Abstract
Polymeric fluids arise in many practical applications in biotechnology, medicine, chemistry, industrial processes and atmospheric sciences. In this article, the Doi model for the suspensions of rod-like molecules in a compressible fluid is investigated. The model under consideration couples a Fokker-Planck-type equation on the sphere for the orientation distribution of the rods to the Navier-Stokes equations for compressible fluids, which are now enhanced by additional stresses reflecting the orientation of the rods on the molecular level. The coupled problem is 5-dimensional (three-dimensions in physical space and two degrees of freedom on the sphere) and it describes the interaction between the orientation of rod-like polymer molecules on the microscopic scale and the macroscopic properties of the fluid in which these molecules are contained. Prescribing arbitrarily the initial density of the fluid, the initial velocity, and the initial orientation distribution in suitable spaces we establish the global-in-time existence of a weak solution to our model defined on a bounded domain in the three dimensional space. The proof relies on the construction of a sequence of approximate problems by introducing appropriate regularization and the establishment of compactness.
Introduction
The evolution of rod-like molecules in both compressible and incompressible fluids is of great scientific interest with a variety of applications in science and engineering. The present article deals with the Doi model for the suspensions of rod-like molecules in a dilute regime. The model under consideration couples a Fokker-Planck-type equation on the sphere for the orientation distribution of the rods to the Navier-Stokes equations for compressible fluids, which are now enhanced by additional stresses reflecting the orientation of the rods on the molecular level. The coupled problem is 5-dimensional (three-dimensions in physical space and two degrees of freedom on the sphere) and it describes the interaction between the orientation of rod-like polymer molecules on the microscopic scale and the macroscopic properties of the fluid in which these molecules are contained. The macroscopic flow leads to a change of the orientation and, in the case of flexible particles, to a change in shape of the suspended microstructure. This process, in turn yields the production of a fluid stress. In this paper, we consider the Doi model for a compressible fluid in a bounded domain. The derivation of the system under consideration is described below.
A smooth motion of a body in continuum mechanics is described by a family of one-to-one mappings X(t, ·) : Ω → Ω, t ∈ I.
The curve X(t, x) represents the trajectory of a particle occupying at time t a spatial position x and this curve is completely determined by a velocity field u : I × Ω → R 3 through ∂ ∂t X(t, x) = u(t, X(t, x)), X(0, a) = a.
Then, the conservation of mass can be formulated as follows: We next obtain equation of motion by applying Newton's second law of motion as follows:
ρ(t, x)F (t, x)dx +
∂X(t,B)
t(t, x,n)dS.
Then, we have For the simplicity, we take F = 0. The stress principle in continuum mechanics can be addressed through the fundamental laws of Cauchy: there is a symmetric stress tensor T(t, x) such that t(t, x,n) = T(t, x)n.
Therefore, (1.2) becomes By applying Green's lemma to (1.3), we finally have
The stress tensor T of a general fluid obeys Stokes' law:
where p is the pressure and S is the stress tensor. Let us determine S and p in our model. The pressure p is of the form
(1.5)
S consists of two parts:
where S 1 is the viscous stress tensor generated by the fluid
and S 2 is the macroscopic symmetric stress tensor derived from the orientation of the rods at the molecular level. The microscopic insertions at time t and macroscopic place x are described by the probability f (t, x, τ )dτ. The suspension stress tensor S 2 is given by an expansion S 2 (x, t) = σ (1) (x, t) + σ (2) (x, t) + σ (3) (x, t), where σ (1) (t, x) = S 2 (3τ ⊗ τ − I 3×3 )f (t, x, τ )dτ, σ (2) (t, x) = −σ (2) ij (t, x)I 3×3 , with σ 
ij (τ 1 , τ 2 )f (t, x, τ 1 )f (t, x, τ 2 )dτ 1 dτ 2 .
