Anisotropically diffused and damped Navier-Stokes equations

The incompressible Navier-Stokes equations with anisotropic diffusion and anisotropic damping is considered in this work. 
For the associated initial-boundary value problem, we prove the existence of weak solutions and we establish an energy inequality satisfied by these solutions. 
We prove also under what conditions the solutions of this problem extinct in a finite time.

where Ω is a bounded domain of R N , N ≥ 2, ∂Ω is the compact boundary of Ω, and T > 0. Here, the vector function u = (u 1 , · · · , u N ) and the scalar function p are the unknowns of the problem. The vector function f = (f 1 , · · · , f N ) is a given problem data, D i u = (∂ i u 1 , . . . , ∂ i u N ) and ∂ i u j = 350 HERMENEGILDO BORGES DE OLIVEIRA that, due to the damping term, one obtains better existence results and some qualitative properties as the extinction in a finite time can be extended to all the range of the power-law exponent. The case of the non-damped problem (1.1)-(1.4), i.e. when in (1.2) κ i = 0 for all i ∈ {1, . . . , N }, was previously considered in [2,3], where it was proved the existence of weak solutions and it were established some properties of finite time extinction and large time behavior of the solutions. With respect to Navier-Stokes equations with anisotropic diffusion, its main attribute is that, with no need of a damping term, the existence results and many qualitative properties of the solutions can be improved (with respect to its isotropic versions) in almost all directions, or for particular choices of all the diffusion coefficients, with the possibility to achieve optimal results, at least in some directions. In this work, we consider the case in which the generalized Navier-Stokes equations are modified, not only by considering an extra damping term in the momentum equation, but also by assuming that this damping, as well the diffusion, might be fully anisotropic.
This article is organized as follows. In Section 2 we introduce the main concepts of the anisotropic function spaces we are going to work with and we define the notion of weak solution to the problem (1.1)-(1.4). The existence of weak solutions to our problem is established in Section 3 and there we also establish an energy inequality satisfied by these solutions. Properties of extinction in a finite time are studied in Section 4 under different conditions on the diffusion or on the damping terms. The notation used throughout this article and the main notions of the considered (isotropic) function spaces are largely standard in the literature of Partial Differential Equations (see e.g. [1,9]).
2. Weak formulation. Let us define the vectors σ and q in R N , whose components are the exponents of the anisotropic terms of damping and of diffusion considered in (1.2), by σ := (σ 1 , · · · , σ N ) and q := (q 1 , · · · , q N ), where 1 < σ i , q i < ∞ for all i ∈ {1, . . . , N }, and let us set In order to emphasize that σ and q are multicomponent, throughout the text we will use the notations − → σ and − → q . We define the anisotropic Banach spaces where L qi (Ω) is the usual Lebesgue space and W 1,qi . An important limitation of the anisotropic Sobolev space W 1, − → q (Ω), is that, for bounded domains Ω, the validity of Sobolev imbeddings is restricted to rectangular domains (see e.g. [7]). In fact, for rectangular domains Ω, the following imbedding is continuous (cf. [13, Theorem 1]) where q * := N q N −q and q := 1 . As a particular case of (2.1), it can be derived the following inequality [14,Theorem 1.2]). Moreover, the imbedding (2.1) is compact (cf. [13,Theorem 2]), and we denote this fact by In some situations it is possible to remove the restrictions on the shape's domain and to enlarge the interval of s for the validity of (2.1)-(2.3). Let us see this fact by defining In this case, we have (cf. Observe that for N = 2, q * > q + and therefore q * a = q * . But, if N > 2, it may well happen that q + > q * . See [2, Remark 2.1]) for more details and useful examples.
In order to introduce the notions of weak solutions we shall consider in this work, let us recall the well-known function spaces of Mathematical Fluid Mechanics. Given q such that 1 < q < ∞, we set V := {v ∈ C ∞ 0 (Ω) : div v = 0}, H := closure of V in the norm · L 2 (Ω) , and V q := closure of V in the norm · W 1,q (Ω) . Let us now define the anisotropic analogue The anisotropic Bochner space considered here is defined by . Note that, for a bounded domain Ω and for a finite T , the continuous imbedding L q is separable and reflexive. The dual spaces of L qi (0, T ; , respectively, where V qi and V − → q stay for the dual spaces of V qi and V− → q . A vector field u is a weak solution to the problem (1.1)-(1.4), if: Note also that v ∈ L θ (Ω) is necessary to control the boundedness of the convective integral term when u merely belongs to L ∞ (0, On the other hand, the assumption (2.5) is needed to control the first integral when u and v solely belong to the mentioned spaces.
3. Existence. First we observe that, according to what is customary in Mathematical Fluid Mechanics, the determination of the pressure p is not a problem. In fact, after we determine u, we can recover p by applying de Rham's theorem (see e.g. [9]). Therefore, with regard to the existence, in this work we shall be concerned only with the existence of the unknown field u.
Proof. The proof of Theorem 3.1 will be split into several steps.
