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

1 Introduction

In this paper we study the uniqueness of “small” (in an appropriate sense) solutions to a family of double-phase steady problems, arising in the analysis of non-Newtonian fluids. 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 Chorfi and Radulescu [8], Baraket, Chebbi, Chorfi, and Radulescu [2], and the review in Radulescu [25]. In the context of fluid mechanics, problems with more than one phase arise especially in the case of fluids 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 differential equations with variable exponents.

Here, instead that regularity or multiplicity of solutions, we focus on some specific 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⊂ℝ³

is that of uniqueness of weak solutions, under the assumption of smallness of the L³(Ω)-norm of u. (In this case the possible non-uniqueness is neither a special feature of the problem in ℝ³, 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 u1,u2∈W01,2(Ω) are weak solutions corresponding to the same external force f ∈ W^−1,2(Ω), then there exists ϵ₀ > 0 such that if ∥u₁∥₃ ≤ ϵ₀, then u₁=u₂; see the review in Galdi [17, Ch. IX.2]. The proof is based on writing the system satisfied by the difference U = u₁ − u₂ 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 H01(Ω)↪L6(Ω) .

In this way from (1.1) one easily gets, after integration by parts, that

which implies U = 0, provided that ϵ₀<ν₀/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 L³(Ω), at least when Ω is bounded.

We stress the critical role played by the exponent “two” of ∥∇U∥₂ 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 ∥∇u₁∥_3/2 ≤ ϵ₁ (for a possibly different small constant ϵ₁ > 0), by using a similar estimate for the convective term and observing that W^1,3/2(Ω)↪ L³(Ω). Note that the two alternative assumptions have the same scaling.

In the case of steady non-Newtonian fluids, the situation becomes more complex. If one considers for 1 < p<∞ the following system, describing a family of shear dependent fluids,

then, for any δ≥0 and for all ν₀ > 0, 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 W01,2(Ω) will be enough to disregard the effect 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 fluids 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

is just the prototypical example of a stress tensor with (p, δ)-structure. These results are reviewed for instance in [16, Sec. 2.2.1(c)].

A similar approach has been recently used by Gasiński and Winkert [18] to treat the following anisotropic double-phase scalar problem

in the case p=2<q < 3, with μ≥0 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 fluids becomes more complex when the linear part is missing (that is system (1.3) with ν₀=0 and, in the case p > 2, also δ=0), 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 fluids 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 definition in the Assumption 2:1) as in the example (1.4), it is well-known that the following point-wise estimate holds:

for all A, B symmetric matrices. For a review see for instance Růžička [24]. Hence, at least if δ>0 and p > 2, it follows that

which allows us to employ the same machinery.

On the other hand, the case 1 < p<2 presents some peculiarities as exploited in Blavier and Mikelić [6], which permits to prove uniqueness for small solutions if 95≤p<2 . (Note that 95 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 fluid

with 1 < p<2 < 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 6 5 ≤ p < 2 and q=q(p) large enough. Then, some extensions to an anisotropic problem are proved in Theorem 4.2.

2 Notation and preliminary results

In the sequel, Ω⊂ℝ³ will be a smooth and bounded open set. As usual, we write x=(x₁, x₂, x₃)=(x′, x₃), for all x ∈ ℝ³. If the boundary ∂Ω is at least of class C^0,1, then the normal unit vector n at the boundary is well defined. We recall that a domain is of class C^k,1, if for each point P ∈ ∂Ω, there are local coordinates such that in these coordinates we have P = 0 and ∂Ω is locally described by a C^k,1-function, i.e., there exist R_P,  RP′∈(0,∞),rP∈(0,1) and a C^k,1-function aP:BRP2(0)→BRP′1(0) such that

1. x∈∂Ω∩BRP2(0)×BRP′1(0)⟺x3=aPx′,

2. ΩP:=x∈R3:x′∈BRP2(0),aPx′<x3<aPx′+RP′⊂Ω,

3. ∇aP(0)=0, and ∀x′∈BRP2(0)∇aPx′<rP,

where Brk(0) denotes the k-dimensional open ball with center 0 and radius r > 0.

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 1 ≤ p ≤ ∞. As usual, p′=pp−1 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 L0p(Ω) denotes the subspace with zero mean value, while W01,p(Ω) is the closure of smooth and compactly supported functions with respect to the ∥ . ∥_1,p norm. We denote by W−1,p′(Ω):=W01,p(Ω)⋆ its dual space, with norm ∥ . ∥_−1,p′.

If Ω is bounded and if 1 < p<∞, the following two relevant inequalities hold:

1. the Poincaré inequality

   (2.1) ