DOMAIN DEPENDENCE OF SOLUTIONS TO COMPRESSIBLE ...

c 2006 Society for Industrial and Applied Mathematics 

SIAM J. CONTROL OPTIM. Vol. 45, No. 4, pp. 1165–1197

DOMAIN DEPENDENCE OF SOLUTIONS TO COMPRESSIBLE NAVIER–STOKES EQUATIONS∗ P. I. PLOTNIKOV† AND J. SOKOLOWSKI‡ Abstract. The minimization of the drag functional for the stationary, isentropic, compressible Navier–Stokes equations (NSE) in three spatial dimensions is considered. In order to establish the existence of an optimal shape, the general result on compactness of families of generalized solutions to the NSE is established within in the framework of the modern theory of nonlinear PDEs [P. L. Lions, Mathematical Topics in Fluid Dynamics. Vol. 2. Compressible Models, Oxford University Press, Clarendon Press, New York, 1998], [E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, Oxford, UK, 2004]. The family of generalized solutions to the NSE is constructed over a family of admissible domains Uad . Any admissible domain Ω = B \ S contains an obstacle S, e.g., a wing profile. Compactness properties of the family of admissible domains in the form of the condition (HΩ ) is imposed. Roughly speaking, the condition (HΩ ) is satisfied, provided that for any sequence of admissible domains {Ωn } ⊂ Uad there is a subsequence convergent both in Hausdorff metrics and in the sense of Kuratowski and Mosco. The analysis is performed for the adiabatic constant γ > 1 in the pressure law p(ρ) = ργ and it is based on the technique used in [P. I. Plotnikov and J. Sokolowski, Comm. Math. Phys., 258 (2005), pp. 567–608] in the case of discretized NSE. The result is a generalization to the stationary equations with γ > 1 of the results obtained in [E. Feireisl, A. H. Novotn´ y, and H. Petzeltov´ a, Math. Methods Appl. Sci., 25 (2002), pp. 1045–1073], [E. Feireisl, Appl. Math. Optim., 47 (2003), pp. 59–78] for evolution equations within the range γ > 3/2 for the adiabatic ratio. Key words. Navier–Stokes equations, compressible fluids, shape optimization AMS subject classifications. 35Q30, 49J20, 76N10 DOI. 10.1137/050635304

1. Introduction. 1.1. Shape optimization. In the present paper, a class of shape optimization problems for stationary isothermal compressible Navier–Stokes equations (NSE) is considered. Usually the mathematical analysis of such problems includes the following: • Proof of the existence of optimal shapes for a sufficiently large class of admissible domains. • Derivation of necessary optimality conditions, which characterize an optimal domain and can be used for the numerical solution of the shape optimization problem. • The convergence proof for the numerical method, which can be used to evaluate an optimal shape. To the best of our knowledge, only the first point is studied in the literature for the compressible NSE and drag minimization. We refer the reader to [10] for the existence results for shape optimization problems in the case of evolution equations for the adiabatic constant γ > 3/2; see also [9] for related results. In the present paper the case of γ > 1 is considered in three spatial dimensions. The technique used ∗ Received by the editors July 6, 2005; accepted for publication (in revised form) March 20, 2006; published electronically September 12, 2006. This paper was prepared in June 2005 during a visit of Pavel I. Plotnikov to the Institute Elie Cartan of the University Henri Poincar´e Nancy 1. http://www.siam.org/journals/sicon/45-4/63530.html † Lavryentyev Institute of Hydrodynamics, Siberian Division of Russian Academy of Sciences, Lavryentyev pr. 15, Novosibirsk 630090, Russia ([email protected]). ‡ Institut Elie Cartan, Laboratoire de Math´ ematiques, Universit´e Henri Poincar´e Nancy 1, B.P. 239, 54506 Vandoeuvre l´es Nancy Cedex, France ([email protected]).

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is introduced in [12], [24], and [25]. The optimality conditions for drag minimization are derived in [27]. We restrict ourselves to the first issue on the above list of problems to be solved, which is already quite difficult, since there are no existence results for the PDE model itself in the range of parameters we are going to consider, i.e., for the adiabatic constant γ > 1 in the law p(ρ) = ργ , which gives the pressure p of the fluid in the function of its density ρ in the isothermal regime. We also point out that the existence of generalized solutions for such equations with nonhomogeneous boundary conditions for a full range of parameters is an open problem; for some results in this direction we refer to [15], [16], [17] for two spatial dimensions and for special geometry of the flow region. We show that, under appropriate assumptions, the convergence of the sequence of admissible domains implies the convergence of the associated sequence of shape functionals. To make our analysis representative we consider the drag functional, which is the standard choice in view of possible applications in the shape optimization of a wing. A related shape optimization problem is considered in [10] for evolution equations, and with the adiabatic constant γ > 3/2. We consider the case of γ > 1 using the same technique as in [12], [25]. The novelty of our results, in comparison with paper [25], is the PDE model; i.e., we consider here the stationary problem and use a farreaching approach to the problem based on the new kinetic formulation to the mass balance equation. In [25] the discretized problem introduced in [18] is considered; such a problem depends on the parameter α > 0 of the time discretization of evolution equations [18], and the limit case of α = 0 is the stationary problem. The outline of the paper is the following. In section 1.2 an example of a shape optimization problem is presented and the existence result of optimal shapes is given. In section 1.3 the stationary NSE are introduced, and generalized solutions are defined. In section 1.4 the main result of the paper on compactness of generalized solutions and solvability of the drag minimization problem is presented. In section 2 local a priori estimates for generalized solutions are derived. In section 3 weak limits of sequences of generalized solutions are characterized. In section 4 the so-called effective viscous flux is defined and some properties of the viscous flux are obtained. In section 5 the oscillation defect measure is introduced for sequences of densities and its boundedness is proved for generalized solutions. In section 6 the kinetic formulation for the mass balance equation is derived in the form of a transport equation for the distribution function, which characterizes the Young measure associated with weakly convergent sequences of densities. In section 7 the renormalization of the kinetic formulation for the mass balance equation is performed, which shows the strong convergence of sequences of densities. Finally, in section 8 the proof of the main result of the paper is completed. Since all of our presented results for the adiabatic ratio γ > 1 seem to be new, we provide the complete proofs for the convenience of the reader. To our knowledge such proofs are not known in the literature. The proof technique was introduced by Lions and by Feireisl, Novotn´ y, and Petzeltov´ a and it is adapted to the specific problem along the lines of [12], [24], and [25]. 1.2. Example of the existence of an optimal shape. We use the notation introduced below. Let (u, ρ) be a generalized solution to the boundary value problem (1.3) posed in the geometrical domain Ω = B \ S, where B ⊂ R3 is a fixed hold-all domain and S is an obstacle. The family of admissible domains Ω ∈ Uad includes

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domains with obstacles of the volume bounded from below, i.e., meas(S) ≥ Vol for all admissible domains Ω = B \ S. Let us consider the problem of drag minimization, in which the shape functional J(Ω) takes the form J(Ω) := J(ρ, u, Ω), where the functional J(ρ, u, Ω) is defined in (1.5), over such a family of admissible domains, and let us assume that the set of generalized solutions to (1.3) over admissible domains Ω ∈ Uad is nonempty. If the family Uad is compact with respect to the convergence defined by the condition (HΩ ), then there is an optimal domain Ω∗ = B \ S ∗ such that J(Ω∗ ) ≤ J(Ω) for all Ω ∈ Uad . We prove the existence result without any regularity assumptions on the boundaries of obstacles. In fact, we prove a compactness result for the family of solutions to the NSE so that the existence of optimal shapes is a simple corollary of the theorem on the compactness of the set of generalized solutions. 1.3. The problem formulation. Suppose that compressible Newtonian fluid occupies the bounded region Ω ⊂ R3 . We will assume that Ω = B \ S, where B is an open sufficiently large hold all with a sectionally smooth boundary containing a compact obstacle S. We could take, e.g., for B a ball of radius R, B = {x||x| ≤ R}. We do not impose restrictions on the topology of the flow region. The cases of S with a finite number of connected components or S = ∅ are taken into consideration. The fluid density ρ : Ω → R+ and the velocity field u : Ω → R3 are governed by the NSE (1.1)

−νΔu − ξ∇div u + ρu∇u + ∇p(ρ) = ρf,

div (ρu) = 0,

where ν, ξ are positive viscous coefficients and f : Ω → R3 is a given continuous vector field. We suppose that the flow is barotropic and p(ρ) = ργ with the adiabatic constant γ > 1. If the viscous stress tensor is defined by the equality (1.2)

Π = ν(∇u + ∇u ) + (ξ − ν)div u I,

then the governing equations can be written in the equivalent divergence form (1.3a)

div (ρu ⊗ u) + ∇p(ρ) − ρf = div Π,

div (ρu) = 0 in Ω.

In view of possible applications, e.g., to the shape optimization problem of a wing, it is assumed that the velocity field satisfies the nonhomogeneous boundary condition (1.3b)

u = 0 on ∂S,

u = U∞ on ∂B,

and the density distribution is prescribed on the entrance set (1.3c)

ρ = ρ∞ on Σ+ = {x ∈ ∂B : U∞ · n(x) < 0}.