This, and more general expansions for S 2 are encountered in the polymer literature (cf. Doi and Edwards [8] ). We refer the reader to the articles by Constantin et al [5] , [6] , where a general class of stress tensors is presented in the context of incompressible fluids. The structure coefficients in the expansion γ
ij are in general smooth, time independent, x independent, and do not depend on f. Assuming for simplicity that
and denoting
the suspension stress tensor S 2 takes the form
In this setting, f describes the time-dependent orientation distribution that a rod with a center mass at x has an axis τ in the area element dτ and it is described by a compressible Fokker-Plank type equation,
where
τ is the projection of ∇uτ on the tangent space of S 2 at τ ∈ S 2 . With ∇ τ and ∆ τ we denote the gradient and the Laplace operator on the unit sphere, while ∇ and ∆ represent the gradient and the Laplacian operator in R 3 . The second term ∇·(uf ) in (1.7) describes the change of f due to the displacement of the center of mass of the rods due to macroscopic advection. The term ∇ τ · (P τ ⊥ ∇uτ f ) is a drift-term on the sphere, which represents the shear-forces acting on the rods. The term D τ ∆ τ f represents the rotational diffusion due to Brownian motion. This effect causes the rods to change their orientation spontaneously, whereas the term D∆f is the translational diffusion due to Brownian effects.
By integrating (1.7) over S 2 , we can obtain the equation of η:
By substituting (1.6) and (1.5) to (1.4), the equation of motion becomes
In sum, after normalizing all the constants by 1 for the sake of simplicity, we have the following system of equations:
where Ω is a bounded domain and we impose Dirichlet boundary conditions to u, f , and η: u = 0, f = 0, and η = 0, on ∂Ω.
In the sequel, we construct a sequence of approximating problems by regularizing the equations by extending functions to be zero outside Ω. Prescribing arbitrarily the initial fluid density, the initial velocity, and the initial orientation distribution in suitable spaces, we establish long-time and large data existence of a weak solution. Since the definition of a weak solution and the main result are rather complicated, they are stated in Section 2.
Related results on the Doi model for the suspensions of rod-like molecules in incompressible fluids have been studied by many authors. We refer the reader to Constantin [5, 6, 7] , Lions and Masmoudi [14, 15] , Masmoudi [16] and Otto and Tzavaras [19] for results on related models on the whole space. In [1] the authors treat the Doi model for an incompressible fluid within a bounded domain in the 3-dimensional space and establish results on the global existence of solutions. For compressible models, related results have been presented in a series of articles. We refer the reader to Carrillo et al [2, 3, 4] , Goudon et al [11, 11, 12] , and Mellet and Vasseur [17, 18] , where asymptotic, analytical and numerical results on related fluid-particle interaction models are discussed. These articles deal with models coupling the Stokes, Navier Stokes or Euler system with either the Smolukowski equation or Fokker-Planck equation. What distinguishes the model presented in this article, besides the general type of the stress tensor under consideration, is the fact that, unlike other models, the Fokker-Planck-type equation presented here takes into consideration in addition to the Brownian effects the presence of the shear forces acting on the rods. This new element yields a new equation for the entropy induced by the probability density function f in the microscopic level and therefore new apriori estimates. We refer the reader to Appendix 1 for the derivation of equation of the entropy ψ.
The paper is organized as follows. In Section 2, we introduce the notion of a weak solution of the system (1.10), and we state the main results; compactness (Proposition 2.3) and existence (Theorem 2.2). In Section 3, we prove various convergence results and we provide a formal proof of the strong convergence of ρ under better assumptions. This formal convergence argument is recasted in Section 4, where the strong convergence of ρ is proved by using suitable cut-off functions in the renormalized equation of ρ. In Section 5, we generate an approximate sequence of weak solutions in three steps. (i) We first regularize the equation of ρ, which corresponds to the regularization of ρ d dt + ρu · ∇ in (1.10b). We also regularize u in the equation of f which requires to regularize η and σ in the right-hand side of (1.10b). Before regularization, we extend equations to zero outside Ω. (ii) Next we add nonlinear damping terms to the equation of ρ and η to increase integrability of ρ and η. (iii) We finally truncate ρ γ and η 2 to increase regularity of {ρ, u, η}. By passing to the limits in sequences, we can prove Theorem 2.2.
