ε)⁰(t)|6 µ0

√t (t >0), whereµ0 is independent ofε. As a consequence,

|h^ε(t)|62µ0

√

T (t∈[0, T]).

We fixq∈(1,2), thus (h^ε) is bounded in W^1,q(0, T;R²) and we have, up to a subsequence,

h^ε→h uniformly in [0, T], (4.2)

with

h∈W^1,q(0, T). (4.3)