Step 1: Existence of approximative solutions. For the smallest integer s > 1 + N 2 , we define V s := closure of V in W s,2 (Ω). Let {v k } k∈N be a set of non-trivial solutions v j of the following spectral problem associated to the eigenvalues λ j > 0: : for k = 1, . . . , m; where C is an independent of m positive constant. Thus, from the Theory of the ODEs, we can take t m = T .
Step 2: Convergence of the approximative solutions. Due to (3.4) and by means of separability and reflexivity, there exists u and a subsequence (still denoted by) u m such that On the other hand, it can be proved that for any i ∈ {1, . . . , N }  By means of reflexivity, we have Now, we observe that by the choice of s, the continuous imbedding holds and, by (2.3) with q * a , the anisotropic compactness imbedding Step 3: Passing to the limit. Fixing k, we pass the equation (3.2) to the limit m → ∞ by using (3.6), (3.8), (3.10), (3.15) and (3.16), and yet observing the definition of the space V s given in Step 1, we obtain for all t ∈ [0, T ] The boundedness follows by using the imbedding V− → q → V q − and the anisotropic Sobolev imbedding (2.1) which holds for Step 4: Use of the monotonicity. Since the diffusion term is the sum of N possible different diffusion terms which are strictly monotonous, the anisotropic diffusion term is strictly monotonous: Consequently, appealing to this strict monotonicity property, we can construct a suitable test function v ∈ L − → q (0, T ; V− → q ) to use the Minty trick in the spirit of Lions [9, pp. 212-215]) and to identify For the application of the this reasoning, one needs to show that In [3] it is proved that (3.19) holds provided that and q − > 2 . (3.20) Finally, we observe that (3.18) implies (3.13) and the requirements of (3.18) and (3.20) follow from the assumption that q * a ≥ q * .
In the next result is established the energy inequality satisfied by the solutions to the problem (1.1)-(1.4).
Note that the assumption f = −div F is made only to simplify the exposition.
Proof. We proceed similarly as in the proof of Theorem 3.1, but by taking f = −div F in the equation (3.2). We multiply this equation by c m k , we add up from k = 1 until k = m and we integrate the resulting equation between t 0 and t 1 , with t 0 < t 1 and t 0 , t 1 ∈ [0, T ). After all, we obtain the following energy equality Then by (3.3) and (3.5)-(3.7), and by a classical property of weak limits, we obtain for all t 0 , t 1 ∈ [0, T ], with t 0 < t 1 . Thus we can write for every t, t + ∆t ∈ [0, T ], with t > 0, In consequence, every term on the right-hand side of the previous inequality has a limit, for all t ∈ [0, T ], as ∆t → 0. This in turn yields the existence of a limit of the left-hand side of this inequality, for all t ∈ [0, T ], as ∆t → 0. Whence we can write for all t ∈ [0, T ] Then making use of Young's inequality, we obtain (3.21). ,
Proof. We first observe that, since (2.5) holds, we can use the anisotropic inequality (2.2) with s = 2. Then making use of the notations (4.1)-(4.2) and using the algebraic inequality A a × B b ≤ (A + B) a+b , which is valid for any real numbers A, B, a, b, with A, B ≥ 0 and a, b > 0, we get Then plugging (4.5) into the energy inequality (3.21), we get for all t ∈ [0, T ]. Due to assumption (4.3), we have 0 < µ < 1. Thus we are precisely in the conditions of [3, Theorem 5.1] and the proof follows in the same way.
Observe that for the validity of Theorem 4.1, the damping term is needless. In this particular case and if we consider the diffusion to be isotropic, i.e. when, in the equation (1.2), κ i = 0 and q i = q for all i ∈ {1, . . . , N }, then, from (4.3), we recover the condition q < 2 that characterizes shear-thinning fluids for which it is well-known the phenomenon of extinction in a finite time (see e.g. [1,11]). Note also that in the important case of q * ≥ q + , the assumptions (2.5) and (4.3) show us that the property of finite time extinction holds provided that The following result gives us the conditions for the validity of this property when 1 < Then the assertions of Theorem 4.1 also hold, but with µ now defined in (4.9) below.
Proof. The same reasoning used to prove (4.5), but now preceded by interpolation between q * a and σ − , yields Observe that θ ∈ (0, 1) provided that q * a > 2 > σ − or that σ − > q * a > 2. In this case, we plug (4.8) into the energy inequality (3.21) to get the differential inequality (4.6), but with the exponent of nonlinearity defined by (4.9) Since θ ∈ (0, 1), we have by assumption (4.7), that θ As a consequence we also have 0 < µ < 1. Again we are in the conditions of [3, Theorem 5.1] and the proof follows in the same way.
If in the proof of Theorem 4.2, we let be θ = 1 or θ = 0, then q * a = 2 or σ − = 2, respectively, and we fall in the case of Theorem 4.1 for which the extinction in a finite time only holds for 1 σi , which, in the isotropic case, i.e. when σ i = σ and q i = q for all i ∈ {1, . . . , N }, reduces to the well-known condition of fast diffusion σ < q (see [1]).