Notations: • L p (0, T ; X) denotes the Banach set of Bochner measurable functions f from (0, T ) to X endowed with either the norm
• A B means there is a constant C such that A ≤ CB.
• X ⊂ comp Y means that X is compactly embedded in Y .
• I X is the indicator function which is 1 for x ∈ X and 0 otherwise.
• C(T ) is a function only depending on initial data and T .
• and → denote weak limit and strong limit, respectively.
2 Definition of weak solution and main results
A priori estimate
Before introducing the concept of a weak solution of the system (1.10), let us present the energy estimate. Multiplying (1.10b) by u and integrating over Ω we get
Next we introduce an entropy induced by f in the microscopic level. Let
Then, ψ satisfies
For the derivation of (2.2), we refer the reader to Section 6.1. Integrating (2.2) over Ω, we obtain
By adding (2.3) to (2.1), we have
, which cannot be obtained derived from (1.8) in the three dimensions. From (2.4), we can obtain various estimates of {ρ, u, f, η, σ}. First,
By expressing ρu as
We next estimate the derivative of σ by using the entropy dissipation.
Definition of weak solution, main result
We now define a weak solution of the system (1.10). By a notational abuse, we include η and σ in the definition of weak solution.
Definition 2.1 We say that {ρ, u, f, η, σ} is a weak solution of the system (1.10) if (i) (1.10a) holds in the sense of renormalized solutions, i.e.,
(ii) (1.10b) and (1.10c) hold in the sense of distributions.
(iii) Moreover, {ρ, u, f, η, σ} satisfies the following energy inequality:
Remark 1 (1) The central difficulty in showing the existence of a weak solution in the theory of compressible fluids is typically the dependence of the pressure on nonlinear terms, for instance ρ γ . From the a priori estimate, we have ρ ∈ L ∞ (0, T ; L γ (Ω)), which is not enough to pass to the limit to ∇ρ γ in the sense of distributions. The issue is resolved by showing that ρ satisfies a better integrability condition by choosing appropriate cut-off functions in the renormalized form (2.10) in the spirit of Feireisl [9] (see also Lions [13] ). Note that in the present context the suspension stress tensor depends on the density of the particles in a nonlinear way as well. In this case, the regularity of η, η ∈ L 2 (0, T ; H 1 (Ω)) enables us to handle the nonlinearity.
(2) The main additional difficulties in the present context involve the presence of two nonlinear terms in the equation of f . In fact, letting χ ∈ C ∞ c (Ω × S 2 ) we obtain from the advection term ∇ · (uf ),
whereas from the shear term
To pass to the limit in (2.12) and (2.13), we need to show that
. This is proved in Section 3. (3) In order to pass to the limit to linear terms in the equation of f , we only need
We now state the main result of the paper.
and Ω be a C 1 bounded domain. Assume that a sequence {ρ 0 , u 0 , f 0 , η 0 } satisfies
Then, there exists a weak solution {ρ, u, f, η, σ} of the system (1.10) satisfying (2.14) at t = 0. Moreover,
The proof of Theorem 2.2 consists of two parts. First, we prove the compactness of an approximate sequence {ρ n , u n , f n , η n , σ n } n≥1 under the assumption γ > 3 2 in Section 3 and 4. We state the detailed statement in Proposition 2.3 below. Secondly, we construct an approximate sequence of solutions through regularizing equations in Section 5. The reader should contrast the approximating scheme presented here with the schemes presented in [4] , [10] and [13] for different models.
