Large Time Behavior for a Simplified N-Dimensional Model of Fluid–Solid Interaction

ABSTRACT In this paper, we study the large time behavior of solutions of a parabolic equation coupled with an ordinary differential equation (ODE). This system can be seen as a simplified N-dimensional model for the interactive motion of a rigid body (a ball) immersed in a viscous fluid in which the pressure of the fluid is neglected. Consequently, the motion of the fluid is governed by the heat equation, and the standard conservation law of linear momentum determines the dynamics of the rigid body. In addition, the velocity of the fluid and that of the rigid body coincide on its boundary. The time variation of the ball position, and consequently of the domain occupied by the fluid, are not known a priori, so we deal with a free boundary problem. After proving the existence and uniqueness of a strong global in time solution, we get its decay rate in L p (1 ≤ p ≤ ∞), assuming the initial data to be integrable. Then, working in suitable weighted Sobolev spaces, and using the so-called similarity variables and scaling arguments, we compute the first term in the asymptotic development of solutions. We prove that the asymptotic profile of the fluid is the heat kernel with an appropriate total mass. The L ∞ estimates we get allow us to describe the asymptotic trajectory of the center of mass of the rigid body as well. We compute also the second term in the asymptotic development in L 2 under further regularity assumptions on the initial data.


Introduction and Main Results
The aim of this paper is to describe the large time asymptotic behavior for a coupled system of partial and ordinary differential equations. The system under consideration is a simplified N -dimensional model for the motion of a rigid body inside a fluid flow.

Munnier and Zuazua
The governing equation for the fluid is merely the heat equation, whereas the motion of the solid is governed by the balance equation for linear momentum. For the sake of simplicity, we assume the solid to be a moving ball of radius 1 occupying the domain B t of N whose center of mass lies in the point h t . Thus the system we shall deal with is the following one: where t = N \B t and m > 0 stands for the mass of the ball. The vector n x t is the unit normal to t at the point x directed to the interior of B t . In the above system the unknowns are u x t (that can be seen as the Eulerian velocity field of the fluid) and h t . The coupling condition (1.1-ii) ensures that the velocity of the body is the same as the one of the fluid on its boundary. Equation (1.1-iii) results from the standard conservation law of linear momentum.
Let us stress the main differences between our model and a full model of fluid-structure interaction, namely where T is the stress tensor in the fluid whose components are defined by T ij x t = −p x t ij + u i x j + u j x i and p stands for the pressure. When N = 2, on account of (1.2-ii) and (1.2-iii), the relation (1.2-iv) can be rewritten as

Large Time Behavior of Fluid-Solid Interaction 379
The model (1.2) and other more complete and complex ones involving several bodies with rotational motions, were extensively studied during the last years. Concerning the existence and uniqueness of weak solutions, see for example Conca et al. (1999Conca et al. ( , 2000 and Esteban (1999, 2000); Desjardins et al. (2001) and Grandmont et al. (2002), Grandmont (2002) and the references therein. Recently and independently of the present work, Takahashi and Tucsnak (2004) in the whole space, and Takahashi (2002) for a bounded domain, proved the existence and uniqueness of a strong solution for a more complete version of the model (1.2), adding the rotational motion of the ball. Moreover, it was shown that the solution is global in time, provided the ball does not collide with the boundary of the domain. Whether finite time collision occurs is one of the most interesting and challenging problems in this field. Recently, the problem was solved in 1D by Vázquez and Zuazua (to appear) showing that finite time collision may not occur.
In a previous paper by Vázquez and Zuazua (2003), the large time behavior for a simplified one-dimensional model of fluid-structure interaction was analyzed. In this paper, a sharp description of the asymptotic behavior as time goes to infinity of a point particle, which floats in a fluid governed by the viscous Burgers equation, was given. More precisely, it was proved that the velocity u of the fluid behaves, for t large, like the unique self-similar solution of the Burgers equation on with source type initial data M 0 . The constant M is defined by M = u 0 dx + mh 1 the functions u 0 and h 1 being the initial velocities of the fluid and of the particle, respectively. The present work is a natural extension of this one to the case of several space dimensions. However, in the present paper, the equation governing u is assumed to be linear although similar results could be proved for a model including a convective nonlinearity in the parabolic equation.
It is also of interest to compare our results with the existing ones on the asymptotic behavior of the Navier-Stokes equations (without rigid bodies) in Carpio (1994Carpio ( , 1996 (and references given there). In these papers it is shown that, roughly speaking, the first order approximation is given by the heat kernel with an appropriate total mass. One can expect the same result to be true for the Navier-Stokes equations coupled with the motion of a rigid body (as in (1.2)), but this result has not been proved so far.
Let us go back now to system (1.1). It is a linear free boundary problem, since the position of B t is to be determined. Applying the change of variables v x t = u x + h t t and g t = h t we can rewrite system (1.1) using v and g as new unknown functions, and the system turns out to be nonlinear but in a fixed domain, independent of t. Indeed, we get Here B stands for the fixed ball of center 0 and radius 1, = N /B, and n x is the unit normal to at the point x directed to the interior of B. In view of (1.4), it is clear that the N components of the fluid field (u or v) are coupled through the unknowns (h or g) describing the motion of the solid.
Thus, although it might seem not to be the case, all the components of u are coupled in (1.1).