Here n is the outward unit normal vector to ∂Ω. It is assumed that U∞ ∈ R3 is a given vector and ρ∞ is a given nonnegative constant. Boundary condition (1.3b) can be written in the form of the equality u = u∞ on ∂Ω, where u∞ (x) is a smooth function defined for any x ∈ R3 , which vanishes in the vicinity of S and coincides with U∞ in an open neighborhood of ∂B. The physical

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quantities which characterize the flow include the total energy E, the volume rate of energy dissipation D, and the drag J, and are defined by     1 ργ 2 ρ|u| + dx, D = (ν|∇u|2 + ξ|div u|2 ) dx, E= (1.4) 2 γ−1 Ω Ω  ∞ J = −U · (Π − p(ρ)I) · n dS. ∂S

The drag J accounts for the reaction of the surrounding fluid on the obstacle S. For our purposes, the formula for the drag can be written in the equivalent form   ∞ J(ρ, u, Ω) = (1.5) (Π − ρu ⊗ u − p(ρ)I) : ∇u dx + (U∞ − u∞ ) · fρdx. Ω

Ω

We will consider the solutions to problems (1.3) for which the density is nonnegative and the total energies are bounded from above by some positive constant CΩ . In what follows we will denote by c various constants depending only on E, data fC(B) , ρ∞ , u∞ C 1 (B) , material constants γ, ν , ξ, and the domain B. In this paper the standard notation is used for the function spaces. The space H 1,r (Ω) is the Sobolev space of functions integrable along with the first order generalized derivatives in Lr (Ω) equipped with its natural norm. For r = 2 we use the notation H 1,2 (Ω) rather than H 1 (Ω); the notation H01,r (Ω) stands for the closure of C0∞ (Ω) in the norm of H 1,r (Ω). Definition 1.1. For given U∞ ∈ R3 and f ∈ C(Ω)3 , a generalized solution to problem (1.3) is the pair (ρ, u), where ρ ∈ Lγ (Ω) is a nonnegative function in Ω and u − u∞ ∈ H01,2 (Ω), which satisfies the following conditions: (a) The scalar function ρ|u|2 is integrable in Ω, i.e., the total energy E of the flow is finite. The mass density and the velocity field satisfy the energy inequality  ν∇(u − u∞ )2L2 (Ω) + ξdiv (u − u∞ )2L2 (Ω) + ρu ⊗ u : ∇u∞ dx  Ω  1 ∞ ∞ (1.6) + p(ρ)div u dx − ρf · (u − u ) dx + (ρ∞ )γ (U∞ · n) dΓ ≤ 0. γ − 1 Γ+ Ω Ω (b) For all vector fields ϕ ∈ C01 (Ω)3 ,      ρu ⊗ u + p(ρ) : ∇ϕ dx + (1.7a) ρf · ϕ dx = Π : ∇ϕ dx. Ω

Ω

Ω

(c) The integral identity       G(ρ)u · ∇ψ + G(ρ) − G (ρ)ρ ψdiv u dx − (1.7b)

ψG(ρ∞ )U∞ · ndΣ = 0

Σ+

Ω

holds for all functions ψ ∈ C 1 (Ω) vanishing on Σ− = ∂B \ Σ+ , and all functions 1 [0, ∞) with the properties G ∈ Cloc lim sup |G(r)|/r < ∞, r→∞

[0, ∞) r → G(r) − G (r)r ∈ R

continuous and bounded. Condition (c) of the above definition means that we consider the renormalized weak solutions of the stationary problem; see [11] for a discussion. Such a definition simplifies further analysis without any loss of generality.

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Remark 1.1. Denote by Σ− = ∂B \ Σ+ the exit part of the boundary ∂B. It follows from the definition of generalized solutions that the extensions of the density and of the velocity vector field onto R3 , given by the equalities (1.8)

ρ(x) = 0 in S,

ρ(x) = ρ∞ in R3 \ (B ∪ Σ− ),

u = 0 in S,

u = U∞ in R3 \ B

satisfy the integral identity      G(ρ)u · ∇ψ + G(ρ) − G (ρ)ρ ψdiv u dx = 0 (1.9) R3

for any functions ψ ∈ C01 (R3 ) vanishing near Σ− . 1.4. Preliminaries and main results. The cost functional for shape optimization problems is the drag J(ρ, u, Ω) defined by formula (1.5). In applications, the drag is usually minimized within the class of admissible shapes. To the best of our knowledge there are no results on the shape optimization problem in the framework of generalized solutions to stationary problems for the adiabatic constant γ > 1; the case of evolution equations for the adiabatic constant γ > 3/2 is considered in [10]. The drag depends on the solution (ρ, u) to problem (1.3); however, such a solution, if it does exist, is not in general unique. We point out that the existence of solutions for the adiabatic constant γ ≥ 1 in three spatial dimensions is in general an open and difficult problem [19]. The case of discretized problems is considered in [25] for γ ≥ 1; however, no dependence of solutions on geometrical domains is considered in [25]. On the other hand, the case of drag minimization in two spatial dimensions is studied in [24]. Furthermore, the drag depends on an admissible shape of the obstacle S. The dependence of the drag on the admissible shapes is twofold: first, it depends directly on Ω since the integrals in (1.5) are defined over Ω, and second, it depends on the generalized solutions to the NSE defined in Ω. The restrictions on the shapes of admissible obstacles S are defined in such a way that the set of admissible shapes and the set of the associated generalized solutions are compact. The precise conditions for admissible shapes are established below in the form of condition (HΩ ). In the present paper we do not derive the necessary optimality conditions for the problem of drag minimization; we prove only the compactness of the set of solutions over the set of admissible shapes, which leads to the existence of an optimal shape, provided the set of generalized solutions is nonempty. The optimality conditions for the drag minimization problem will be given in forthcoming paper [27]. We are now in a position to formulate the main result of the paper. Suppose that a sequence of flow domains Ωn = B \ Sn satisfies the following three conditions, which we refer to collectively as condition (HΩ ). Condition (HΩ ). (a) There is a compact KS  B such that ∪n Sn ⊂ KS . (b) If a compact set K  Ω, then K  Ωn for all large n. (c) If wn → w weakly in H 1,2 (B) and wn ∈ H01,2 (Ωn ), then w ∈ H01,2 (Ω). In this case we will write H

Ωn → Ω. It is easily seen that the convergence of the sequence Ωn both in the Hausdorff metrics and in the sense of Kuratowski and Mosco implies all three conditions listed in (HΩ ).

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We deal with the complicated system of nonlinear PDEs. Therefore, we need stronger conditions on the convergence of geometrical domains, compared to the classical Kuratowski–Mosco convergence [14], which is adapted to the linear elliptic case. In particular, our conditions are more restrictive when compared to the Kuratowski– Mosco convergence of domains Ωn . The first main result of the paper is the following compactness theorem. Theorem 1.2. Assume that there are a sequence (Ωn ) of domains which conH verges Ωn → Ω and a sequence of generalized solutions {ρn , un } to the compressible NSE (1.10a)

div (ρn un ⊗un ) + ∇p(ρn ) − ρf = div Π,

div (ρn un ) = 0 in Ωn ,

satisfying the boundary conditions (1.10b)

un = 0 on ∂Sn ,

un = U∞ on ∂B,

ρn = ρ∞ on Σ+ .

Suppose also that the total energies of the sequence (ρn , un ) of generalized solutions to problem (1.10) are uniformly bounded by a constant cΩ ,    1 1 2 γ En = (1.11) ρ|un | + ρ dx ≤ cΩ . 2 γ−1 n Ωn Then there is a subsequence of the sequence {ρn , un }, still denoted by {ρn , un }, such that for any r < γ, ρn → ρ in Lr (B),

p(ρn ) → p(ρ) in L1loc (Ω),

un → u weakly in H 1,2 (B).

The pair of functions (ρ, u) serves as a generalized solution to problem (1.3) for the limit domain Ω; furthermore, the shape functionals converge for n → ∞, (1.12)

J(ρn , un , Ωn ) → J(ρ, u, Ω).

1.5. Shape optimization. We provide an existence result for the drag minimization. We also list, for the convenience of the reader, the related recent results on the compactness of the solutions to compressible NSE. In order to compare our results with the recent results given in the literature on the subject, we note that for fixed S and γ > 5/4, compactness results were obtained in [12]. In the general case, the compactness of solutions to boundary value problems for NSE with γ > 3/2 was proved in [10] under the assumption that ∂Sn satisfy the “uniform thickness” condition. Theorem 1.2 leads to the following result on the solvability of the shape optimization problem, which is the second main result of the paper. In order to formulate the result we introduce some notation. Choose an arbitrary continuous function h : R3 → R+ which is positive with a possible exception of a null capacity set. Choose also a positive constant V and a compact K  B with meas K ≥ V . Denote by O(h, K, V ) the family of all compact obstacles S ⊂ K with meas K ≥ V satisfying the following conditions. Condition S. For each x ∈ S, are a± (x) ∈ R3 so that S contains the plane there − triangle x + Δ(x) with Δ(x) = sa + ta+ : s, t ≥ 0, s + t ≤ 1 and

(1.13) min |a± (x)|, |a+ (x) × a− (x)| ≥ h(x).

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  In particular, the angle ϑ(x) = arcsin |a+ (x) × a− (x)||a− (x)|−1 |a+ (x)|−1 satisfies the inequalities 0 < ϑ(x) < π quasi everywhere (q.e.) in S. This condition is fulfilled, for example, if an obstacle admits a finite partition, such that every component of the partition is starlike with respect to two distinct points. For positive C and compact S  B, denote by E(C, S) the family of generalized solutions to problem (1.3) in domain Ω = B \ S with the total energies bounded from above by a constant, E ≤ C. The following theorem gives sufficient conditions for the solvability of the drag minimization problem in the class O(h, K, V ). Theorem 1.3. Suppose that there exist a compact S ∈ O(h, K, V ) and a generalized solution {ρ, u} ∈ E(C, S) such that J(ρ, u, B \ S) < ∞. Then there are S ∗ ∈ O(h, K, V ) and {ρ∗ , u∗ } ∈ E(C, S ∗ ) such that (1.14)

J(ρ∗ , u∗ , B \ S ∗ ) =

inf

S∈O(h,K,V )

inf

{ρ,u}∈E(C,S)

J(ρ, u, B \ S).

Proof. Choose a minimizing sequences Sn , (ρn , un ) so that J(ρn , un , B \ Sn ) →

inf

S∈O(h,K,V )

inf

{ρ,u}∈E(C,S)

J(ρ, u, B \ S) as n → ∞.

Without any loss of generality, we can assume that the sequence Sn converges in the Hausdorff metric to some compact set S ∗ ∈ O(h, K, V ). By Theorem 1.2, it suffices to prove that H

Ωn = B \ Sn −→ B \ S ∗ = Ω∗ . Note that conditions (a) and (b) in (HΩ ) follow from the definition of the class O(h, K, V ) and from the convergence of the sequence Sn in the Hausdorff metric. In order to prove (c) in (HΩ ), let us consider an arbitrary sequence wn ∈ H01,2 (B) such that wn → w weakly in H 1,2 (B) and wn ∈ H01,2 (Ωn ). Without any loss of generality, we can assume that wn vanishes on R3 \ Ωn and w is quasi continuous in R3 . It is necessary to prove that w ∈ H01,2 (Ω). By the Hedberg approximation theorem (see [13]), it is sufficient to show that w = 0 q.e. on S ∗ . To this end, we choose an arbitrary x∗ ∈ S ∗ with h(x∗ ) > 0 and note that x∗ = limn→∞ xn ∈ Sn . After passing to a subsequence we can assume that the sequence a± (xn ) converges to vectors a± (x∗ ) satisfying (1.13). Denote by On : R3 → R3 the orthogonal linear mapping, which maps span{a± (xn )} onto the coordinate plane Π = {x · e3 = 0}, and set Wn (y) = wn (xn + On−1 y),

b± = Oa± (xn ).

Since On → O∗ , the functions Wn converge weakly in H 1,2 (R3 ) and strongly in L2 (Π) to W (y) = w(x∗ + O∗ −1 y). Noting that Wn vanishes on the triangle On Δ(xn ) ⊂ Π we can conclude from this that W = 0 on O∗ Δ(x∗ ), and hence w = 0 on x∗ + Δ(x∗ ). Since the capacity of the plane disk of radius r is equal to 2r/π 2 , the inequality cap Δ(x∗ ) ∩ B(x∗ , r) ≥ rϑ(x∗ )/π 3 holds true for all r < h(x∗ ), and hence lim inf r→0

cap Δ(x∗ ) ∩ B(0, r) ≥ c(x∗ ) > 0. cap B(0, r)

From this and the above quasi-continuity property of w we can deduce w(x∗ ) = 0. Therefore, w = 0 q.e. in S and the proof of the theorem is completed.

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Remark 1.2. The minimal value of the drag in the right-hand side of (1.14) depends on V and a choice of a compact K and energy bound C. It increases in V and decreases in C and K. Remark 1.3. A typical obstacle satisfying the conditions of Theorem 1.3 is a union of starlike bodies endowed with a system of plane wings. The case of curvilinear wings also can be included if we replace Δ(x) with g(Δ(x)), where g : R3 → R3 is an arbitrary bi-Lipschitz mapping with g(0) = 0. To the best of our knowledge, there are no global results on the solvability of spatial stationary problems for compressible NSE with nonhomogeneous boundary conditions even for large γ. In this connection we note that there is a significant difference between the question on existence of solutions to boundary value problem (1.3) and the question on solvability of shape optimization problem (1.14). Thus, due to Remark 1.2, the conditions of Theorem 1.3 will be satisfied if there exists at least one obstacle S for which problem (1.3) has a finite energy solution. Such a solution can be constructed in the case when both B and S have a spherical symmetry or in the case when ∂S is a smooth surface sufficiently close to ∂B. This condition is also satisfied for sufficiently small data, since in such a case the existence of local solutions is proved at least for the homogeneous boundary conditions. We refer also to [15], [16], [17], [21] for the local existence results in two spatial dimensions with nonhomogeneous boundary conditions. The remaining part of the paper is devoted to the proof of Theorem 1.2. We begin with the derivation of certain local a priori estimates for the generalized solutions to problem (1.3). 2. A priori estimates. In this section we prove the following theorem on local integrability of generalized solutions; cf. [12], [25], [26]. Theorem 2.1. Let (ρ, u) be a generalized solution to problem (1.3), and let Ω be a subdomain of Ω with dist (Ω , ∂Ω) > d > 0. Then for κ = 2(γ − 1)/(γ + 2) > 0, ρu2 L1+κ (Ω ) ≤ cd−1 ,

(2.1)

ρLγ(1+κ) (Ω ) ≤ cN (Ω ),

where the constant N depends only on Ω . The proof of Theorem 2.1 is divided into a sequence of lemmas. The first lemma gives the estimate of the rate of energy dissipation in terms of the total energy of the fluid. Lemma 2.2. Under the assumptions of Theorem 2.1, the velocity vector field is bounded, uH 1,2 (Ω) ≤ c,

(2.2)

where the constant c depends only on the data of the boundary value problem. Proof. The proof follows from condition (a) in the definition of generalized solutions to problem (1.3) and from the obvious inequalities 

u||H 1,2 (Ω) ≤ c∇(u − u∞ )L2 (Ω) + u∞ C 1 (Ω) , ρ|f · (u − u∞ )| dx ≤ c(γ, B)f C(Ω) (E + 1)(u∞ C 1 (Ω) + 1).

Ω

Choose a domain Ω0 with a smooth boundary so that Ω  Ω0  Ω,

dist (∂Ω, Ω0 ) ≥ d/3,

dist (∂Ω0 , Ω )) ≥ d/3 .

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The second lemma shows that the Newtonian potential of the pressure is uniformly bounded on Ω0 . Lemma 2.3. Under the assumptions of Theorem 2.1,  ργ (y) dy ≤ cd−1 . ess sup (2.3) x∈Ω0 Ω |x − y| Proof. Introduce the symmetric, nonnegative, matrix-valued function E = (Ei,j )3×3 ,

Ei,j = ρui uj + p(ρ)δi,j ,

Tr E = ρ|u|2 + 3p(ρ),

and rewrite integral identity (1.7a) in the form   (2.4) (E − Π) : ∇ϕ = ρf · ϕ dy for all ϕ ∈ H01,2 (Ω)3 . Ω

Ω

Fix an arbitrary x ∈ Ω0 and set  M0 (s) = n ⊗ n : E(y) dy,

 M1 (s) =

B(x,s)

Tr E(y) dy, B(x,s)

where n = |y − x|−1 (y − x) . Next choose positive r < R ≤ d/3, δ ∈ [0, R − r), s ∈ [r, R − δ), and a sequence of continuous functions hδ (t) with hδ (t) = 1 for t ≤ s, hδ (s) = 0 for t ≥ s+δ, and hk (t) = −1/δ for t ∈ (s, s + δ). Substituting the vector function ϕ(y) = hδ (|y − x|)(y − x) into (2.4) and taking into account that the functions Mj (s), j = 0, 1, are absolutely continuous on the interval [r, R], followed by the limit passage δ → 0, leads to     Tr Π dy − s M1 (s) − sM0 (s) = n ⊗ n : Π dS − ρ(y − x) · f(y)dy B(x,s)

∂B(x,s)

B(x,s)

for almost everywhere (a.e.) s ∈ (0, R]. Multiplying both sides of this equality by s−2 and integrating the result over the interval (r, R], we find that     1 M1 (s) M0 (s) ds = (2.5) − Tr Π dy s2 s r B(x,r) (r,R]   1 − Tr Π dy + SΠ(x, r, R) − kr,R (y − x)f(y)ρ(y) dy R B(x,R) B(y,R) with kr,R (x) = min{r−1 , |x|−1 } − R−1 ,  1 SΠ(x, r, R) = (I − n ⊗ n) : Π(y) dy. |x − y| r 3. Recall (see [11]) that for a given function q ∈ C0∞ (Ω) with  q dx = 0, [q] := meas (Ω0 )−1 Ω0

there exists the vector field ϕ ∈ C ∞ (Ω0 )3 such that div ϕ = q in Ω0 ,

(2.15)

ϕ = 0 at ∂Ω0 ,

ϕH 1,s (Ω0 ) ≤ N (Ω0 )qLs (Ω0 ) .

Thus, we get 

 η(ρu ⊗ u − Π) : ∇ϕ dx +

qηp(ρ) dx = Ω0



Ω0

ηϕΦ dx. Ω0

Since the embedding H01,s (Ω0 ) → C(Ω0 ) is bounded, we have |ϕ| ≤ cϕH 1,s (Ω0 ) ≤ N (Ω0 )qLs (Ω0 ) , which yields    

Ω0

  qηp(ρ) dx ≤ N (Ω0 )(ρ|u|2 L1+κ (Ω0 ) + ΠL1+κ (Ω0 ) )ϕH 1,s (Ω0 ) + ΦL1 (Ω) ϕC(Ω) ≤ cN (Ω0 )d−1 qLs (Ω0 ) .

It follows from this that ηp(ρ) − [ηp(ρ)]Ls/(s−1) (Ω0 ) ≤ cN (Ω0 ). Note that s/(s − 1) = 1 + κ, η = 1 on Ω , and [ηp(ρ)] ≤ c. Hence pL1+κ (Ω ) ≤ cN (Ω )d−1 , and the proof of Theorem 2.1 is completed. 3. Weak convergence. Since the notion of weak limits plays a crucial role in our analysis, we begin with a short description of some basic facts concerning weak convergence and weak compactness. We refer the reader to, e.g., [5] for proofs of basic results. Let A be an arbitrary bounded, measurable subset of R3 and 1 < r ≤ ∞. Then for every bounded sequence {gn }n≥1 ⊂ Lr (A), there exist a subsequence, still denoted by {gn }, and a function g ∈ Lr (A) such that for n → ∞,   gn (x)h(x)dx → g(x)h(x)dx for all h ∈ Lr/(r−1) (A). A

A

We say the sequence converges gn → g weakly in Lr (A) for r < ∞ and converges star-weakly in L∞ (A) in the limit case of r = ∞. In the very special case of r = 1, it is known that the sequence of gn contains a weakly convergent subsequence in L1 (A) if and only if there is a continuous function Φ : R → R+ such that lim Φ(s)/s = ∞ and

s→∞

sup Φ(gn )L1 (A) < ∞. n≥1

DOMAIN DEPENDENCE FOR NAVIER–STOKES EQUATIONS

1177

If the sequence of gn is only bounded in L1 (A) and A is open, then after passing to a subsequence we can assume that gn converges star-weakly to a bounded Radon measure measure μg , i.e.,   gn (x)h(x)dx = h(x)dμg (x) for all compactly supported h ∈ C(A). lim n→∞

A

A

In what follows, the linear space of compactly supported functions on a set A is denoted by C0 (A), and its dual is denoted by C0 (A)∗ . Ball’s version [3] (see also [20], [23]) of the fundamental Tartar theorem on Young measures gives a simple and effective representation of weak limits in the form of integrals over families of probability measures. The following lemma is a consequence of Ball’s theorem. Lemma 3.1. Suppose that a sequence {gn }n≥1 is bounded in Lr (A), 1 < r ≤ ∞, where A is an open, bounded subset of R3 . Then we have the following characterizations of weak limits. (i) There exists a subsequence, still denoted by {gn }n≥1 , and a family of probability measures σx ∈ C0 (R)∗ , x ∈ A, with a measurable distribution function Γ(x, λ) := σx (−∞, λ] so that the function λ → Γ(x, λ) is monotone and continuous from the right and admits the limits 1, 0 for λ → ±∞. Furthermore, for any continuous function G : A × R → R such that lim G(·, λ)C(A) /|λ|r = 0 for r < ∞ and sup G(·, λ)C(A) < ∞ for r = ∞,

|λ|→∞

|λ|

the sequence of G(·, gn ) converges weakly in L1 (A) to a function  (3.1) G(x) = G(x, λ)dλ Γ(x, λ). R

Moreover, the function  A x→

R

|λ|r dλ Γ(x, λ) ∈ R

belongs to L1 (A). (ii) If G(x, ·) is convex and the sequence gn converges weakly (star-weakly for r = ∞) to g ∈ Lr (A), then G(x) ≤ G(x, g(x)). If the functions gn satisfy the inequalities gn ≤ M (resp., gn ≥ m), then Γ(x, λ) = 1 for λ ≥ M (resp., Γ(x, λ) = 0 for λ < m). (iii) If Γ(1−Γ) = 0 a.e. in A, then the sequence gn converges to g in measure, and hence in Ls (A), for positive s < r. Moreover, in this case Γ(x, λ) = 0 for λ < g(x) and Γ(x, λ) = 1 for λ ≥ g(x) Let us consider the sequence of generalized solutions {(ρn , un )}n≥1 to problem (1.3) satisfying all assumptions of Theorem 1.2. Assume that the functions (ρn , un ) are extended onto R3 by formulae (1.8), and fix an arbitrary bounded smooth domain D with B  D. For such extended functions, by inequalities (1.11) and formulae (1.8), it follows that the sequence (ρn , un ) contains a subsequence, still denoted by (ρn , un ), such that (3.2)

ρn → ρ weakly in Lγ (D),

un → u weakly in H 1,2 (D),

ρn → ρ weakly in L(1+κ)γ (Ω ) for all Ω  Ω.

