On the uniqueness for weak solutions of steady double-phase fluids


 We consider a double-phase non-Newtonian fluid, described by a stress tensor which is the sum of a p-Stokes and a q-Stokes stress tensor, with 1 < p<2 < q<∞. For a wide range of parameters (p, q), we prove the uniqueness of small solutions. We use the p < 2 features to obtain quadratic-type estimates for the stress-tensor, while we use the improved regularity coming from the term with q > 2 to justify calculations for weak solutions. Results are obtained through a careful use of the symmetries of the convective term and are also valid for rather general (even anisotropic) stress-tensors.


Introduction
In this paper we study the uniqueness of "small" (in an appropriate sense) solutions to a family of doublephase steady problems, arising in the analysis of non-Newtonian uids. The interest in double-phase problems started with the celebrated result by Zhikov [29] concerning the Lavrentiev phenomenon. The problem was set in the framework of functionals with (p, q)-growth, for which we refer also to the pioneering paper by Marcellini [22]; recent results can be found in the works of Esposito, Leonetti, and Mingione [15], Baroni,Colombo,and Mingione [3], and Colombo and Mingione [9], especially in the context of regularity of minimizers. For results regarding the applications to spectral analysis and multiplicity of solutions, see Chor and Radulescu [8], Baraket, Chebbi, Chor , and Radulescu [2], and the review in Radulescu [25]. In the context of uid mechanics, problems with more than one phase arise especially in the case of uids with complex rheologies, as introduced by Růžička [23], where the modeling leads to a problem with variable exponent. See also the review in Rădulescu and Repovš [26] for general problems involving partial di erential equations with variable exponents.
Here, instead that regularity or multiplicity of solutions, we focus on some speci c problems concerning uniqueness in the class of small weak solutions.
In order to introduce the problem, we recall that a basic result for the steady Navier-Stokes equations in a smooth and bounded domain Ω ⊂ R −ν ∆u + (u · ∇) u + ∇π = f in Ω, div u = in Ω, u = on ∂Ω, (1.1) is that of uniqueness of weak solutions, under the assumption of smallness of the L (Ω)-norm of u. (In this case the possible non-uniqueness is neither a special feature of the problem in R , nor coming from the fact that it is a nonlinear system with the divergence-free constraint. Possible multiplicity of solutions is a common result even for semi-linear elliptic scalar equations.) More precisely, for the steady Navier-Stokes equations (1.1), it holds that if u , u ∈ W , (Ω) are weak solutions corresponding to the same external force f ∈ W − , (Ω), then there exists ϵ > such that if u ≤ ϵ , then u = u ; see the review in Galdi [17,Ch. IX.2]. The proof is based on writing the system satis ed by the di erence U = u − u and testing by U itself (a procedure which is legitimate in the steady case also for weak solutions). Next, one uses the inequality with a constant C depending only on Ω, obtained by application of the Hölder inequality and of the Sobolev embedding H (Ω) → L (Ω). In this way from (1.1) one easily gets, after integration by parts, that ν ∇U ≤ Cϵ ∇U , which implies U = , provided that ϵ < ν /C. Observe that from the energy estimate valid for weak solutions one knows a priori just that hence, uniqueness for small forces/large viscosities follows by using a Sobolev embedding to ensure smallness in L (Ω), at least when Ω is bounded. We stress the critical role played by the exponent "two" of ∇U in the energy estimate (1.2). It appears at the same time as exponent in: i) the lower bound for the dissipative term; and ii) the upper bound for the quantity obtained by estimating of the convective term. If there is a mismatch in the powers, then this easy but powerful argument may fail.
The same argument can be also applied if ∇u / ≤ ϵ (for a possibly di erent small constant ϵ > ), by using a similar estimate for the convective term and observing that W , / (Ω) → L (Ω). Note that the two alternative assumptions have the same scaling.
In the case of steady non-Newtonian uids, the situation becomes more complex. If one considers for < p < ∞ the following system, describing a family of shear dependent uids, then, for any δ ≥ and for all ν > , the same argument valid for the Navier-Stokes equations can be applied. In fact, the additional stress tensor is monotone and the following inequality holds: The above inequality and the fact that weak solutions still belong to W , (Ω) will be enough to disregard the e ect of the additional non-linear stress tensor. It is possible to treat problem (1.3) in the same way as for the unperturbed Navier-Stokes equations, at least for what concerns uniqueness. The argument applies also to uids such that the stress tensor has, in addition to the linear part, a nonlinear stress tensor S(Du) = S p,δ (Du) with a (p, δ)-structure (see Section 2). We recall that the stress tensor A similar approach has been recently used by Gasiński and Winkert [18] to treat the following anisotropic double-phase scalar problem in Ω, u = on ∂Ω, (1.5) in the case p = < q < , with µ ≥ Lipschitz continuous, under appropriate growth conditions on f . Our results improve also those in [18], since we consider a vector valued problem, with the constraint of divergence-free, and with term F(x, u, ∇u) = f(x) − (u · ∇) u.
In addition to the results above, we observe that the problem of uniqueness for non-Newtonian uids becomes more complex when the linear part is missing (that is system (1.3) with ν = and, in the case p > , also δ = ), since one cannot take advantage of the classical estimates. In fact, the convective term is still quadratic as in (1.1); this has to be balanced in some way by a non-quadratic term. Moreover, in the context also of electro-rheological uids uniqueness for small (and smooth) solutions is proved in Crispo and Grisanti [10,11], at least in the non-degenerate case.
Note that for a stress tensor with (p, δ)-structure (see precise de nition in the Assumption 2.1) as in the example (1.4), it is well-known that the following point-wise estimate holds: On the other hand, the case < p < presents some peculiarities as exploited in Blavier and Mikelić [6], which permits to prove uniqueness for small solutions if ≤ p < . (Note that is the critical value to apply the classical monotonicity argument and to use the solution itself as a test function in the weak formulation.) We will review these results and provide more details in Section 2.1, explaining how we improve them.
Here, we will then consider as basic example the following boundary value problem for a double-phase non-Newtonian uid with < p < < q, proving uniqueness of small solutions (recall again that if at least one among p or q is equal to 2 the result is trivial). Our main aim is to consider the degenerate case and to exploit the interplay between the p-growth and the q-growth in double-phase problems, as a way to enforce uniqueness, which cannot be proved in the single-phase cases. In addition, a non-standard estimation of the convective term using its symmetries is also employed. The main result we prove, see Theorem 3.1, is the uniqueness of small solutions, for < p < and q = q(p) large enough. Then, some extensions to an anisotropic problem are proved in Theorem 4.2.