Notations
Throughout this article, we shall use boldface for N -dimensional vectors like x, y, u, v, , whereas we keep the usual characters for real-valued functions: u, v, . The generic notation v will be used for any of the components v i of the vector v.
In the same way, L 2 , H 1 will stand for L 2 N and H 1 N respectively, being a measurable subset of N . A N × N matrix is denoted M . Its entries are M ij , 1 ≤ i j ≤ N , and M i stands for the i-th row. However, to shorten notation, we sometimes drop the index i and denote generically by M any row of the matrix M . For instance, according to these simplifications, the matrix identity M = UV T leads to the vectors equality (equality of the rows of the matrices) M = U V and can also be rewritten as N scalar equalities M i = UV i . For vectors and matrices, the classical Euclidean norms are defined, V = Nonnegative constants shall be denoted by C along the computations. The value of C can change from one line to the other. We sometimes use C 1 and C 2 when its values need to be followed along the computations. The notation C p allows us to emphasize the dependence with respect to p, the exponent of the Sobolev or L p space we are working in. Finally, in some equalities, C t will stand for a real-valued function such that C t ≤ C for all t > 0. L 2 F and H 1 F stand for weighted spaces where F is a positive function (the weight) on the subset of N . They are endowed with the scalar products uvF x dx and u · vF x dx + uvF x dx, respectively. To shorten notation, we will write L 2 F and H 1 F instead of L 2 F N and H 1 F N , respectively, when = N .
Finally,Ḣ 1 is the closure of C 1 c (the space of C 1 functions with compact support in ) for the norm u 2 dx 1/2 .

The Scalar Version of System (1.1)
Any component (v i g i ), i = 1 N of the solution v g of system (1.4), which we shall merely denote by v and g, is a vector-valued function with two scalar components, which solves Note in particular that, in (1.5), v satisfies a scalar heat equation. However, all the scalar equations satisfied by the components v i , i = 1 N are coupled through the convective term and in particular through the vector field g describing the motion of the solid. As far as the first term in large-time asymptotic development is concerned, we shall prove that the term g · v can be neglected.
To simplify notations, we will sometimes work with these scalar functions v g .

Main Results
Theorem 1.1 (Existence and Uniqueness of Solutions). For any v 0 g 0 ∈ L 2 × N , there exists a unique global strong solution v g of system (1.4) such that The proof of this theorem is quite classical and follows the same ideas as in Vázquez and Zuazua (2003). A complete proof can be found in the self-contained version of the present article Munnier and Zuazua (2004). If we integrate the first equation of system (1.4), using the Stokes formula and the transmission condition (1.4-iii) on the boundary of the ball, we deduce that is independent of time. This first momentum plays a crucial role in the description of the large time behavior of v. This idea will be made more precise in the following theorem. Let us introduce the weight function K x = exp x 2 /4 and the constant N , the area of the unit sphere, necessary to state the main results of this paper. Note that N /N is therefore the volume of the unit sphere. Theorem 1.2 (First Term in the Asymptotic Development). Assume that v 0 ∈ L 2 K and g 0 ∈ N . Then there exist constants C p > 0 depending on the dimension N , on the mass m of the solid and on p such that the following inequalities hold: In these estimates, G stands for the heat kernel on N defined by G t x = 4 t − N 2 exp − x 2 /4t , and the first asymptotic momentum M 1 is given by M 1 = v 0 dx + mg 0 . The error functions R 1 and R 2 are given in the tables, Remark 1.2. Note that the decay rates we obtain for g are the same as those for v in the L -norm. This is perfectly natural in view of the coupling condition in (1.4-ii).
Remark 1.3. Theorem 1.2 provides different decay rates in the case m = N /N and m = N /N . This appears naturally along the proof when analyzing the behavior of More precisely, the estimates on mḡ + n · vd x depend on the mass m of the solid ball. This term decays as t −1/2−2 when m = N /N but only as t −N/2−1 when m = N /N . According to Theorem 1.2, in a first approximation, v behaves, as t → , as the fundamental solution G of the heat equation. Note that this Gaussian profile is multiplied by M 1 = v 0 dx + mg 0 which indicates that the fluid component of the system absorbs asymptotically as t → , the initial momentum introduced by the solid mass.
The values of R 1 in (1.9) and (1.10) of Theorem 1.2 are sharp for p = 2 and all N ≥ 2. This clearly appears when exhibiting the second term in the asymptotic development of v in L 2 in Theorem 1.3 below. We do not know yet if the error estimates for the L p -norms with p > N are sharp or not. In this respect it is important to observe that, although the estimates we obtain for p ≤ N are similar to those that one obtains for the linear heat equation, where one gains an extra t −1/2 factor of decay when subtracting the fundamental solution, our estimates deteriorate as p increases beyond the exponent p = N owing to the additional factor t N p . Remark 1.4. In Theorem 1.2 the dynamics of g is rather simple since, for t large, the action of the fluid on the ball can be neglected. This can be easily predicted by a scaling argument. According to the scaling properties of the heat equation, given v g solution of (1.4), it is natural to introduce v x t = N v x 2 t and g t = N g 2 t for all > 0. Then v g is a solution of the system where B is the ball centered at the origin and of radius 1/ and = N \B . Formally, as → the convective term in the first equation vanishes, and the equation for the acceleration of the ball tends to the trivial identity. Taking this into account, the rescaled solution of the heat equation can be shown to converge to the Gaussian kernel with an appropriate mass. Thus, denoting byṽ andg the limits of v and g as → , one expects as well thatṽ where M 1 can be identified by the property of conservation of momentum.
In view of Theorem 1.2, the solution u of system (1.1) behaves as follows: in all the L p spaces. Moreover, Theorem 1.2 yields precise estimates of the velocity of the ball, g = h . Integrating these relations, we deduce that • When N = 2, h t = M 1 /4 log t + O 1 , and then the ball goes to infinity as t → . • When N ≥ 3, h t ≤ C 0 for all t ≥ 0, where C 0 depends on the initial data.
The ball remains in a bounded domain as t → .