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P. I. PLOTNIKOV AND J. SOKOLOWSKI

The behavior of the functions p(ρn ) is more complicated. Since they are uniformly bounded in L1 (D), we can assume, after passing to a subsequence if necessary, that p(ρn ) converge weakly to some finite Borel measure μp on D. On the other hand, the sequence {p(ρn )} is bounded in L1+κ (K) for any compact K  Ω. Using the diagonal process, we obtain the existence of a subsequence which converges weakly on each compact K  Ω to some function p ∈ L1+κ loc (Ω). Since pL1 (K) ≤ lim inf p(ρn )L1 (K) ≤ c n→∞

the function p is integrable over Ω. Next, we note that the functions p(ρn ) = (ρ∞ )γ are bounded and independent of n on D\B. Hence the extended function p(x), x ∈ D, defined as p(x) = p(x) in Ω,

p(x) = (ρ∞ )γ in D \ B,

∞ belongs to the class L1 (D \ S) ∩ L1+κ loc (Ω) ∩ L (D \ B), which implies the equality     h(x)dμp = h(x)p(x)dx + h(x)dμp + h(x)dμp for all h ∈ C0 (R3 ), D

D\S

∂B

S

where the compact obstacle takes the form S = B \ Ω. Applying Lemma 3.2 to the sequence of gn := ρn and to the sets A = D, Ω leads to the following result on the representation of weak limits of a sequence of generalized solutions to problem (1.3). Lemma 3.2. There exists a subsequence of the sequence {ρn , un }, still denoted by {ρn , un }, and a distribution function Γ : D × R → [0, 1] such that (i) Γ(x, λ) meets all requirements of Lemma 3.1 and satisfies the equalities Γ(x, λ) = 0 for λ < 0 a.e. in D, Γ(x, λ) = 0 for λ < ρ∞ , Γ(x, λ) = 1 for λ ≥ ρ∞ a.e. in D \ B . (ii) for any continuous function G : D×R such that limρ→∞ ρ−γ G(·, ρ)C(D) = 0, the sequence G(·, ρn ) converges weakly in L1 (D) to the function  G(x) = (3.3) G(x, λ) dλ Γ(x, λ) a.e. in D. [0,∞)

In particular, the weak limit of ρn takes the form   (3.4) λ dλ Γ(x, λ) ≡ (1 − Γ(x, λ)) dλ a.e. in D. ρ(x) = [0,∞)

[0,∞)

(iii) the function p admits the representation   (3.5) p(x) = λγ dλ Γ(x, λ) ≡ γ λγ−1 (1 − Γ(x, λ)) dλ a.e. in D \ S . [0,∞)

[0,∞)

Since the embedding H01,2 (D) → Lr (D) is compact for r < 6, we can expect that ρn un converge weakly to ρu. The corresponding result is given by the following lemma. Lemma 3.3. For ι = (γ − 1)/(γ + 1) > 0 and for any Ω  Ω, ρn un → ρu converges weakly in L1+ι (D)3 , ρn un ⊗ un → ρu ⊗ u weakly in L1+κ (Ω )9 .

DOMAIN DEPENDENCE FOR NAVIER–STOKES EQUATIONS

1179

Proof. Since ρn and un are bounded and independent of n in the exterior of B, it suffices to show that ρn un → ρu converges weakly in L1+ι (B). We have 1+ι

1+ι

|ρn un |1+ι = (ρn ) 2 (ρn |un |) 2 1+γ  (1+γ)/γ  ≤ ρ(1+ι)/2 + (ρn |un |2 )(1+ι)/2 = ργn + ρn |un |2 , n which implies the estimate   1+ι |ρn un | dx ≤ (ργ + ρn |un |2 )dx ≤ cEn ≤ c. B

B

Hence, it suffices to prove that ρn un converges to ρu in D , i.e., in the sense of distributions. For any positive N , set ρN n = min{ρn , N }. Next note that the functions ∞ N ρN n converge star-weakly in L (B) to some function ρ . It is easily seen that the N sequence of nonnegative functions ρ − ρ decreases for N → ∞. Moreover, since 

 |ρn −

(3.6)

ρN n | dx

≤

B

1/γ ργn

dx

meas {x : ρ(x)γ ≥ N γ }(γ−1)/γ ≤ cN −(γ−1) ,

B

we can pass to the limit for N → ∞,   (3.7) (ρ − ρN )dx = lim (ρn − ρN n ) dx → 0. n→∞

B

B

N Since un converge strongly in Lr (B), for r < 6, the sequence ρN n un converges to ρ u  ∞ 3 in D (B). It follows from this that for any h ∈ C0 (B) ,      (3.8) lim sup  (ρn un − ρu)h dx n→∞ B         N N    ≤ lim sup  (ρn − ρn )un h dx +  (ρ − ρ)uh dx . n→∞

B

B

By the Cauchy–Schwarz inequality we have     N   (3.9)  (ρn − ρn )un h dx B   (γ−1)/2 N −(γ−1)/2 ≤ hC(B) N |ρn − ρn | dx + hC(B) N ρn |un |2 dx. B

B

Combining (3.6), (3.8), and (3.9) leads to the inequality (3.10)

     lim sup  (ρn un − ρu)h dx ≤ chC(B) N −(γ−1)/2 + hC(B) (ρ − ρN )|u| dx. n→∞

B

B

It follows from (3.7) that the sequence of nonnegative integrable functions (ρ − ρN )|u| decreases and converges to 0 a.e. in B for N → ∞. From this, and by the Fatou theorem, we conclude that the right-hand side of (3.10) tends to 0 as N → ∞, which yields the weak convergence of the sequence ρn un .

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P. I. PLOTNIKOV AND J. SOKOLOWSKI

It remains to prove the weak convergence of the sequence ρn un ⊗ un . We begin with the observation that since this sequence is bounded in L1+κ (Ω ), it suffices to show that it converges in D (Ω ). Arguing as before, we obtain      (3.11) lim sup  (ρn un ⊗ un − ρu ⊗ u) : h dx n→∞ B      N  ≤ lim sup  (ρn − ρn )un ⊗ un : h dx + (ρ − ρN )|u|2 |h| dx n→∞

B

B

for all h ∈ C0∞ (Ω )9 . Using the Young inequality and taking into account that ρn ≥ ρN n , we obtain (γ−1)/2 −κ(γ−1)/2 (ρn − ρN ρn |un |2(1+κ) |(ρn − ρN n )un ⊗ un | ≤ N n)+N   −κ(γ−1)/2 |un |6 + (ρn |un |2 )(1+κ) . ≤ N (γ−1)/2 (ρn − ρN n)+N

Since, by Lemma 2.2, un L6 (Ω ) ≤ cun H 1,2 (B) ≤ c, we conclude from this and inequality (3.6) that      − ρ )u ⊗ u : h dx lim sup  (ρN n n n n  n→∞ B

≤ chC(B) (N −(γ−1)/2 + N −κ(γ−1)/2 ) → 0 for N → ∞. It is sufficient to observe that, by (3.7) and the Fatou theorem, the last term in the right-hand side of (3.11) tends to 0 for n → ∞. The proof of Lemma 3.3 is completed. 4. The effective viscous flux. Following [18], we introduce the quantity V (ρ, u) = p(ρ) − (ξ + ν)div u, which is called the effective viscous flux. As was shown in [18], [6], [7], [8], the effective viscous flux enjoys many remarkable properties. The most important is the multiplicative relation ϕ(ρ)V = ϕ(ρ) V for weak limits, which was proved in [18] for all γ > 3/2 . The simple proof of this result, based on the new version of the compensated compactness principle, was given in papers [6], [8]. In our case, by Theorem 2.1, the critical estimate ρn |un |2 L(1+κ) (Ω ) ≤ c(Ω ) holds for every Ω  Ω, which leads to the following local version of the compensated compactness result from [8]; see also [11], [22]. Lemma 4.1. Let there be given function Φ ∈ C ∞ (Ω) so that Φ(x, λ) vanishes near ∂Ω and ∂λ Φ(x, λ) = 0 for sufficiently large λ. Then   (4.1) ΦV (ρ, u)dx = Φ V dx, where V = p − (2 + ν)div u. Ω

Ω

Proof. It suffice to prove the lemma for Φ in the form Φ = h(x)ϕ(λ) with h ∈ C0∞ (Ω), and ϕ ∈ C ∞ (R) so that ϕ (λ) = 0 for all large λ. Denote by 1Ω the extension operator, for functions defined in Ω, such that 1Ω u = u in Ω and 1Ω u = 0 in Ωc = R3 \Ω. The adjoint operator 1∗Ω assigns to every function its restriction to Ω. Introduce the linear operators Ai = 1∗Ω Δ−1 ∂xi 1Ω ,

Rij = 1∗Ω ∂xi Δ−1 ∂xj 1Ω ,

1 ≤ i,

j ≤ 3.

From the classical potential theory, it follows that they admit the following integral representations (the integrals for Rij , Rii are determined in the sense of Cauchy’s

DOMAIN DEPENDENCE FOR NAVIER–STOKES EQUATIONS

1181

principal value):  xi − yi 1 u(y) dy, Ai u(x) = 4π Ω |x − y|3  (xi − yi )(xj − yj ) 3 u(y) dy for i = j, Rij u(x) = − 4π Ω |x − y|5  |x − y|2 − 3|xi − yi |2 1 1 u(y) dy + u(x). Rii u(x) = 5 4π Ω |x − y| 3 We denote by A the vectorial operator with the entries Ai and by R the matrix operator with the entries Rij , i, j = 1, 2, 3. Recall that the operators Rij : Lr (Ω) → H

Lr (Ω), Ai : Lr (Ω) → H 1,r (Ω) are bounded for every r > 1. Since Ωn → Ω , there is n0 such that spt h  Ωn for all n ≥ n0 . Multiplying the moment balance equation for (ρn , un ), n ≥ n0 , by h we arrive at   div hρn un ⊗ un − Π(hun ) + hp(ρn )I + νun ⊗ ∇h + ν∇h ⊗ un + (ξ − ν)∇h · un I   − ρn un ⊗ un − Π(un ) + p(ρn )I ∇h = hρn f in D (Ω). Next, apply to both sides of this identity the operator A to obtain (4.2)

R : {hρn un ⊗ un − Π(hun ) + hp(ρn )I + 2νun ⊗ ∇h + (ξ − ν)∇h · un I}    = A · (hρn f) + A · ρn un ⊗ un − Π(un ) + p(ρn )I ∇h in L1 (Ω).