Notation and preliminary results
In the sequel, Ω ⊂ R will be a smooth and bounded open set. As usual, we write x = (x , x , x ) = (x , x ), for all x ∈ R . If the boundary ∂Ω is at least of class C , , then the normal unit vector n at the boundary is well de ned. We recall that a domain is of class C k, , if for each point P ∈ ∂Ω, there are local coordinates such that in these coordinates we have P = and ∂Ω is locally described by a C k, -function, i.e., there exist R P , R P ∈ ( , ∞), r P ∈ ( , ) and a C k, -function a P : where B k r ( ) denotes the k-dimensional open ball with center and radius r > . For our analysis, we will use the customary Lebesgue (L p (Ω), . p) and Sobolev spaces (W k,p (Ω), . k,p ) of integer index k ∈ N, with ≤ p ≤ ∞. As usual, p = p p− denotes the conjugate exponent. We do not distinguish between scalar and vector valued function spaces, we just use boldface for vectors and tensors. We recall that L p (Ω) denotes the subspace with zero mean value, while W ,p (Ω) is the closure of smooth and compactly supported functions with respect to the . ,p norm. We denote by W − ,p (Ω) := (W ,p (Ω)) * its dual space, with norm . − ,p .
If Ω is bounded and if < p < ∞, the following two relevant inequalities hold: 1) the Poincaré inequality where Du denotes the symmetric part of the matrix of derivatives ∇u. For ≤ p < we have, as a combination of (2.1)-(2.2), also the Sobolev-type inequality where p * := p −p . When working with incompressible uids it is natural to incorporate the divergence-free constraint directly in the de nition of the function spaces. These spaces are built upon completing the space of solenoidal smooth functions with compact support (here denoted by C ∞ ,σ (Ω)) in an appropriate topology. For < p < ∞, we de ne For the nonlinear stress tensors, we make the following assumption of being with (p, δ)-structure, which is a generalization of the example from (1.4).
, satis es S(P) = S P sym , and S( ) = . Moreover, we assume that S has (p, δ)-structure, i.e., there exist p ∈ ( , ∞), δ ∈ [ , ∞), and constants C , C > such that are satis ed for all P, Q ∈ R × with P sym ≠ and all i, j, k, l = , . . . , . The constants C , C , and p are called the characteristics of S.