Theorem 1.3 (Second Term in the Asymptotic Development
Then as far as the L 2 -norm is concerned, we can improve Theorem 1.1: there exist two constants a > 0 and b > 0 such that where a = a N , b = b N and where the second asymptotic momenta M 1 2 , M 2 2 and M 2 are N × N matrices defined by • When N = 2, is a bounded quantity. Moreover, all the integrals involved in the definition of M 2 2 for N = 2 and M 2 for N ≥ 3 are well-defined. Remark 1.5. The second term in the asymptotic expansion of the solution contains some terms that may not be explicitly computed in terms of the initial data. This is the case both in dimension N = 2 and N = 3. When N = 2, the definition of M 2 2 contains several time integrals that involve the solution v g for all time t ≥ 0. The same phenomenon occurs (see Theorem 3 in Zuazua, 1993) for scalar convectiondiffusion equations on the whole space N .
It is convenient to display the results of Theorem 1.3 as an asymptotic development as t → in L 2 : For the solutionṽ of the heat equation on the whole space N , with initial datã v 0 ∈ L 1 N 1 + x 2 , we have the asymptotic expansion in L 2 N : where M 1 = Nṽ0 dx and M 2 = − Nṽ0 x dx. Comparing (1.15a) for N = 2 and (1.15b) for N = 3 with the known results for the heat equation (1.15c), we observe some slight differences owing to the presence of the solid mass. In dimension N = 2 the main difference is due to the presence of a time logarithmic multiplicative factor on the second term of the asymptotic expansion involving G. This was already observed to be the case in Zuazua (1993) for the quadratic convective nonlinearity in dimension N = 2. We also see the presence of this time logarithmic factor on the error term. The main difference in the case N = 3 comes from the definition of the factor M 2 multiplying the second term G, which reflects the coupling between the heat equation and the solid mass.

Sketches of the Proofs of Theorems 1.2 and 1.3
The first step in proving Theorem 1.2 consists in establishing the decay rate of the solution v g of the system (1.4) in L p (1 ≤ p ≤ ). We get this result componentwise by multiplying the heat equation by nonlinear functions of v, integrating by parts and using the Hölder, Sobolev, and interpolation inequalities. The problem is then reduced to solving an ordinary differential inequality, and the conclusion arises by exhibiting a suitable supersolution.
In the second step, we introducev In order to prove that M 1 G is the first term in the asymptotic development of v, we have to prove thatv decreases faster than v and G separately do. The decay rate for v is obtained by using the same arguments employed when analyzing the decay rate of v. However, the proof is technically more involved owing to the presence of the correcting terms on the right hand side of (1.16) and (1.18).
In the third step, we rewrite equations (1.16) and (1.18), using the so-called similarity variables and rescaled functions. Working in weighted Sobolev spaces, we determine the decay rate ofv = v − M 1 G in these similarity variables. Expressing this result in the classical variables, we prove, in particular, the decay of the L 1 norm ofv.
The conclusion of Theorem 1.2 follows by interpolation of the L p estimate with the L 1 decay of the solution.
The outline of the proof of Theorem 1.3 is the following: we begin by determining the expressions of M 2 distinguishing the dimension N = 2 and N ≥ 3, using scaling arguments and similarity variables. Then, following in similarity variables, we compute the decay rate in The expressions of these results in classical variables yield the conclusion of Theorem 1.3.

Plan of the Paper
This article is organized as follows: at the beginning of the following section, we give some basic estimates like, for example, the energy dissipation law. Then we study the decay rate in L p of a solution of a generalized version of system (1.4).
This system is similar to (1.4) but is a little more complex because it contains some additional nonlinear terms. As an application of these results, we deduce the decay rate of the solution v of system (1.4), as well as the decay rate ofv = v − M 1 G. The decay of the L 1 norm is proved in Section 3 by classical parabolic techniques, using similarity variables and scaling arguments. However, in our case, the presence of the second unknown g requires special care. These arguments allow us to perform the proof of Theorem 1.2, combining the decay rate of the L 1 norm with the results of Section 2.1. Afterwards, in Section 4, we identify the second term in the asymptotic development in similarity variables and give the proof of Theorem 1.3.

Decay Rates
From now on, we shall work with the scalar functions v and g introduced in Section 1.2 to denote any of the components v i g i of the vectors v, g.

Basic A Priori Estimates
Energy Dissipation: Multiplying by v and integrating by parts the first equation of system (1.5), we find L p Estimates: In the same way, we multiply the equation by j v , with j a realvalued convex function and we integrate with respect to x to obtain If we choose for j v an approximation of the function v p , we deduce that the quantity v p dx + m g p (2.2) decreases in time whenever v 0 ∈ L p for all 1 ≤ p < . The first step in the analysis of the large time behavior of (1.5) consists in establishing the decay rate of the solution. But, instead of studying directly (1.5), we prefer considering the following more general framework in which the same decay properties hold.

General Decay Results
We consider, in this section, any smooth global in time solution v g : where U x t is a matrix valued function and V t , 1 t , and 2 t , three are vector-valued functions that will be specified later.
In the sequel, we will apply the results obtained for the general system (2.3) in the following particular cases.