Since h is compactly supported in Ω, we have (4.3)

R : {−Π(hun ) + hp(ρn )I} = hV (ρn , un ) + (ξ + ν)∇h · un , R : ((∇h · un )I) = ∇h · un .

Multiplying both sides of (4.2) by ϕn = ϕ(ρn ), integrating the result over Ω, and using in addition relations (4.3), we obtain     (4.4) hϕn V (ρn , un )dx + ϕn Pn + R : (hρn un ⊗ un ) dx = 0, Ω Ω

Pn = 2ξ∇h · un + 2νR : (un ⊗ ∇h) − A · (ρn un ⊗ un − Π(un ) + pn I)∇h + ρn hf . On the other hand, multiplying both sides of the renormalized mass balance equation for (ρn , un ) by h and setting G = ϕ, we get   (4.5) div (hϕn un ) + h ϕn (ρn )ρn − ϕn div un − ϕn ∇h · un = 0. Introduce the vector field (m) v(m) = ρ(m) = min{m, ρn }, m ≥ 1. n n un , where ρn (m)

Applying the operator vn · A to both sides of (4.5) and integrating the result over Ω, we arrive at the equality  (m)

vn Pn + v(m) (4.6) n R(hϕn un ) dx = 0, Ω

1182

P. I. PLOTNIKOV AND J. SOKOLOWSKI

where we denote

  Pn = A · h(ϕ (ρn )ρn − ϕn )div un − ϕn ∇h · un .

Combining (4.4), (4.6), and the equality    v(m) n R(hϕn un ) − ϕn R : (hρn un ⊗ un ) dx Ω     (m) (m) = vn,i Rij (hϕn un,j ) − ϕn Rij (hvni un,j ) dx+ ϕn R : {(ρ(m) − ρn )hun ⊗ un }dx n Ω Ω     (m) (m) = hun,j ϕn Rij vni − vn,i Rij ϕn dx + ϕn R : {(ρ(m) − ρn )hun ⊗ un }dx, n Ω

Ω

we obtain the equality     (m) v(m) (4.7) hϕn V (ρn , un )dx = n (Pn − ϕn Pn + Rn dx + n , Ω

Ω (m)

in which components of the vector Rn and the scalar n are defined by   (m) (m) Rn,i = hun,j ϕn Rij vn,i − vn,i Rij ϕn , (4.8)  (m) n = (4.9) ϕn R : {(ρ(m) − ρn )hun ⊗ un }dx, n Ω

respectively. Recall that ρn un ⊗ un ∇h → ρu ⊗ u∇h and pn ∇h → p∇h weakly in L1+κ (Ω) for n → ∞. Hence, there are limits for n → ∞, Pn → P in Lr (Ω) for some r > 1,

(4.10) where

  P = 2ξ∇hu + 2νR : (u ⊗ ∇h) − A · { ρu ⊗ u − Π(u) + pI ∇h + hρf},

furthermore, (4.11)

Pn → P ≡ A · {h(ϕ ρ − ϕ)div u − ϕ∇h · u} in L2 (Ω) .

Since the sequences ϕn and vn are bounded in L∞ (Ω) and L6 (Ω), respectively, it follows from the compensated compactness Lemma from [8] that (m)

(m)

(m)

ϕn Rij vn,i − vn,i Rij ϕn → ϕRij v(m) i − v(m) i Rij ϕ weakly in L2 (Ω). Therefore, Rn converges weakly in L3/2 (Ω) to R = {uj (ϕRij v(m) i − v(m) i Rij ϕ)}. Passing to the limit in (4.7) and using (4.10)–(4.11), we obtain   (4.12) hϕV dx = (vP − ϕP + R)dx + (m) Ω

Ω

(m) lim sup |n |.

with |(m) | ≤ On the other hand, passage to the limit in equalities (4.2) and (4.5) gives   div hρu ⊗ u − Π(hu) + hpI + νu ⊗ ∇h + ν∇h ⊗ u + (ξ − ν)∇h · u   − ρu ⊗ u − Π(u) + pI ∇h = hρf, (4.13) (4.14)

div (hϕu) + h(ϕ ρ − ϕ)div u − ϕ∇h · u = 0.

1183

DOMAIN DEPENDENCE FOR NAVIER–STOKES EQUATIONS

Applying the operators ϕA and v · A, respectively, to both sides of the moment and the renormalized mass balance equations and arguing as before, we obtain     (4.15) hϕV dx = v(m) P − ϕP + R dx + (m) , Ω Ω  (m) where  = ϕR : [h(v − ρu) ⊗ u]dx. Ω

Combining (4.12) and (4.15), we finally obtain   (4.16) hϕV dx − hϕV dx = (m) − (m) . Ω

Ω

By Lemma 3.3 the sequence hρn un ⊗ un converges weakly in L1+κ (Ω) to hρu ⊗ u; (m) (m) obviously ρn un = vn converges weakly in L6 (Ω) to v(m) . From this we conclude that |(m) | ≤ R : [h(v(m) − ρu) ⊗ u]L1 (Ω) ≤ ch(v(m) − ρu) ⊗ uL1+κ/2 (Ω) ≤ c lim sup h(vn − ρn un ) ⊗ un L1+κ/2 (spt h) . n→∞

(m)

Similar arguments can be used to obtain the same bound for lim supn→∞ |n |. Since the sequence ρn un ⊗ un is bounded in L1+κ loc (Ω) and the sequence ρn is bounded in L1 (Ω), we have h(v(m) − ρn un ) ⊗ un L1+κ/2 (spt h) ≤ c(h)(ρ(m) − ρn )un ⊗ un L1+κ/2 (spt h) n n  2/(2+κ) ≤c (ρn |un |2 )1+κ/2 dx {ρn >m}∩spt h

≤ cρn |un |2 L1+κ (spt h) mes {ρn > m}β ≤ cm−β , where β = κ(1 + κ)−1 (2 + κ)−1 > 0. Hence |(m) | + |(m) | → 0 for m → ∞. It remains to note that the left-hand side of (4.16) does not depend on n, and the proof of Lemma 4.1 is completed. Corollary 4.2. Assume that the function λ → ϕ(λ) belongs to the class C ∞ (R) and vanishes for sufficiently large λ. Let ϕp ∈ L∞ (D) be an L∞ -star-weak limit of the sequence {ϕ(ρn )p(ρn )}, ϕdiv u ∈ L2 (D) be L2 - weak limit of the sequence {ϕ(ρn )div un }. Then (4.17)

1 1 ϕp − ϕdiv u = ϕp − ϕdiv u in D \ S, ξ+ν ξ+ν

where ϕ and p are given by Lemma 3.2. Proof. Recall that D \ S = (D \ B) ∪ Ω. Since the characteristic function of any compact subset of Ω is a pointwise limit of a sequence of smooth functions compactly supported in Ω, relation (4.17) holds true on each compact subset of Ω. It remains to note that both sides of (4.17) belong to L1 (D \ S) and vanish outside of B. 5. The oscillation defect measure. The notion of the oscillation defect measure was introduced in [6] to justify the existence theory for isentropic flows with small values of the adiabatic constant γ. Following [6], [11] the r-oscillation defect measure associated with the sequence {ρn }n≤1 is defined as oscr [ρn → ρ ](K) := sup lim sup Tk (ρn ) − Tk (ρ)rLr (K) , k≥1 n→∞

1184

P. I. PLOTNIKOV AND J. SOKOLOWSKI

where Tk (z) = kT (z/k), with T (z) a smooth concave function, is equal to z for z ≤ 1 and is a constant for z ≥ 3. The smoothness properties of Tk are not important and we can take the simplest form Tk (z) = min{z, k}. Note that the total energy estimates provide the boundedness of the γ-oscillation defect measure on the whole domain D. The unexpected result was obtained by Feireisl [6] and Feireisl, Novotn´ y, and Petzeltov´ a [8], who showed that the (1 + γ)-oscillation defect measure associated with the sequence {ρn } is uniformly bounded on all compact subsets of Ω. Note that in the shape optimization problem we cannot replace the compact subsets K  Ω with the domain Ω itself, since the oscillation defect measure is not any regular set additive function on the family of compact subsets of Ω, i.e., it is not any measure in the sense of measure theory. To bypass this difficulty we observe that the finiteness of the oscillation defect measure on compacts gives some additional information on the properties of the distribution function Γ. Our task is to first extract this information and then to use it in the proof of Theorem 1.2. In order to formulate the appropriate auxiliary result, we define the function Tϑ (x) by the equality Tϑ (x) = min{ρ, ϑ}(x) − min{ρ(x), ϑ(x)}

for each ϑ ∈ C(Ω) .

Lemma 5.1. Under the assumptions of Theorem 1.2 and Lemma 3.2, there is a constant c independent of ϑ and K such that the inequalities  Tϑ 1+γ ≤ lim (5.1) | min{ρn (x), ϑ(x)} − min{ρ(x), ϑ(x)}|1+γ dx ≤ c 1+γ L (K) n→∞

Ω

hold for all ϑ ∈ C(Ω) and K  Ω. We point out that the limit in (5.1) does exist by the choice of the sequence ρn . Proof. The proof imitates the proof of Lemma 4.3 from [8]. It can be easily seen that  Tϑ 1+γ (5.2) ≤ lim sup | min{ρn (x), ϑ(x)} − min{ρ(x), ϑ(x)}|1+γ dx. L1+γ (K) n→∞

K

Hence it suffices to show that the right-hand side of this inequality admits a bound independent of ϑ. From the properties of min{·, ·}, it follows that | min{s , ϑ}−min{s , ϑ}|1+γ ≤ (min{s , ϑ}−min{s , ϑ})(s −s ) for all s , s ∈ R+ ; γ

γ

furthermore, for the weak limits we have the inequalities ργ ≥ ργ and min{ρ, ϑ} ≤ min{ρ, ϑ}, and therefore, for any compactly supported, nonnegative function h ∈ C(Ω), we get  lim h| min{ρn , ϑ} − min{ρ, ϑ}|1+γ dx (5.3) n→∞ Ω  ≤ lim h(min{ρn , ϑ} − min{ρ, ϑ})(ργn − ργ ) dx n→∞ Ω  ≤ lim h(min{ρn , ϑ} − min{ρ, ϑ})(ργn − ργ ) dx n→∞ Ω  + (ργ − ργ )(min{ρ, ϑ} − min{ρ, ϑ}) dx Ω  = lim h(ργn min{ρn , ϑ} − ργ min{ρ, ϑ}) dx n→∞ Ω  = lim h(p(ρn ) min{ρn , ϑ} − pmin{ρ, ϑ}) dx. n→∞

Ω

DOMAIN DEPENDENCE FOR NAVIER–STOKES EQUATIONS

1185

By Lemma 4.1 with Φ(ρ, x) = h(x) min{ρ, ϑ(x)}, the right-hand side of (5.3), divided by (ξ + ν), is equal to  lim (5.4) h(min{ρn , ϑ}div un − min{ρ, ϑ}div u) dx n→∞ Ω  = lim h(min{ρn , ϑ} − min{ρ, ϑ})div un dx n→∞ Ω  − lim h(min{ρn , ϑ} − min{ρ, ϑ})div u dx n→∞ Ω  h| min{ρn , ϑ} − min{ρ, ϑ}|1+γ dx ≤ δ lim sup n→∞ Ω  −γ +δ lim h(|div un | + |div u|)(1+γ)/γ n→∞ Ω  ≤ δ lim h| min{ρn , ϑ} − min{ρ, ϑ}|1+γ dx + cδ −γ h||C(Ω) . n→∞

Ω

Combining (5.4) and (5.3), choosing h = 1 on K, and choosing δ > 0 with δ sufficiently small, we obtain (5.1). We reformulate this result in terms of the distribution function Γ. Recall that the functions min{ρn , λ} are uniformly bounded in R3 and that min{ρn , λ}div un converges weakly in L2 (D) for all nonnegative λ. Introduce the functions   Vλ = min{ρ, λ}div u − min{ρ, λ}div u ∈ L2 (D),  H(x) = Γ(x, s)(1 − Γ(x, s)) ds, H ∈ Lγ (D).