Remark 2.1.
We would like to emphasize that, if not otherwise stated, the constants depend only on the characteristics of S, but are independent of δ ≥ .
De ning for t ≥ a special N-function φ by we can replace C i δ + |P sym | p− on the right-hand side of (2.3a) and (2.3b) by C i φ |P sym | , for i = , . Next, the shifted functions are de ned for t ≥ by In the following proposition, we recall several useful results, which will be frequently used in the paper. The proofs of these results and more details can be found in [4,13,14,27]. Many inequalities can be written in a compact form by means of the following tensor valued function Since the range of the allowed p ∈ ( , ) will play a relevant role, we rst recall that the restriction p > / is quite natural for the problem, at least for what concerns existence of weak solutions. In the case ν = , the weak formulation of (1 To properly de ne the quadratic term, one needs (at least) that u ∈ L loc (Ω), which follows, for instance, if u ∈ W ,p (Ω) for p ≥ . The basic a priori estimate obtained testing with u itself shows that, if f ∈ W − ,p (Ω), then Du p ≤ C. Next, Korn inequality and the embedding W ,p (Ω) → L (Ω) (which holds for p ≥ / in three space-dimensions) give that u ⊗ u ∈ L (Ω). This restriction on p is intrinsic to the problem, due to the growth of the convective term. The limiting case p = / is excluded from the existence theorem due to some technical compactness arguments which are used in the proof.
The following existence result holds true.
for some constant C depending on the domain and on the characteristics of the stress tensor.
The proof of existence of weak solutions requires a precise use of the monotonicity of the operator, but for p < the usual Browder-Minty approach (as employed by Lions [21] and Ladyžhenskaya [20] also for values of p > ) is not directly applicable and one has to resort to more technical arguments with bounded or even Lipschitz truncation, plus an appropriate divergence correction, see the reviews in Breit [7] and Růžička [24].
Here, we do not discuss the existence of weak solutions, which easily follows also for the problem (1.6), by employing similar arguments without any relevant change. Hence, we can state the following theorem.

7)
has at least a weak solution u ∈ W ,q ,σ (Ω), which satis es and which satis es the following inequality:

8)
for some constant C depending only on the characteristics of Sq and Ω.

. On problems without linear part: some known uniqueness results
When considering the problem (1.3) in the case ν = δ = and p > , the problem of uniqueness (even of small solutions) is completely open, since the (sharp) estimates from Proposition 2.1 seem to be not suitable to "absorb" the convective term. This is determined by a "gap" between the powers in the lower bound for the stress tensors and those in the upper bound for the convective term.
On the other hand, in the case ν = and for some p < , the following result of uniqueness for weak solutions is proved in Blavier and Mikelić [6], under appropriate smallness of both weak solutions. The available proofs of uniqueness go through considering the equation satis ed by the di erence and testing with the di erence of the solutions, U := u − u . To this end, one needs to be able to rigorously write the integrals (where the equality derives from integration by parts and from the divergence-free constraint) and to prove proper upper bounds (similar to (1.2), already used in the Newtonian case). By using the information u i ∈ W ,p ,σ (Ω), and by the Hölder inequality, we get that if then the following estimate holds: Hence, by using Sobolev and Korn inequalities, we get The uniqueness result from [6] exploits this estimation, together with the lower bound