Application 2. In view of equations
In the following proposition, we describe the decay rate in L p of the solution v g of the general system (2.3).
Proposition 2.1. Let us denote: Fix 1 < p < and assume also that there exists C p > 0 and p > 0 such that the functions Then any smooth solution v g of system (2.3) satisfies the decay properties where p is a positive constant defined by Moreover, the constant C p in estimates (2.7) depend on p and N only. Assume furthermore that C p and p in (2.6) are uniformly bounded for all p large enough. (2.9) In this case, estimates (2.7) remain valid for p = with p as in (2.8) with p = .
Remark 2.1. The following comments are in order.
• In view of the definitions (2.5), it is obvious that 1 and 2 depend on p. Nevertheless, to shorten notation, we have not made this dependence explicit. • We do not make any assumption on the decay properties of the potential V because the term V · v vanishes in all the estimates, since V depends only on t. • The decay rate (2.7) we obtain for v coincides with the one of the solution of the heat equation on N and with those of the 1-d model for fluid-solid interaction in Vázquez and Zuazua (to appear).
Proof of Proposition 2 1. We treat only the case 1 < p < . The case p = is obtained applying an iterative argument inspired by Véron (1979) and used in Vázquez and Zuazua (2003) for a fluid-solid interaction model. We refer to Munnier and Zuazua (2004) for details. We proceed componentwise, using the rules of notation of Section 1.1: v and g stand for any component v i and g i of v and g. The corresponding first momentum will be denoted by M 1 although it stands for the quantity M 1 i .
Multiplying the equation (2.3-i) by v v p−2 and integrating by parts, the term V · v v p−2 v dx vanishes according to Green's formula and we get where I t = g · Uv v p−2 dx + 1 v v p−2 dx + 2 g g p−2 = I 1 t + I 2 t + I 3 t can be estimated as follows.
Lemma 2.1. There exists a constant C > 0 depending on m and N only, such that, for all t ≥ 0: Proof of Lemma 2 1. Concerning I 1 , we have
Going back to equation (2.10), we now give estimates for the term involving the gradient of v p 2 .
where the constant C > 0 depends on N and m only.
The proof of this Lemma is quite similar to the proof of Lemma 1 in Escobedo and Zuazua (1991). The complete proof is given in Munnier and Zuazua (2004). Observe now that, by a convexity argument, Using the inequalities (2.14) and (2.15), one obtains with C uniform with respect to p. In view of the definition (2.4) of 1 , we get v 1 + m g ≤ 1 /2N and then where C does not depend on p. In all the sequel we will be very careful on how the constants in the estimates depend on 1 and p. Introducing the functions we can summarize (2.10), (2.11) and (2.18) by This last inequality holds for each component v and g of v and g. Adding together these N inequalities, we get for all t ≥ 0, Relations (2.22) together with (2.21) yield, for all t ≥ 0: According to the notations (2.5) of Proposition 2.1, (2.20) reads with C p = C p − 1 /p 2 . We introduce then the function Z p defined on 0 by Z p t = C p p N/2 + 1 t − p N/2 1−1/p t −N/2 1−1/p , where p and p are the constants Large Time Behavior of Fluid-Solid Interaction 391 defined by (2.8) and in the hypothesis (2.6), respectively, and 1 and C p are C p and C being the constants in (2.24). The assumption (2.6) of the proposition ensures that 1 is well defined. Direct computations yield We are going to prove that any solution Y p of (2.24) is a subsolution of (2.26), and hence, by Theorem 1.5.3 of Lakshmikantham et al. (1989), that in the definition of Z p is the term we need in the right-hand side of (2.28) in order to conclude the proof of the decay rate (2.7). The term added in the expression of Z p in which t − p appears has no influence on the asymptotic behavior as t → , but it is required to get (2.28) in the neighborhood of t = 0. Thus it is sufficient to prove that Note that the term t −1 Z p , in the definition (2.27) of F t , is positive. Since N/2 1 − 1/p / N/2 + 1 t − p ≤ 1 we obtain also that On the other hand, N/2 + 1 t − p − N 2 1− 1 p ≤ 1 because N ≥ 2, and C −1 p ≤ 1 (obvious with the definition (2.25)) and N/2 ≥ 1. Hence to get (2.29) it is sufficient to prove that

Munnier and Zuazua
Dividing in (2.30) by max N/2 C p C ≥ 1, the problem is reduced to prove that for all t ≥ 0, where 0 < C < 1 is defined by We proceed in two steps proving first that (2.31) holds on 1 and then on 0 1 .
We deduce that, for all t ≥ 1: Combining (2.33a) and (2.33b), we get, for all t ≥ 1: Comparing the definitions (2.25) and (2.32) of C p and C, one remarks that C p = 3 C −1 N/2 and therefore that CC 2p/N p−1 p = 3 3 C −1 1/ p−1 . Since 0 < C < Summing together the three relations (2.36), we get: We deduce also, from (2.36c) that C −1 1 C p p 2p N p−1 − p ≥ 0 for all t > 0, what allows us to conclude that (2.31) is true for all t ≥ 1.
The case t ∈ 0 1 . We must now establish the estimate (2.31) on the interval 0 1 . Since t p > 0, (2.31) is equivalent to According to (2.36c) we have C C p p / 1 2p/N p−1 − 1 ≥ 0 as well as From the definition (2.8) of p , we deduce straightforwardly that p ≥ 1 . Hence it remains only to check that which is obvious in view of the definition (2.25) of 1 . The proof is then completed for p < .