(5.5)

[0,∞)

Lemma 5.2. There is a constant c independent of λ such that HL1+γ (D\S) + sup Vλ L1 (D\S) ≤ c.

(5.6)

λ

Proof. Recall that D \ S = (D \ B) ∪ Ω and that H = Vλ = 0 on D \ B. Hence it is sufficient to prove that for all compacts K  Ω, we have HL1+γ (K) + sup Vλ L1 (K) ≤ c,

(5.7)

λ

with the constant c independent of K. We begin with the observation that by Lemma 3.2,   Tϑ (x) =

(5.8)

min{λ, ϑ(x)} dλ Γ(x, λ) − min

λ dλ Γ(x, λ), ϑ(x)

[0,∞)

[0,∞)

for all functions ϑ ∈ C(D). From this and the identity ρ(x) = we conclude that (5.9)

 Tϑ (x) = 0

ϑ(x)

 Γ(x, s) ds for ϑ(x) ≥ ρ(x) and Tϑ (x) =

∞

ϑ(x)

 [0,∞)

(1 − Γ(x, λ)) dλ,

(1 − Γ(x, s)) ds otherwise.

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P. I. PLOTNIKOV AND J. SOKOLOWSKI

Next, choose a sequence of continuous nonnegative functions {ϑk }k≥1 which converge for k → ∞ to the density ϑk → ρ a.e. in D \ S. By Lemma 5.1 the norms in L1+γ (K) of functions Tϑk are uniformly bounded by a constant independent of k and K. Moreover, Tϑk converges a.e. in K to the function  ρ(x)  ∞ Tρ (x) = Γ(x, s) ds = (1 − Γ(x, s)) ds, ρ(x)

0

which yields the estimates Tρ L1+γ (K) ≤ c with the constant c independent of K. It remains to note that estimate (5.7) for H obviously follows from the inequality H ≤ 2Tρ . To estimate Vλ note that     Vλ = w-lim (min{ρn , λ} − min{ρ, λ})div un − w-lim min{ρn , λ} − min{ρ, λ} div u, n→∞

n→∞

where w-lim is denotes the weak limit in L (D \ S). From this and the boundedness of norms div un L2 (D) , we obtain 1

Vλ L1 (K) ≤ lim sup(div un L2 (K) + div uL2 (K) ) n→∞

×  min{ρn (x), λ} − min{ρ(x), λ}L2 (K) , which along with (5.1) implies (5.7). The proof of Lemma 5.2 is completed. 6. Kinetic formulation of the mass balance equation. In this section we show that the distribution function Γ(x, λ) of the Young measure, associated with a given sequence of solutions to problem (1.3), satisfies some integrodifferential transport equation, which is called the kinetic equation. This result is given by the following lemma. Fix an arbitrary function ζ(x, λ) satisfying the conditions ζ ∈ C0∞ (D × R),

(6.1)

spt ζ  D \ (Σ− ∪ S).

We use the notation ∂λ for the partial derivatives with respect to the variable λ, e.g., ∂ζ ∂λ ζ := ∂λ . The absolutely continuous measure is denoted by dλ ζ := ∂λ ζdλ. Recall that the compact obstacle is of the form S = B \ Ω and that Σ− ⊂ ∂B is the exit set. Lemma 6.1. Suppose that all assumptions of Theorem 1.2 are satisfied and Γ is a distribution function of the Young measure associated with a given sequence {ρn } of solutions to problem (1.3). Then   Γ(x, λ)∇x,λ ζ · w dλ dx + (6.2) λM(x, λ) dλ ζdx = 0. (D\S)×R

(D\S)×R

Here w is the solenoidal vector field of the form w(x, λ) = (u(x), −λdiv u), and the function M is defined by the equalities   1 1 (6.3) M(x, λ) = − (sγ − p) ds Γ(x, s) = (sγ − p) ds Γ(x, s), ξ + ν (−∞,λ) ξ + ν [λ,∞)  in which the weak limit for the pressure p(x) = R λγ dλ Γ(x, λ) is defined as in Lemma 3.2. Integral identity (6.2) is equivalent, in the sense of distributions, to the kinetic equation    ∂ ∂  λdiv u(x)Γ(x, λ) − div Γ(x, λ)u(x) − (6.4) [λM(x, λ)] = 0 ∂λ ∂λ   in D (D \ (S ∪ Σ− )) × R .

DOMAIN DEPENDENCE FOR NAVIER–STOKES EQUATIONS

1187

Remark 6.1. Since the kinetic equation is understood in the sense of distributions, the equation remains valid if we replace the intervals of integration (−∞, λ) and [λ, ∞) in formulae (6.3) with (−∞, λ] and (λ, ∞) respectively; however, this creates some discomfort. To avoid such ambiguity, we observe that (6.2) also holds true if we replace the function M with its invariant form  1 lim M(x, s) + lim M(x, s) . M(x, λ) := (6.5) s→λ−0 2 s→λ+0 ∞ ∞ Proof. Choose  − an arbitrary ϕ ∈ C0 (R), and note that for all h ∈ C (D) with spt h  D \ Σ ∪ S ,      h ϕ(ρn ) − ϕ (ρn )ρn div un + ϕ(ρn )∇h · un dx = 0. D\S

Taking the limit n → ∞, we obtain      h ϕ − ϕ ρ div u + ϕ∇h · u dx = 0, D\S

which along with (4.17) gives the integral identity for weak limits, 

   1  h ϕ − ϕ ρ div u + ϕ∇h · u + (6.6) (ϕ − ϕ ρ)p − (ϕ − ϕ ρ)p dx = 0. ξ+ν D\S Next choose  ∞ an arbitrary smooth function η(λ) which vanishes near +∞ and set ϕ(λ) = λ η(s)ds. Identity (3.3) from Lemma 3.2 combined with the formula of integration by parts for Stieltjes integrals yields the representations      ϕ − ϕ ρ (x) = η(λ)Γ(x, λ) dλ + λη(λ)dλ Γ(x, λ) = η dλ (λΓ) , [0,∞)

[0,∞)

[0,∞)

which are substituted into (6.6) and lead to the integral identity      hdiv u dλ λΓ(x, λ) + Γ∇hudλ η dx (6.7) (D\S)×R    1 + h (ϕ − ϕ ρ)p − (ϕ − ϕ ρ)p dx = 0. ξ + ν D\S Recall that Γ vanishes for λ < 0, the function G(x, λ) = (ϕ(λ) − ϕ (λ)λ)h(x) satisfies all conditions of Lemma 3.2, and h is compactly supported in D \ S. It follows from this and Lemma 3.2 that       h (ϕ − ϕ ρ)p − (ϕ − ϕ ρ)p dx = hηλ(λγ − p)dλ Γ(x, λ)dx  +

D\S

(D\S)×R



h λ



(D\S)×R

 η(s)ds (λγ − p)dλ Γ(x, λ)dx = −(ξ + ν)

∞

  hη dλ λM(x, λ) dx.

(D\S)×R

By substitution of this identity into (6.7), since the linear hull of the set of functions in the form hη is dense in C0∞ ((D \ S) × R), we obtain (6.2), which completes the proof of Lemma 6.1. The next lemma describes the basic properties of the function M(x, λ), which are important for further analysis.

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P. I. PLOTNIKOV AND J. SOKOLOWSKI

Lemma 6.2. For a.e. x ∈ D \ S, (i) M(x, ·) is nonnegative and vanishes on R− . Moreover, if the Borel function M(x, .) given by (6.5) vanishes σx -a.e. on the interval (ω, ∞) with ω = p(x)1/γ , then σx = dλ Γ(x, ·) is a Dirac measure and Γ(x, λ) = 0 for λ < p(x)1/γ ,

Γ(x, λ) = 1 for λ ≥ p(x)1/γ .

(ii) for all g ∈ C0∞ (0, ∞),   (6.8) g(λ)M(x, λ)dλ = − Rλ

g  (λ)Vλ (x)dλ,

[0,∞)

where Vλ is defined by (5.5). Proof. Let θ be a standard mollifying kernel in R,  θ ∈ D+ (R), θ(t) dt = 1, spt θ  {|t| ≤ 1}, R

with the corresponding mollifier  (Sk Γ)(x, λ) := k

R

θ(k(λ − t))Γ(x, t)dt.