On the double-phase problem
In this section, we consider the problem (2.7) with δp = δq = and we observe that -on one hand-the "good" estimates (lower bound) for the stress tensor of the di erence are valid if < p < , hence one would like to have one phase with this properties; On the other hand, to handle the convective term larger (than 2) values of the exponent q will provide the estimates needed to rigorously de ne the integrals involving the convective term. First, we have the existence Theorem 2.2 for weak solutions. Next, we observe that the same argument as in [6] can be directly adapted to the problem (2.7) (of which system (1.6) represents a particular case), to produce the following elementary -but original-uniqueness result. Proof of Proposition 3.1. The proof is a simple adaption of results in [6]. Taking the di erence of two solutions u i ∈ W ,q ,σ (Ω) corresponding to the same force f ∈ W − ,q (Ω), one gets for the di erence U the following estimates: obtained by using Hölder, Korn, and Sobolev inequalities, and also (2.10). The calculations are justi ed if which follows if q is as in (3.1). Then, if one assumes the smallness of the L q -norm of the gradient or u , the following estimate holds: which allows again to conclude uniqueness (exactly as in the previous case) by the same argument developed in [6].
In order to improve this result, here we follow a slightly di erent approach which is based on more precise point-wise estimates for the stress tensor corresponding to the "p-phase." This will allow us to require less restrictive conditions on q. The main result we prove is the following one: Let u , u be weak solutions of (2.7) corresponding to the same f ∈ W − ,q (Ω). Then, there exists a constant ϵ = ϵ (p, q, Ω, f − ,q ) > such that if at least one solution satis es Before proving the theorem, we observe that the signi cant case is when the "q-phase" is such that δq = .
Also the possible degeneracy δp = of the p-phase improves previously known results. Moreover, we think that an interesting further development would be the study of the uniqueness for a single-phase uid by using the results of "higher regularity" from Crispo and Maremonti [12]. This will hold provided that: a) the results from [12] can be adapted to the non-modi ed p-Stokes system; and b) an explicit estimation of some constants related with singular integrals is available. We make some remarks to explain the improvements in the various ranges, with respect to the results in Proposition 3.1. A rst signi cant di erence comes into play if q is smaller or larger than the space dimension, since this will imply estimates involving the Sobolev exponent q * = q −q or, alternatively, estimates in L ∞ (Ω).

Remark 3.2. Concerning the case
we can see that the restriction on the range of p is made to enforce q > , since the calculations leading to this estimate are valid for q > (cf. the proof of Lemma 3.2). Moreover, we also have that hence the condition (i) is less restrictive than (3.1), coming from Proposition 3.1 (in the same range of p).

Remark 3.3. Concerning the case
the restrictions are made to impose that ≤ p( −p) p− < , hence to make possible the choice of some < q < . The lower bound on q is requested to have a non-trivial result, and the upper bound is requested to use Lemma 3.2.
Moreover, we have again and this condition is less restrictive than that from Proposition 3.1 (in the same range of p).
Remark 3.4. The restriction p < / is already present in Proposition 3.1 since beyond that value the result is included in Proposition 3.1. Again, the condition (iii) is less restrictive than the one previously obtained, since the following inequality holds: The proof of Theorem 3.1 is heavily based on the following inequality coming from an application of Proposition 2.1: and then using the following two lemmas about the convective term. The estimation of the tri-linear term we use is not the direct one as in (2.9). We bound the convective term exploiting some of its symmetries. This fact is relevant since in the stress tensors present in the equations (1.6) the velocity enters only through the symmetric gradient. We rst have a result which follows by integrating by parts: Lemma 3.1. The following equalities hold: Proof. The proof of the rst equality is simply obtained by interchanging the dummy variable in the double summation and using that Du is, by de nition, a symmetric tensor. The second equality follows by integrating by parts: due to the vanishing of the boundary trace of u we obtain Then adding the null term the second identity follows.
The same result clearly holds also for u, u in spaces in which C ∞ ,σ (Ω) is dense, provided that the integrals are well-de ned (as an example Lemma 3.1 is valid for u, u ∈ W ,q ,σ (Ω), with q ≥ / ). The nonlinear term is now estimated with the following inequalities: Lemma 3.2. Let U, u ∈ W ,q ,σ (Ω) for some q ≥ and let δ > be any positive number. If q < , (hence if W ,q (Ω) ⊂ L q * (Ω) for q * = q −q < ∞), then the following inequality holds: If q > , then W ,q (Ω) ⊂ L ∞ (Ω), and the following estimate holds: Finally, if q = , then W ,q (Ω) ⊂ L s (Ω) for all s < ∞, and the following estimate holds: Proof. We observe that the integral Ω (U · ∇) u · U dx is nite by (2.9) since all functions belong to W ,q ,σ (Ω), hence all calculations we perform are completely justi ed.
We start from the case q < and we observe that the estimate involves a "natural quantity" related to a stress tensor with (p, δ)-structure. We use directly Lemma 3.1, and since δ > , we can freely multiply and divide by δ + |Du | + |DU| −p ≠ . We use the Hölder inequality which is valid if and Sobolev inequalities to get The proof in the case q > follows more or less the same lines and if the following inequality is satis ed We can now give the proof of the main result of this paper.
Proof of Theorem 3.1. We write the di erence between the two weak solutions of (2.7) and we use U = u −u ∈ W ,q ,σ (Ω) as test function to get Since the operator S q, ( · ) is monotone, we get (3.4) and the right-hand side is nite due to the fact that q > . We now estimate the left-hand side by Proposition 2.1 and, for u i at least in W ,p ,σ (Ω), the following inequality holds: Next, the last term from the right-hand side of the inequalities proved in Lemma 3.2 can be estimated for δ = δp by observing that since for α ∈ ( , ) (x + y) α ≤ x α + y α ∀ x, y ≥ , by the Minkowski inequality the following holds: and, if δp ∈ [ , δ ] for some δ > , then Hence, in the case of the problem with a non-degenerate stress tensor S p,δp , that is if δp > , by collecting the lower bound for the left-hand side of (3.4) and the upper bound for the right-hand side (using Lemma 3.2 with δ = δp) and simplifying similar terms, we get We use now the hypothesis ∇u q ≤ ϵ and, after squaring both sides,w we get where C depends on the data of the problem, since we used the a priori estimate in W ,q (Ω) for both solutions. The proof follows now as in the previous case. In fact, by using (2.10) and (2.11), we get implying uniqueness, provided that ϵ is small enough.
In the degenerate case δp = , we observe that from the (p, )-structure we get hence the integral from the left-hand side is well de ned. Then, we show that the inequalities from Lemma 3.2 are valid also with δ = . This can be seen by writing the estimates in Lemma 3.2 for some < δ ≤ and taking the inferior limit of the right-hand side as δ → . In fact, since p < we have the following monotonic increasing convergence which implies convergence of the corresponding integrals. This shows convergence of the second term from the right-hand side of estimates from Lemma 3.2. Moreover, since p < , we have the uniform bound which allows us to use the Lebesgue dominated convergence, to handle the last term in the right-hand side of estimates from Lemma 3.2. After having justi ed the limiting step δ → , one obtains again (3.5) and the proof proceeds as in the non-degenerate case.