We begin with the compactness result. Suppose there is a an approximate sequence of solutions {ρ n , u n , f n , η n , σ n } n≥1 such that
Then, we can extract a subsequence, using the same notation,
e ij in the sense of measures and e ij is a bounded measure,
and Ω be a C 1 bounded domain. Assume that the energy inequality (2.11) holds for a sequence {ρ n , u n , f n , η n , σ n } n≥1 . Then, limit functions in (2.17) satisfy the followings.
(iv) ρ and u solve (1.10a) in the sense of renormalized solutions.
(v) If in addition we assume that ρ n 0 converges to ρ 0 in L 1 (Ω), then {ρ, u, f, η, σ} is a weak solution of (1.10) such that
(vi) Finally, we have the following strong convergence:
We will prove this proposition in Section 3 and 4.
3 Proof of Proposition 2.3 (i), (ii), (iii), formal proof of (v), and (vi)
Proof of Proposition 2.3. (i)
We begin with the proof of Proposition 2.3 (i). To this end, we need the following lemma.
Lemma 3.1 [13] Let g n and h n converge weakly to g and h respectively in
Then, g n h n converges to gh in the sense of distributions.
We would like to apply Lemma 3.1 to h n = u n with q 1 = 2 and q 2 ∈ [2, 6), and
Next, from the equation of motion,
Here, we use the fact that (η n ) 2 , σ n , and η n are bounded in
, we conclude as above that v = √ ρu.
Proof of Proposition 2.3. (ii)
To show the strong convergence of η n and σ n , we need the following lemma.
Lemma 3.2 [20] Let X, B, and Y be Banach spaces such that X is compactly embedded in B and B is a subset of
• Strong convergence of η n : First,
• Strong convergence of σ n : First,
• Strong convergence of
We note that these two terms are of the form
, and
ςf n dτ and their time and spatial derivatives satisfy the same bound of σ. Therefore, these two terms converge strongly in L 2 (0, T ; L 2 (Ω)) as well.
Proof of Proposition 2.3. (iii)
To show (iii), we need to show (η n ) 2 converges strongly in L p (0, T ; L q (Ω)) for some p > 1 and q such that
, we take > 0 such that
,
Let p =p 2 and q =q 2 . Then,
where 1 q + 1 γ ≤ 1 by taking δ < 3 . Therefore, ρ n (η n ) 2 converges to ρη 2 in the sense of distributions.
Proof of Proposition 2.3 (v): formal proof on the whole spaces
Before proving Proposition 2.3 (iv) and (v), we provide a formal proof of the convergence of ρ n on the whole spaces under a stronger assumption:
For details of the proof, see Section 6.2. From (1.10a),
Next, we take (−∆) −1 ∇· to (1.10b). Then,
from which we have
By (3.6) and (3.7),
we rewrite (3.8) as follows.
(ρ log
Suppose that (3.9) also holds for {ρ n , u n , f n , η n , σ n } n≥1 .
(ρ
(3.10)
Lets be a weak limit of ρ n log ρ n . By taking the limit of (3.10) for n → ∞,
where we use Proposition 2.3 (iii). Next, we take the limit to (1.10b).
where we use Proposition 2.3 (ii). Let s = ρ log ρ. By following the same calculations above, we have from which we obtain that
(3.13)
Comparing (3.11) and (3.13), we have
Therefore, s = s almost everywhere, and ρ n converges strongly to ρ in
Remark 2 In Section 4, we show the strong convergence of ρ n in C([0, T ]; L 1 (Ω)) using (2.10) with appropriate cut-off functions approximating ρ log ρ.
Proof of Proposition 2.3 (vi)
We now prove Proposition 2.3 (vi) assuming that we have already resolved Proposition 2.3 (v). We will prove Proposition 2.3 (iv) and (v) in the next section. Let us begin with the convergence of ρ n u n . First, we show that (ρ n u n ) converges to ρ n u n in L 2 (0, T ; L 1 (R 3 )). Here, we extended functions to zero outside Ω, and g = g k and k is the usual mollifier. Since
we have
. Now, we choose p >
converges to 0 as goes to 0.