We will write simply Γk instead of Sk Γ. The mollified distribution function Γk (x, ·) belongs to the class C ∞ (R) and generates the absolutely continuous Stieltjes measure σkx of the form dσkx = ∂λ Γk dλ. It is easily seen that for k → ∞ the sequence of measures σkx converges star-weakly to the measure σx = dλ Γ in the space of Radon’s measures on R. In particular, for all λ with σx {λ} := lims→λ+0 Γ(x, s) − lims→λ−0 Γ(x, s) = 0, we can pass to the limit to obtain   (6.9) (tγ − p)∂t Γk (x, t) dt → (tγ − p) dt Γ(x, t) for k → ∞. [0,λ)

[0,λ)

In other words, relation (6.9) holds true for all λ, except possibly for some countable set. Since ∂λ Γk ≥ 0, the function on the left-hand side of (6.9) increases on (−∞, ω) and decreases on (ω, ∞). From this and (6.9) we conclude that M(x, ·) does not decrease for λ < ω and does not increase for λ > ω, which along with the obvious relations limλ→±∞ M(x, λ) = 0 yields the nonnegativity of M. To prove the second part of (i), note that M(x, λ) = limk→∞ Sk M(x, λ) belongs to the first Baire class and hence is measurable in σx . It follows from the monotonicity of M(x, ·) on the interval (ω, ∞) that if M(x, α) = 0 for some α > ω, then M(x, λ) = 0 and Γ(x, λ) = 1 on (α, ∞). Assume that M(x, ·) vanishes σx -a.e. on (ω, ∞), and consider the set   O = α > ω : σx (ω, α) ≡ lim Γ(x, s) − lim Γ(x, s) = 0 . s→α−0

s→ω+0

Let us prove that O = (ω, ∞). If the set O is empty, then there is a sequence of points λk  ω with M(x, λk ) = 0, which yields Γ(x, ·) = 1 on (ω, ∞), and thus O = (ω, ∞). Hence O = ∅. If m = sup O < ∞, then there is a sequence λk  m with M(x, λk ) = 0, which yields Γ(x, ·) = 1 on (m, ∞). By construction, Γ(x, λ) = c = constant on (ω, m). In other words, restriction of σx to (ω, ∞) is the monoatomic measure (1 − c)δ(· − m). Hence M(x, m) = 2−1 (1 − c)(mγ − ω γ ) = 0, which yields

DOMAIN DEPENDENCE FOR NAVIER–STOKES EQUATIONS

1189

c = 1. From this we can conclude that Γ(x, ·) = 1 on (ω, ∞), and σx is a probability measure concentrated on [0, ω]. Recalling that ω γ = p(x), we obtain  (ξ + ν)M(x, 0) = (λγ − ω γ ) dλ Γ(x, λ) ≥ 0. [0,ω]

Hence dλ Γ(x, λ) is the Dirac measure concentrated at ω, which implies (i). The proof of (ii) is straightforward. It is easily seen that        γ −(ξ + ν) g(λ)M(x, λ) dλ = g (s) ds (t − p)dt Γ(x, t) dλ R

[0,∞)

 g  (s)

 = 

[0,∞)

 g  (s)

= [0,∞)



=

[λ,∞)

 dλ

[0,s)

[λ,∞)

 (tγ − p) dt Γ(x, t) ds

[λ,∞)

 min{t, s}(tγ − p) dt Γ(x, t) ds

[0,∞)

g  (s)(min{ρ, s}p − min{ρ, s}p) ds.

[0,∞)

On the other hand, Lemma 4.1 yields min{ρ, λ}p − min{ρ, λ}p = (ξ + ν)Vλ (x). The proof of Lemma 6.2 is completed. 7. Renormalization of the kinetic equation. The notion of a renormalized solution, introduced in the pioneering paper [4], plays an important role in the theory of compressible NSE developed by Lions and by E. Feireisl and coworkers. Moreover, the kinetic equation itself is a result of the renormalization procedure. Formally we can renormalize (6.4) by multiplying both sides by a function Ψ (Γ), which leads to the transport equation for the function Ψ(Γ), but whether this construction is justified a delicate question. The corresponding result is given by the following lemma. Set Ψ(Γ) = Γ(1 − Γ). Lemma 7.1. For all functions h ∈ C0∞ (D \ S) with spt h  D \ (S ∪ Σ− ) and for all functions η ∈ C ∞ (R) vanishing near +∞, we have the integral identity     (7.1) F(x, λ)dx dλ = 2 η(λ)λM(x, λ)dλ Γ(x, λ) h(x) dx, (D\S)×R

(D\S)

[0,∞)

where F ≡ η(λ)Ψ(Γ)u(x)∇h(x) − λh(x)Ψ(Γ)η  (λ)div u(x) + λh(x)Ψ (Γ)M(x, λ)η  (λ). In other words, the function Ψ(Γ) satisfies the transport equation      ∂   ∂Γ λΨ (Γ)M + 2λM = 0 in D D \ (Σ− ∪ S) × R . div λ,x Ψ(Γ)w) + ∂λ ∂λ Proof. Note that by assertion (i) of Lemma 3.2, the function Γ is well defined on each domain in R4 , and hence on the whole space R4 . The same conclusion can be 1,2 drawn for the vector field u ∈ Hloc (R3 ). Let Θ be the regularizing kernel in R3 , i.e.,  Θ ∈ D+ (R3 ), Θ(x)dx = 1, spt Θ  {|x| ≤ 1}, R3

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P. I. PLOTNIKOV AND J. SOKOLOWSKI

and define the mollifier for scalar or vector functions,  Tm u(x, λ) = m3 Θ(m(x − y))u(y, λ) dy. Ω

We will simply write Γk,m and Γk instead of Sk Tm Γ and Sk Γ. Obviously, Γk are smooth functions of the variable λ and |∂ α Γk (x, λ)| ≤ k α . Recall that for any f ∈ L2 (D ) with D  D and dist (D , ∂D) > m−1 , Tm f L2 (D) ≤ f L2 (D ) ,

(7.2)

Tm f → f in L2 (D).

Substituting the test function ζ(x, λ) = m3 Θ(m(x0 − x))θ(k(λ0 − λ)),

dist (x0 , ∂D) > m−1 ,

λ0 ∈ R,

into (6.2), we arrive at the equality ∂λ [λΓk,m div u] − div (Γk,m u) − ∂λ Sk (λTm M) + r1 + r2 + r3 = 0,

(7.3) (7.4)

which holds true in any subdomain of D \ (S ∪ Σ− ) × R for sufficiently large m. Here the remainders are given by

  r1 = ∂λ {Tm (λΓk div u) − λdiv uTm Γk }, r2 = Tm div u∂λ Sk (λΓ) − λSk Γ ,   r3 = div (Tm Γk )u − Tm (Γk u) . Recall that Γk,m are smooth functions of the variables x and λ. Multiplying both sides of (7.3) by hηΨ (Γk,n ) and integrating the result over D \ S, we arrive at the equality    η(λ)Ψ(Γk,m )u(x)∇h(x) − λh(x)Ψ(Γk,m )η  (λ)div u(x) dx dλ (7.5) (D\S)×R

+ J k,m +

3 

Ilk,m = 0,

l=1

where

Iik,m

 F k,m ∂λ Sk (λTm M), J k,m = − (D\S)×R  = F k,m ri with F k,m = hηΨ (Γk,m ) . (D\S)×R

Our first task is to find the limits for m → ∞ of the sequences {J k,m }, {Ilk,m }. It follows from inequality (7.2) that for i = 1, 2, ri L2 ((D\S)×I) ≤ k c (I) < ∞ for all compact I ⊂ R. On the other hand, since η vanishes near ∞ and Γk,m vanishes for λ < 0, the functions F k,m are uniformly compactly supported, and their absolute values are uniformly bounded. Hence for i = 1, 2, (7.6)

F k,m ri L2 ((D\S)×R) ≤ c < ∞ for all m ≥ 1.

DOMAIN DEPENDENCE FOR NAVIER–STOKES EQUATIONS

1191

  Noting that ∂λ λΓk (·, λ) ∈ L∞ (D), taking into account the form of limits in (7.2), we obtain for fixed λ and k, and for m → ∞,

  Γk,m → Γk , Tm ∂λ (λΓk )div u − div uTm ∂λ (λΓk ) → 0,

    Tm div u∂λ Sk (λΓ) − λSk Γ → div u∂λ Sk (λΓ) − λSk Γ , a.e. in D \ S, which along with (7.6) yield for m → ∞,

  (7.7) F k,m r1 → 0, F k,m r2 → F k ∂λ div u Sk (λΓ) − λSk Γ in L1 ((D \ S) × R). The next from [4]: Let u ∈ H 1,2 (D) and  evaluations  are based on the following result ∞  g ∈ L D × (a, b) with b − a < ∞. Then for all D  D,   div (Tm g)u − Tm (gu) → 0 in L1 (D × (a, b)). Setting g(x, λ) = Γk (x, λ) since F k,m are uniformly bounded and compactly supported in (D \ S) × R, that for m → ∞ the functions F k,m r3 tend to zero  we can conclude  1 strongly in L (D \ S) × R . Combining this result with (7.7), we obtain lim I m,k m→∞ 1 

lim I2m,k = I2k :=

m→∞

= lim I3m,k = 0, m→∞

  F k div u∂λ Sk (λΓ) − λSk Γ dx dλ.

(D\S)×R

Similar arguments show that (7.8)



F k ∂λ Sk (λM) dx dλ with F k = h(x)η(λ)Ψ (Γk ).

lim J m,k = J k := −

m→∞

(D\S)×R

Taking the limits, first for m → ∞ and then for k → ∞ in (7.5), we arrive at 

  η(λ)Ψ(Γ)u(x)∇h(x) − λh(x)Ψ(Γ)η  (λ)div u(x) dx dλ

(7.9) (D\S)×R

+ lim J k + lim I2k = 0. k→∞

k→∞

Let us show that the last term in the left-hand side equals 0. It is easily seen that  ˜ ˜ = tθ (t) + θ(t). − t))Γ(x, t) dt, θ(t) ∂λ [(λΓ)k − λΓk ](x, λ) = k θ(k(λ R

 ˜ = 0, the sequence of functions ∂λ [(λΓ)k − Since θ˜ is compactly supported and R θ(s)ds λΓk, ](x, λ) is bounded and converges to 0 a.e. in (D \ S) × R. Recalling that the functions F k are uniformly supported, i.e., the supports spt F k are included in the given compact subset of (D \ S) × R, and uniformly bounded in (D \ S) × R, we can conclude that limk→∞ I2k = 0. Next, integration by parts in the formula for J k leads to  Jk = (7.10) hη  Ψ (Γk )Sk (λM)dx dλ − 2W k (D\S)×R

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P. I. PLOTNIKOV AND J. SOKOLOWSKI

with

 k

W =

(7.11)

hηSk (λM)∂λ Γk dx dλ. (D\S)×R

Integration by part in the Stieltjes integral results in   ∂λ Γk (x, λ) = k∂λ θ(k(λ − t)Γ(x, t)dt = k θ(k(λ − t)dt Γ(x, t); R

therefore, (7.12)



R

 hSk ηSk (λM) dλ Γ(x, λ) dx

k

W = (D\S)×R

 ≡





h(x) R

D\S





+

ηS2k (λM) dλ Γ(x, λ)

h(x) D\S

R

dx

 [Rk Sk (λM)](x, λ) dλ Γ(x, λ) dx

  with the linear operator Rk defined, for any g ∈ L∞ (D \ S) × R , by  Rk g = rk (λ, t)g(x, t)dt, where rk (λ, t) = k(η(t) − η(λ))θ(k(λ − t)). R