Some remarks for anisotropic (weighted) double-phase problems
In the analysis of double-phase problems, it is also interesting to consider, as done in [9,18,25], families of anisotropic double-phase problems. We also adapt our uniqueness results to the following problem where µ is a non-negative regular function and r > . Stress-tensors of this type occur in uid mechanics in certain turbulence models introduced by Baldwin and Lomax [1]. The mathematical analysis in terms of existence of weak solutions has been recently developed by one of the authors and D. Breit in [5]. In this case, it makes sense (from the modeling point of view based on Prandtl mixing length), to use as function µ the distance of x from the boundary of being written in rotational form. The system resembles the p-curl system studied in certain problems in electromagnetism, especially in mathematical model for superconductors, see Yin [28]. The value for the power of the distance function is critical in terms of being able to recover estimates on the full gradient from those valid for the weighted curl, by means of the theory of weighted Sobolev spaces and Muckenhoupt weights. The expression (4.1) is the rotational form of the stress tensor − div(d x)|Du(x)|Du(x) , which is a weighted version of a stress tensor with "( , )-structure," within the notation of Assumption 2.1.
In this section, we will see how to generalize the uniqueness results when S q,δ (Du), the stress tensor in (1.6), is replaced by − div d α (x)|Du(x)| r− Du(x) for some r > , with < α < r − .
To handle this term we rst recall a well-known lemma about the distance function d(x), see for instance Kufner [19].

Lemma 4.1.
Let Ω be a domain of class C , . There exist constants < c , c ∈ R such that From the above result we get the following weighted estimate We can now state the following existence theorem: Theorem 4.1. Let Ω ⊂ R be smooth and bounded and let S p,δp satisfy assumption 2.1 for some δp ≥ and for some p > / . Let r > be given and α ≥ . Then, for all f ∈ W − ,p (Ω), the problem − div S p,δp (Du) − div d α (x)|Du| r− Du + (u · ∇) u + ∇π = f in Ω, div u = in Ω, u = on ∂Ω, (4.2) has at least a weak solution u ∈ W ,p ,σ (Ω), such that for some constant depending only on p and Ω, but not on the solution. Moreover, if < α < r − , then