Moreover, (2.18) implies that sup
converges to 0 as goes to 0. Therefore,
) as goes to 0, uniformly in n. Next, we deduce that (ρ n u n ) converges to (ρu) as n goes to ∞. Since (ρ n u n ) is smooth in x and
The second statement in (2.19) is immediate consequence of the uniform bound of u n in L 2 (Ω × (0, T )).
Since
converges to 0 in L 1 (Ω), the third statement is proved. Finally, three previous results implies ρ n u n i u n j converges to ρu i u j almost everywhere, and
Therefore, the last statement in (2.19) holds.
Proof of Proposition 2.3 (iv) and (v)
The strong convergence of {ρ (n) } n≥1 can be proved by introducing a family of cut-off functions approximating ρ log ρ in the renormalized solution setting. First, we show that ρ satisfies higher integrability condition.
Higher integrability of ρ
We first define the inverse of the divergence operator. We denote the solution v of
by v = T g. This operator T = (T 1, T 2 , T 3 ) is the inverse of the divergence operator such that
with the following boundedness property:
If in addition g can be written as g = ∇ · h for a certain h ∈ L r with h ·n = 0 on ∂Ω, then
We will use this operator to obtain higher integrability of ρ. By extending (2.10) to zero outside Ω and regularizing it, we have,
where b(ρ) = b(ρ) g . As proved in [13] , we have
We are now ready to prove the following result.
Lemma 4.1 Let γ > 3 2 and {ρ, u, f, η, σ} be a weak solution of the system (1.10). Then, there exists θ > 0, depending only γ, such that
Proof: We take a test function of the form
and test it against (1.10b). Then, with the aid of (4.1),
We now estimate I 1 , · · · , I 11 . For details, see [10] .
• I 1 C(T ).
• I 2 ρu
.
• I 4 ρu 0,T ) ) .
•
.
• ,T ) ) .
•
.
In sum,
By taking the limit in → 0,
(4.4)
We approximate z → z θ by a sequence of {b n } in (2.10), and approximate χ to the identity function of (0, T ). Then,
We note that which completes the proof.
Remark 3 From (4.6), the best possible θ is 2 3 γ − 1, and higher integrability of ρ can be obtained by choosing appropriate cut-off functions in (2.10) in the spirit of Feireisl [10] . If γ ≥ 9 5 , then γ + θ ≥ 2. In [13] , Lions used this idea in order to show higher integrability of ρ by multiplying (3.7) by ρ θ .
Limit of the effective viscous flux
We now study the limit of the so called the effective viscous flux, ρ γ −2∇·u. In the formal proof, we take (−∆) −1 ∇ to (1.10b) and multiply by ρ. In this section, we instead take χ(t)φ(x)∆ −1 ∇T k (ρ n ) as test functions to (2.10) to obtain a better convergence result on the effective viscous flux. Here, the cut-off function T k is defined as
where T ∈ C ∞ (R) satisfies
and T (−z) = −T (z). From (2.10),
holds in the sense of the distributions. Passing to the limit in (4.9), we have
in the sense of distributions.
To take the limit to the effective viscous flux, we need the following lemma. For the proof, we refer the reader to Feireisl [10] .
Lemma 4.3 Under the condition in Proposition 2.3, we have
for χ ∈ D(0, T ) and φ ∈ D(Ω).
Proof: We take χ(t)φ(x)∆ −1 ∇T k (ρ n ) as test functions to (1.10b). Then,
We now take the limit of (4.11) for n → ∞.
where we use lemma 4.2 to show
(4.13)
We now take χφ∆ −1 ∇T k (ρ) as test functions to
and do the same calculation using (4.10). Then,
(4.14)
Therefore, by comparing (4.12) and (4.14), we complete the proof.