Since the function η(λ) is smooth and vanishes for λ > N = sup spt η, the kernel rk (λ, t) vanishes for λ > N + 1 and satisfies the inequality R |rk (λ, t)| dt ≤ ck −1 . Thus, we get   |Rk g| ≤ ck −1 ess sup sup |g(x, s)| , x

s≤N +1

which along with |Sk (λM)|(x, λ) ≤ sup sM(x, s) ≤ (λ + 1) sup M(x, s) ≤ 2(λ + 1)p(x) R

s≤λ+1

implies |Rk Sk (λM)| ≤ c(η)k −1 p(x). Thus,      [Rk Sk (λM)](x, λ) dλ Γ(x, λ) ≤ c(η)k −1 p(x).   R

From this and the Lebesgue dominant convergence theorem we conclude that the last term in the right-hand side of (7.12) vanishes for k → ∞. Recall that S2k is the standard mollifying operator with the kernel θ ∗ θ. Hence for a.e. x ∈ D \ S and for all λ ∈ R, we have limk→∞ [S2k (λM)](x, λ) = λM(x, λ). From this and the Lebesgue dominant convergence theorem related to the measure dλ Γ(x, ·), we obtain that for a.e. x ∈ D \ S,   2 lim η[Sk (λM)](x, λ) dλ Γ(x, λ) = ηλM dλ Γ(x, λ). k→∞

R

R

Since the integrands in the right-hand side of (7.12) are uniformly bounded, we have    k h(x) ηλM dλ Γ(x, λ) dx. lim W = W := k→∞

D\S

R

DOMAIN DEPENDENCE FOR NAVIER–STOKES EQUATIONS

1193

On the other hand, for k → ∞ the sequences Γk and Sk M converge a.e. in (D \S)×R to Γ and M, respectively; thus, by the limit passage k → ∞ in (7.10), we obtain  k hη  Ψ (Γ)λM dx dλ − 2W. lim J = k→∞

(D\S)×R

Substituting this equality into (7.9) and recalling that limk→∞ I2k = 0, we obtain (7.1). The proof of Lemma 7.1 is completed. 8. The proof of Theorem 1.2. The theorem will be proved if we show that any sequence of generalized solutions to problem (1.3) satisfying the hypotheses of Theorem 1.2 and Lemmas 3.2 and 3.3 converges a.e. on D \ S. In light of Lemmas 3.2 and 3.3, it suffices to verify the equality Ψ(Γ) = 0 in (D \ S) × R. We begin with proving that renormalized integral identity (7.1) after substituting h = 1 turns into the integral inequality    λΨ (Γ)Mη  − λΨ(Γ)η  div u dx dλ (8.1) (D\S)×R





≥2 (D\S)

 η(λ)λM(x, λ)dλ Γ(x, λ) dx,

[0,∞)

which holds true for all nonnegative functions η ∈ C ∞ (R) vanishing near +∞. The proof is based on the following approximation result, which is shown by an application of the Hedberg approximation theorem [13]. Lemma 8.1. For each k > 1, there exist a function ζk ∈ C ∞ (D) and a constant c independent of k such that ζk vanishes in a vicinity of S and  0 ≤ ζk ≤ 1, meas Ak + (8.2) |∇ζk u| dx ≤ 1/k, D

where Ak = {x ∈ D \ S : ζk (x) ≤ 1 − 1/k} ⊂ D \ S. Proof. Introduce the function f (x) = |u(x)|+dist (x, S). Since u−u∞ belongs to 1,2 H0 (D) and u∞ vanishes near S, it follows from the general theory of Sobolev spaces (see [28]) that f has a quasi-continuous representative which vanishes everywhere on S. By the Hedberg approximation theorem (see [13]), for each N > 0 there is ωN ∈ C0∞ (D) so that ωN = 1 in some vicinity of S, 0 ≤ ωN ≤ 1, and (1 − ωN )f H 1,2 (D) ≤ N −1 . On the other hand, by the absolute continuity property of the integral, there exists a positive continuous function μ(s) such that, whenever meas A ≤ δ, then  (f 2 + |∇f |2 )dx ≤ μ(δ) and μ(δ)  0 for δ  0. A

For any δ > 0, set Sδ = {x : dist (x, S) < δ}. Since ∩Sδ = S, the measure of the set Sδ \ S tends to 0 as δ → 0, which yields     (f 2 + |∇f |2 )dx = (f 2 + |∇f |2 )dx ≤ μ meas (Sδ \ S) := r(δ)  0 as δ  0. Sδ

Sδ \S

Obviously, for Bk = {x ∈ D \ S : 1 − ωN (x) > 1/k}, we have   δ2 dx ≤ |f |2 (1 − ωN )2 dx ≤ N −2 , k 2 Bk \Sδ D

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P. I. PLOTNIKOV AND J. SOKOLOWSKI

which implies (8.3)

meas {Bk \ Sδ } ≤ k 2 (δN )−2 ,

meas Bk ≤ k 2 (δN )−2 + meas (S \ Sδ ).

Next, note that     1 (1 − ωN )2 |∇f |2 dx ≤ 2 |∇f |2 dx + |∇f |2 dx + |∇f |2 dx k D\Bk D Bk \Sδ Sδ ≤ ck −2 + μ(k 2 (δN )−2 ) + r(δ),   which along with the identity f ∇ωN = (1−ωN )∇f −∇ (1−ωN )f yields the estimate     |∇ωN |f dx ≤ ∇ (1 − ωN )f L1 (D) + (1 − ωN )|∇f | dx D

D

 1/2 2 2 1,2 ≤ c(1 − ωN )f H (D) + c (1 − ωN ) |∇f | dx ≤N

−1

+ ck

−1

1/2

+ cμ



D

 k (δN )−2 + cr1/2 (δ). 2

Fix δ > 0 sufficiently small such that r1/2 (δ) + meas (S \ Sδ ) ≤ k −1 ; next, choose N sufficiently large such that μ1/2 (k 2 (δN )−2 ) + k 2 (δN )−2 < k −1 . In view of (8.3) and taking into account that |u| ≤ f , we arrive at  |∇ωN ||u| dx + meas Bk ≤ ck −1 . D

Hence ζk = 1 − ωN with Ak = Bk satisfies (8.2). The proof of Lemma 8.1 is completed. Following [10] let us consider the sequence of functions χk (x) = χ(k dist (x, Σ− ∪ ∂D)) with an arbitrary, smooth, monotone function χ such that χ(z) = 0 for z ≤ 1/2 and χ(z) = 1 for z ≥ 1. Since Ψ(Γ)(·, λ) vanishes outside of B, we have for all sufficiently large k, (8.4)

∇χk (u − u∞ ) → 0 in L1 (D \ S),

Ψ(Γ)(·, λ)∇χk u∞ ≤ 0 in D.

Set hk = ζk χk and note that ∇hk u ≤ |∇ζk u| + |∇χk (u − u∞ )| + ζk ∇χk u∞ . Recalling the inequality η ≥ 0 and relations (8.2), (8.4), we obtain  (8.5) Ψ(Γ)η∇hk u dx dλ ≤ 0. lim sup k→∞

(D\S)×R

Moreover, the functions hk converge to 1 in measure on D \ S. The functions hk are Lipschitz continuous in R3 and vanish in the vicinity of Σ− ∪ S; therefore, they can be used as test functions in (7.1). Substitution of hk into (7.1), followed by the limit passage k → ∞ in the resulting integral identity, leads to desired inequality (8.1). Next, we claim that the right-hand side of (8.1) equals zero. To thisend, choose an arbitrary nonnegative function υ ∈ C ∞ (R) with spt υ ⊂ (−1, 1) and R υ(λ)dλ = 1. ∞ For fixed t > 2, set η(λ) = λ υ(s−t) ds. Since η  (λ) = 0 for |λ−t| ≥ 1, and η(λ) = 1 for λ ≤ t − 1, we can use (8.1) to obtain

DOMAIN DEPENDENCE FOR NAVIER–STOKES EQUATIONS

(8.6)  2 (D\S)



  λMdλ Γ dx ≤ −(t + 1)

[0,t−1)

1195

  M + Ψ(Γ)|div u(x)| η  dx dλ.

(D\S)×R

Using identity (6.8) and the relation  − Ψ(Γ) η  (λ) |div u| dx dλ (D\S)×R



≡

η





[0,∞)

D\S



  Ψ(Γ(x, s)) ds |div u| dx dλ

Ω×[0,λ)

we can rewrite inequality (8.6) in the form     2 (8.7) λMdλ Γ dx ≤ (1 + t) (D\S)

[0,t−1)

η  (λ)℘(λ)dλ ,

[1,∞)

where the function ℘ : [0, ∞) → R is given by   ℘(λ) = Ψ(Γ(x, s))|div u(x)| dx ds + Vλ (x)dx. (D\S)×[0,λ)

Since



Ω

 Ψ(Γ(x, s)) ds ≤

[0,λ)

Ψ(Γ(x, s)) ds = H(x), [0,∞)

Lemma 5.2 implies the boundedness of ℘ on R+ , |℘(λ)| ≤ cuH 1,2 (D\S) HL2 (D\S) + Vλ L1 (D\S) ≤ c. Taking into account that η  (λ) = ∂t υ(λ − t), inequality (8.7) can be rewritten in the form    d 2 λMdΓ(x, λ) dx ≤ (1 + t) (υ ∗ ℘)(t). (8.8) dt (D\S) [0,t−1) Since the smooth function (υ ∗ ℘)(t) is uniformly bounded on R+ , there is a sequence d tk → ∞ such that limk→∞ (tk + 1) dt (υ ∗ ℘)(tk ) ≤ 0. Substitution of t = tk into (8.8) followed by the limit passage k → ∞ in (8.8) leads to  λMdΓ(x, λ) = 0 for a.e. x ∈ D \ S. [0,t−1)

In other words, M(x, ·) vanishes σx -a.e. on (0, ∞), which along with Lemma 6.2 implies the equality Γ(1 − Γ) = 0 a.e. in (D \ S) × R. Hence ρn converges a.e. in D \ S. Estimates (1.11) imply the strong convergence of the sequence ρn in Lr (D \ S) for all r < γ. Since (ρn , un ) satisfy all assumptions of Theorem 2.1, we can make use of estimate (2.1) from this theorem, which yields the strong convergence ρn in Lrloc (Ω) for r < γ(1+κ). In particular, the sequence p(ρn ) converges to p = p(ρ) in L1loc (Ω). After substituting (ρn , un ) into integral identities (1.7a) and (1.7b), followed by the limit passage n → ∞, by Lemma 3.3 we can conclude that the pair (ρ, u) is a generalized solution to problem (1.3). Finally, since ∇u∞ is compactly supported in Ω, the limit passage in (1.12) for the sequence of drag functionals follows by Lemma 3.3 and by the strong convergence of the sequence p(ρn ) in L1loc (Ω).

1196

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