Decay Rates
We are now in a position to prove the following proposition, as announced in Application 2: for all t > 0 and for all 1 ≤ p ≤ . The constant C > 0 in these estimates depends on p, m, and N but is independent of the initial data.
Remark 2.2. The complete asymptotic analysis will show that these decay estimates are sharp. The decay rate of g is a consequence of the L estimate of v, because of the transmission condition v = g on the interface .
Proof. As explained in Application 1, we only have to apply Proposition 2.1, setting V = 0, U = 0 . Condition (2.6) is trivially satisfied. The decay property (2.2) ensures that, since v 0 is in L 1 , we have v 1 + m g ≤ v 0 1 + m g 0 . Therefore, in this case, we can set 1 = 2N v 0 1 + m g 0 and the proof of the proposition is then completed.

The First Term in the Asymptotic Expansion
This section is devoted to the proof of Theorem 1.2.
As we have pointed out in Section 1.3, the first momentum M 1 , defined by M 1 = v t dx + mg t is constant in time. The role played by this quantity in the description of the large time behavior of v is made precise in Theorem 1.2. Actually, M 1 G t is the first term in the asymptotic development of v.
In application 2 we have definedv = v − M 1 G andḡ, the trace ofv on the boundary of . The pair, v ḡ solves: where 1 = M 1 g · G and 2 = −M 1 (see (1.17)). The following proposition concerns the decay rate ofv andḡ.
Proposition 3.1. Assume that the initial data v 0 g 0 ∈ L 2 × N of (1.4) are such that v 0 ∈ L 1 . Then define, for all t ≥ 0, and also set, for all t ≥ 1 and all 1 < p ≤ , distinguishing the values of the mass m of the ball: When m = N N , Then 1 t is bounded on 0 and for any t 0 ≥ 1, the solution v ḡ of (3.1) satisfies the decay properties Remark 3.1. We shall prove later that v t 1 + m ḡ t goes to 0 as t → and hence that also 1 t and p t go to 0 as t → . Then, choosing t 0 = t/2 we will be able to improve the decay rate of v t p and ḡ t .
Let us define t x = exp x 2 4 t + 1 (3.5) and recall that K x = 0 x . Since K is radially symmetric, we will sometimes use the notation K r with r = x ∈ + instead of K x .
In the sequel, we will perform the proof of the following proposition, improving the decay rate ofv given in Proposition 3.1 for particular values of p.