Corollary 4.4 Let ρ be a weak limit of the sequence {ρ n }. Then,
Proof: As z → z γ is convex, T k is concave, and
we have lim sup
By lemma 4.3, the last term in (4.16) can be estimated as follows. 
Proof of Proposition 2.3. (iv)
To show that ρ and u solve (1.10a) in the sense of renormalized solutions, we regularize (4.10) to get 18) where r → 0 in L 2 (R 3 × (0, T )). We multiply (4.18) by b T k (ρ) and take → 0. Then,
in the sense of distributions. By the interpolation,
,
as k → ∞. This completes Proposition 2.3 (iv).
Proof of Proposition 2.3. (v)
We introduce a family of functions L k such that
we have
On the other hand, by (1.10a),
Passing to the limit in (4.22) for n → ∞,
By approximating φ to the identity function of Ω,
We approximate z log z by L k (z). By corollary 4.4, with γ + 1 > 2, the right-hand side of (4.24) tends to 0 as k → ∞. Therefore,
which implies the strong convergence of {ρ n } in L 1 (Ω × (0, T )). This completes the proof of Proposition 2.3 (v).
Construction of approximate sequences
We now construct an approximate sequence of solutions to the system (1.10) so that we can apply the compactness argument (Proposition 2.3) to obtain a weak solution. This section consists of three parts: the regularization of ρ d dt + ρu · ∇ in (1.10b), nonlinear damping to the equation of ρ and η, and the truncation of ρ γ and η 2 . Collecting all these steps, we can prove Theorem 2.2.
Smoothing
We consider the following system of equations:
In Section 5.2 will be shown that there exists a solution {ρ, u, f, η, σ} to (
we can obtain the following a priori estimates:
If u ∈ L 2 (0, T ; H 1 (Ω)), (5.1c) has a smooth solution, and the entropy ψ and its equation make sense. Thus, we only focus on the existence of a solution {ρ, u, η} under the assumption that f is smooth. By taking θ =
Nonlinear damping
We wish to build solutions of (5.1) as a limit of the following system of equations:
where q > γ + 1 and m > 3, with m ≥ q, will be determined later. We note that we add a nonlinear damping to the equation of η, not the equation of f . We will prove the existence of a solution {ρ, u, f, η, σ} to (5.5) satisfying
in Section 5.3. From (5.5), we have the following energy identity:
Therefore, we have
Moreover, for a sufficiently small δ,
holds. To prove this, we integrate (5.5a) over Ω.
where 0 < α < γ − 1. By taking a sufficiently small δ, we have
Next, we obtain higher integrability of ρ by using (3.9), with replacing the multiplication of ρ by ρ θ to (3.8) . We also have additional terms from the damping terms.
(5.11)
Integrating (5.11) over Ω × (0, T ), we have
(5.12)
Let θ = 2 3 γ. We only provide estimations involving δ. The rest of them are easily by the regularization. First, 14) where 0 < 
by taking q > γ + 1, which is close to γ + 1. Therefore, as δ → 0,
so we can pass to the limit to (5.5) to obtain a solution of (5.1) by the compactness argument in Section 4. Indeed,
so the analysis is easier than the case without .
Truncation of the pressure
In this section, we construct a solution of (5.5) by
where T (R) = (ρ ∧ R) γ + (η ∧ R) 2 and f ∧ R = min{f, R}. We will show at the end of this section that there exists a smooth solution to (5.17) such that
We begin with the energy identity of (5.17).
where
By taking s =
We can estimate η by using the same method. We set a =
We now estimate u by using the estimation of the heat equation ( [13] ). By writing (ρu)
for any s ≤ l ≤ ∞. Since q > γ+1 and m > 3, (m−1)a > 2s and (q−1)s > γs. Therefore, we can obtain (5.21) by (5.22), (5.23), and (5.24) provided that we can bound ρu
in (5.24). This can be done by a bootstrap argument. First, we take s = 2, l = ∞. Then,
We note that we do not have ∇ · σ − ∇η in the right-hand side of (5.26) due to the regularization. If γ < 2, then we take s =
. Therefore, we can conclude as the previous case. Now, we want to pass to the limit in R → ∞. Since
Therefore, we can recover a solution of (5.5) satisfying (5.6) in Section 5.2.