Proposition 3.2.
Let v 0 be in L 2 K . Then the solution v g of system (1.4) satisfies the estimate for all 1 ≤ p ≤ 2. The constant C in these estimates depends on p, N , and m.
Remark 3.2. Estimates (3.6) fit exactly those of the heat equation on the whole space N , the case N = 2 being excepted, where a logarithmic term appears in the decay rate. We shall show that this logarithmic term is due to the contribution of the solid mass in the system and that estimates (3.6) are sharp when p = 2.
Proof of Theorem 1.2. Assuming that Proposition 3.2 holds, let us proceed to complete the proof of Theorem 1.2. Relation (3.6) with p = 1 provides the estimates: From Proposition 2.2 we deduce that ḡ t ≤ Ct − N 2 . Therefore the positive constant 1 t of Proposition 3.1 can be estimated by On the other hand, (3.4) ensures that, for all the constant p t 0 being defined by (3.3).
The Case m = N /N . Since the quantity v 1 + m g decreases in time (2.2) and 1 + N/2 N/2 1−1/p ≤ C N = 1 + N/2 N/2 , from (3.3) we deduce that, for all 1 < p ≤ , Since 0 ≤ N p−1 2p+N p−1 ≤ N N +2 ≤ 1, we can assume that the constant C N p−1 2p+N p−1 is independent of p and rewrite the inequality (3.9) as According to (3.7), • When N = 2, and, because N ≥ 2, basic computations yield Therefore, from (3.10) and (3.7), we deduce for all t large enough, -For all 1 < p ≤ N , • When N ≥ 3: the only difference with the case N = 2 comes from (3.7), that is to say, from the absence of a logarithmic term. Consequently, we get the following estimates for p t , for all t large enough: -For all 1 < p ≤ N , The Case m = N /N . The definition of p t is different and according to (3.3), we must turn (3.9) into The same kind of computations as above lead to • When N = 2 and for all 1 < p ≤ and all t large enough, • When N ≥ 3 and for all 1 < p ≤ and all t large enough, The estimates of Theorem 1.2 arise straightforwardly when combining (3.8) with (3.11) and (3.12) and specifying t 0 = t/2.
We perform now the proof of Proposition 3.1.
Proof of Proposition 3 1. It is quite easy to check that 1 t is bounded for all t ≥ 0. Indeed, according to the definition ofv andḡ in Application 2, we have v 1 + m ḡ ≤ v 1 + m g + M 1 G 1 + M 1 m J . Explicit computations give G 1 ≤ 1 and J ≤ 4 t −N/2 e −1/4t and M 1 ≤ v t 1 + m g t . On the other hand, relation (2.2) ensures that v t 1 + m g t ≤ v 0 1 + m g 0 so that v 1 + m ḡ ≤ C v 0 + m g 0 .
The proof of estimates (3.4) derives from Proposition 2.1. We have G p ≤ Ct −N/2 1−1/p −1/2 for all 1 ≤ p ≤ , where the constant C does not depend on p. On the other hand, Proposition 2.2 ensures that: (3.13) Therefore, since 1 = M 1 g · G, we deduce that The definition of 2 leads to the estimates Note that the decay rate of the correcting term 2 is of order t −N/2−2 when m = N /N and only of order t −N/2−1 when m = N /N . This leads us to distinguish these two cases in Theorem 1.2. From (3.14) and (3.15) and according to the definition (2.5) of 2 , we deduce that In model (3.1), U = 0 and V = g with the notations of system (2.3). Thus 1 = 0, and according to (3.16), the hypotheses (2.6) and (2.9) are fulfilled with p = N/2 + 1. Note in particular that p is independent of p. Therefore Proposition 2.1 applies, and relation (3.4) holds with t 0 = 1 for all 1 ≤ p ≤ . The constant p is defined by (2.8) and by (2.8) specifying p = . To get estimates (3.4) for any t 0 ≥ 1, remark that the proof above applies for the functionsv t + t 0 andḡ t + t 0 and the initial conditionsv t 0 andḡ t 0 . Indeed, all the estimates, (3.13), (3.14), (3.15), and (3.16), remain valid, replacing v 0 1 + m g 0 by v t 0 1 + m g t 0 , because this quantity decreases in time (see (2.2)). According to (3.16), we can simplify the expression of p t 0 and turn (2.8) into (3.3).
Proof of Proposition 3 2. We use the so-called similarity variables (we refer to Kavian, 1987, 1988;Zuazua, 1991 andZuazua, 1993 for details). Equivalently, we can express v and g with respect to and : The vector valued functions and solve the following system: where the operator L s is defined componentwise by L s = − − y/2 · . Note that the domain s , where (3.20) holds, evolves in the new time variable s. Thus L s (which is, apparently, time independent) has to be viewed as a time-dependent unbounded operator in L 2 K s with domain H 2 K s ∩ H 1 0 K s . We will denote merely by L this unbounded operator in L 2 K N with domain H 2 K N (see Escobedo and Kavian, 1987). We introduce also This function 1 corresponds to the heat kernel in similarity variables. In addition, the function 1 solves i.e., it is an eigenfunction associated with the eigenvalue 1 = N/2 of L. In fact, 1 is simple, and it is the first eigenvalue of L, that has a discrete spectrum that can be computed explicitly (see Escobedo and Kavian, 1987). The quantity M 1 1 is expected to be the first term in the large time expansion of . Hence we are mainly interested in the large time behavior of y s = y s − M 1 1 y y ∈ s and¯ s = s − M 1 1 y y ∈ B s These functions play the role ofv andḡ in similarity variables. They are bounded, and this is a consequence of the decay properties of g and v in Proposition 2.2. Since 1 is also bounded on N , one deduces that Combining (3.20) and (3.22), one deduces that the pair ¯ ¯ solves and r s = e − s 2 is the radius of the ball B s . In (3.24-iii), the quantity e −s N 2 s is a correcting term due to the contribution of 1 . Remark that this system can also be derived from (3.1) in a straightforward way by making the change of variables (3.17).
From now on, we will work componentwise, using the rules of notation of Section 1.1. We shall use in the sequel the notation · · s for the scalar product of L 2 K s , namely f g s = s fgK dy and · s the associated norm. Moreover, s stands for K r s and hence where C s is a positive function such that 0 < C 1 ≤ C s ≤ C 2 < , for all s > 0.
Multiplying componentwise the first equation of system (3.24) in the weighted Sobolev space L 2 K s by¯ we obtain for all s > 0. Integrating by parts, it comes to L s¯ ¯ s = ¯ 2 s − s ¯ n¯ K d y . Then according to the coupling condition on the interface s we can rewrite (3.27) as In order to analyze the first term in (3.28) involving the time derivative, we need the following identity: This lemma derives from the Reynolds formula in fluid mechanics (see, for example, Arnold, 1978, Lemme 1, p. 69, or the complete proof in Munnier and Zuazua, 2004). Applying the above lemma to the function¯ 2 K in the domain s , we deduce that 1 2 d ds On the other hand, a simple computation gives the following identity for the term of (3.28) involving the time-derivative of¯ : Combining together the relations (3.28), (3.30), and (3.31) and introducing the function X s = ¯ 2 s + m e −sN/2¯ 2 , we get Taking into account that 1 = − y/2 1 = − y/2 4 −N/2 K −1 we deduce that M 1 · 1 ¯ s = −M 1 4 −N/2 · y/2 K −1 ¯ s . Keeping in mind that is bounded, it comes to On the other hand, we have the obvious inequalities Combining (3.32), (3.33), and (3.34), we obtain Taking into account once again the fact that¯ (see (3.23)), , and are bounded, we can simplify the above estimate to In order to obtain an ordinary differential inequality for ¯ 2 s + m e −s N 2¯ 2 , one needs an estimate for the term ¯ 2 s . First of all, let us recall some classical results about the operator L: this is a self-adjoint unbounded operator in L 2 K with domain D L = H 2 K . Its eigenvalues are k = N + k − 1 /2 k ∈ * , and the first eigenvalue is simple. Its eigenspace, denoted E 1 , is spanned by 1 (we refer to Escobedo and Kavian, 1987, for details). Moreover, we can express the eigenvalues by means of the Rayleigh principle, which reads, for 1 and 2 : inf ∈L 2 K 2 2 = 1 and inf where · stands for the natural norm of L 2 K . Note that the condition ∈ E ⊥ 1 means precisely that dy = 0. Thus 1 and 2 are the minima of the Rayleigh quotient on H 1 K and on the subspace of H 1 K of functions of null mass, respectively.
However, we are dealing with L s on s and not with L on N . But because of the coupling condition (3.20-ii) on the interface s , any function of H 1 s can be extended to be in H 1 N by setting When s is large, we are going to show that¯ = − M 1 1 is "almost" in E ⊥ 1 since it tends to zero as t → in L 1 N . This together with the definition of 2 shows that ¯ 2 s ≥ N + 1 /2 ¯ 2 s up to a small correcting term. The task consists in evaluating sharply this correcting term. The ideas we shall apply are quite close of those of (Duro and Zuazua, 1999, Lemma 2). We state the lemma in a more general framework, in order to apply it in other cases as well: Proof. We extend to be a function defined in the whole space N by setting: y s = s − 1 1 y y ∈ B s (3.41) We introduce then r 1 s = 1 − s 1 / 1 2 . Remark that r 1 s = 0 if and only if ∈ E ⊥ 1 . According to the expression (3.39) of 1 , and since 1 2 = 4 −N/2 , it comes to r 1 s = s dy + me −sN/2 s − N dy = me −sN/2 s − B s dy = e −sN/2 m − N /N . Since is bounded, it follows that If we set now 1 = + r 1 1 , then 1 ∈ E ⊥ 1 , and according to (3.37), 1 2 ≥ N + 1 /2 1 2 . Therefore we obtain 2 + r 2 1 1 2 + 2r 1 1 ≥ N + 1 /2 2 + r 2 Consequently, we can turn (3.43) into 2 ≥ N + 1 /2 2 + 1/2 r 2 1 1 2 + r 1 1 . Observe that = 1 − r 1 1 and 1 ⊥ 1 . Thus the inequality above can be rewritten as 2 ≥ N + 1 /2 2 − 1/2 r 2 1 1 2 . We denote · B s the scalar product in L 2 K B s and · · B s the associated norm. We get then 2 ≤ Ce −sN . From these three inequalities, we deduce that R s ≤ Ce −sN/2 . This last estimate together with (3.45), yields the conclusion of the lemma.
One plugs now the estimate (3.40) into (3.36) to obtain ∀s > 0 and after grouping together the terms involving ¯ 2 s we obtain that Recalling that ¯ s as well as s are bounded, the inequality (3.46) can be turned into X s This differential inequality yields the desired result by applying a suitable Gronwalltype inequality.
Lemma 3.3. Let X be a nonnegative function on 0 that satisfies where ≥ 0, > 0, ≥ 0, and C 1 , C 2 , and C 3 are given constants. Then, X s satisfies the decay properties • The case N = 2, • The case N ≥ 3, The proof is linked with the so-called Bihari-type inequality (Lakshmikantham et al., 1989, Section 1.3). We refer to Munnier and Zuazua (2004) for the proof.
With the estimates of Lemma 3.3 applied to (3.47), we obtain When N = 2, ¯ 2 s + m e −s N 2¯ 2 ≤ Cs 2 e −s ∀s ≥ 1 (3.50) In particular − M 1 1 2 s ≤ Cs 2 e −s and, according to the definition (3.18) of , s e sN/2 v e s − 1 e s/2 y − M 1 e −sN/2 1 e s/2 y 2 K y e sN/2 dy ≤ Cs 2 e −s for all s ≥ 1. Getting back to the variables x and t, and since dx = e sN/2 dy, it follows In this case, the logarithmic term does not appear and we find simply Remark 3.3. Comparing (3.51) with the estimates of Proposition 3.1, namely with v 2 ≤ Ct −N/4 , we have gained a decay rate of the order of log 1 + t t −1/2 in dimension N = 2 and t −1/2 in dimension N ≥ 3. Moreover, the result is also improved by the presence of the weight x t into the norm in (3.51). Note that similar results are true for the pure heat equation. In that case, when subtracting the fundamental solution of an appropriate mass, solutions gain a decay rate of the order of t −1/2 in any space dimension.
Let us finally prove that, from the estimates (3.51), one can deduce the relations (3.6) of Proposition 3.2.
Fix p in 1 2 and t > 0. Then, by Hölder's inequality, we obtain v p p ≤ v 2 t dx

Large Time Behavior of Fluid-Solid Interaction 405
A straight computation gives t t N/2 1−p/2 for all t > 0. This relation, together with the estimates (3.51), yields the conclusion of the Proposition 3.2.

Second Term in the Asymptotic Development
In this section, we will make more precise the conclusions of Theorem 1.2 for p = 2, analyzing the large time behavior of in L 2 and L 2 K s respectively.

Identification of the Second Term
In this subsection, we identify the quantities entering in the second term of the asymptotic expansion and we show that they are well-defined.
Observe that the decay rate of¯ when N = 2, in (3.50), namely the fact that ¯ s ≤ Cse −s/2 , does not guarantee that 1 is bounded in L 2 s K . Nevertheless, s −1 1 is bounded. However, for N ≥ 3, 1 itself is bounded in L 2 K s . This fact will be relevant all along this section.
To begin, define 1 as the trace of 1 on s . Since 1 1 = e s 2 ¯ ¯ and ¯ ¯ solves system (3.24), we deduce that ( 1 1 ) solves where 1 s = e s/2 s and s as in (3.25). We denote 2 1 2 2 2 N the N eigenfunctions of L associated to 2 = N + 1 /2, that span the eigenspace E 2 , 2 i y = 1 y i y = − y i 2 1 y y ∈ N 2 = 1 (4.2) As we shall see, 1 behaves for large s as M 2 s 2 where M 2 s is as follows: • When N = 2, M 2 s = s M 1 2 + M 2 2 is an affine function with M 1 2 and M 2 2 two constant matrices that we shall identify. • When N ≥ 3, M 2 is a constant matrix to be determined.
To shorten notation and avoid distinguishing dimensions N = 2 and N ≥ 3, we will sometimes use the notation The fact that 2 enters into the large time behavior of the solution 1 of (4.1) can easily be motivated. For instance, when dealing with the solutions of w s + Lw − N + 1 /2 w = 0 in 0 × N , by the Fourier expansion of the solution w on the basis of eigenfunctions of L, it can be easily seen that, when w is of zero mass, the leading term is the projection onto E 2 . System (4.1) can be viewed as a perturbation of this ideal situation. Its dynamics, although it is essentially of the same nature, is more complex.
When N = 2, the projection of 1 over 2 grows linearly with s > 0 and therefore this case needs a distinguished treatment.
As we shall see, the matrices M 1 2 , M 2 2 (N = 2), and M 2 (N ≥ 3) entering into the second term of the asymptotic expansion are The following proposition guarantees that the limits above are well-defined. By direct computation, and since 2 i = −y i 1 /2 and 1 = −y 1 /2, we get on s 2 i n = 2 i · n = − n i 2 1 − y i 2 1 · n = y i 2 1 e s/2 − e −s/2 2 (4.6) because n = −e s/2 y on s . Since 1 is radially symmetric, we get with (4.6) s 2 i n d y = 0. On the other hand, taking into account the fact that 1 = 1 is constant on the boundary s , we deduce that The extra regularity for the initial data assumed in the statement of Theorem 1.3 is needed to prove the following lemma (see the proof in Munnier and Zuazua, 2004). Applying this lemma, we obtain e s/2 s n y i d y ≤ Ce − 0 s (4.12) Let us address now the second term of the right-hand side of equality (4.10). In view of the explicit form (4.2) of 2 i , we get 2 i K = − 1/2 4 −N/2 y i and then, integrating by parts, we obtain where i stands for the ith component of the vector . But, in similarity variables, according to (3.19), M 1 = s dy + me −sN/2 for all s ≥ 0, so (4.13) can be rewritten as · 2 i s = 1/2 M 1 4 −N/2 i − m/2 4 −N/2 e −sN/2 i . Finally, integrating (4.10) in time from 0 to s, we obtain Estimate (4.12) ensures that, for all N ≥ 2, 0 e 2 n · y i d y d < (4.15) On the other hand, according to Theorem 1.2 (once again, the estimate below can be slightly improved when m = N /N , but it is sufficient for our purpose), s + M 1 4 − N 2 with = s as in (1.14d). Relation (4.17) means that −1 N s is bounded for all s ≥ 0. We obtain then, for all > 0, Integrating now from 0 to s, we obtain Then, combining (4.14) and (4.18), it comes to 2 4 N/2 s 1 2 i K dy Then, dividing by s, taking into account (4.15), and letting s go to , we obtain that the definition (4.3a) of M 1 2 leads to the equality (1.14a). We get also When N ≥ 3, we find the expression Letting s → in (4.20) and (4.21), we find that the expressions (1.14b) and (4.3b), and (1.14c) and (4.3c) coincide, respectively. That concludes the proof of Proposition 4.1.

Proof of Theorem 1.3
We shall proceed in several steps, establishing a sequence of preliminary lemmas. Theorem 1.3 will then hold immediately.
We recall the definition of M 2 s (see As we did for¯ in the proof of Proposition 3.2, we are going to show that ¯ 1 s solves an ordinary differential inequality, and then apply a Gronwall-type inequality. where N 1 = B x 2 1 d x = B x 2 i d x for any i = 1 N due to the symmetry of the ball. Since 0 < C 1 < < C 2 , and taking into account the definition (4.22) of M 2 s , we deduce that (4.25) holds.
Let us now address the second term of (4.24). Integrating by parts, we get This term is the one involving the most important technical difficulties. We shall treat separately the two terms on the right-hand side. Denoting by the orthogonal projection from L 2 K onto the subspace E 1 ∪ E 2 ⊥ , where E 1 and E 2 are the eigenspaces of the operator L associated with the eigenvalues 1 = N/2 and 2 = N + 1 /2, respectively. In particular, E 1 is spanned by 1 and E 2 by 2 1 2 N . We have¯ 1 = 1 ¯ 1 / 1 2 1 + N i=1 2 i ¯ 1 / 2 i Therefore, we get ¯ 1 =¯ 1 + e s 2 r 1 s 1 + r 2 · 2 (4.31) Since the third eigenvalue of L is 3 = N +2 2 , and ¯ 1 ∈ E 1 ∪ E 2 ⊥ , we have ¯ 1 2 ≥ N + 2 2 ¯ 1 2 (4.32) The following relations of orthogonality in L 2 K : 1 2 i = 0, since f 2 = f 2 s + f 2 B s for all function f ∈ L 2 K and all s > 0. It remains to estimate each term of the right-hand side of (4.33).
First term. From the definition of¯ 1 on B s (see (4.29)), we deduce that ¯ 1 = −e s/2 M 1 1 − N i=1 M 2 i s 2 i on B s , and hence Indeed, straight computations yield 1 = − y/2 1 = 2 and 2 i = − e i /2 1 + y i /4 y 1 , where e i is the vector whose components are e i j = ij , j = 1 N . Thus 1 2 i B s = B s y i /2 1 − y 2 /2 2 1 K dy = 0, because 1 and K are radially symmetric functions. We also have, since e i · e j = 0, 2 i 2 j B s = − 1/4 B s y i y j 1 − y 2 /4 2 1 K dy = 0 when i = j. From 1 = − y/2 1 , we