It remains to show (5.18). First, we show that ρ ∈ L ∞ (Ω × (0, T )). From the equation of ρ,
Moreover,
. By (5.28) and (5.29),
By the maximum principle, log ρ + Φ ≤ C. Therefore, ρ ∈ L ∞ (Ω × (0, T )). Next, we estimate ∇ρ.
We need to estimate derivatives of u. From standard parabolic estimates,
By taking r > 3,
. This completes the proof.
Appendix

Appendix 1: derivation of the equation of ψ
We take the time derivative to ψ = S 2 f ln f dτ . Then,
We obtain (I) and (II) by using the equation of f (1.10c).
f dτ.
(6.1)
2)
The proof of last equality in relation (6.2) requires the expression of these quantities in spherical coordinates. We refer the reader to [19] for further details.
(c) = where β is a C 1 function from [0, ∞] to R such that
We note that (6.7) makes sense because u ∈ L 2 (0, T ; H 1 loc (Ω)). We approximate ρ log ρ by β δ (ρ) = ρ log(ρ + δ). Then,
which verifies (3.6) in the sense of distributions. Next, we want to verify (3.8) by showing that each term in the right-hand side of (3.7) can be multiplied by ρ to be in L 1 (Ω × (0, T )).
• ρ × ρ γ is from ρ ∈ L γ+1 (Ω).
• Since ρ ∈ L ∞ (0, T ; L s ), s > 3 and η 2 ∈ L 1 (0, T ; L 3 ), the product ρη 2 is in L 1 (Ω × (0, T )).
• ρ(−∆) −1 ∇ · (∇ · σ − ∇η) ∼ ρ(σ − ηI) ∈ L ∞ (0, T ; L s (Ω)) × L 1 (0, T ; L 3 (Ω)). Since s > 3,
Since ρ ∈ L ∞ (0, T ; L s ), ρρu ⊗ u ∈ L 1 (Ω × (0, T )), with Finally, we verify (3.11) by passing to the limit in (3.10).
• ρ n log ρ n u n : We take g n = ρ n log ρ n and h n = u n . To verify the condition of g n , we can use (4.6).
• ρ n (−∆) −1 ∇(∇ · σ n − ∇η n ): We take g n = ρ n , h n = (−∆) −1 ∇ · (∇ · σ n − ∇η n ). This is possible because ∇ · σ n , ∇η n ∈ L 1 (0, T ; L 3 (Ω)) and ρ n ∈ L ∞ (0, T ; L s (Ω)), s > 3.
• We already show that ρ n (η n ) 2 converges to ρη 2 in the sense of distributions.
• ρ n (−∆) −1 ∇ · (ρ n u n ) and ρ n u n (−∆) −1 ∇ · (ρ n u n ): Since
we have (−∆)
loc (Ω)). Therefore, we can take g n = ρ n , ρ n u n and h n = (−∆) −1 ∇ · (ρ n u n ).
• ρ n (−∆) −1 ∂ i ∂ j (ρ n u n i u n j ) − u n · ∇(−∆) −1 ∇ · (ρ n u n ) = ρ n [u n j , R i R j ](ρ n u n ): First, we control the commutator. Since ∇u n ∈ L 2 (0, T ; L 2 (Ω)) and ρ n u n ∈ L 2 (0, T ; L p (Ω)), with
Let g n = ρ n u n i and h n = u n j . Then,
in the sense of distributions. Now, let g n = ρ n and h n = U n . Then, ρ n U n converges to ρU in the sense of distributions. This completes the proof of the formal argument in Section 3.4.

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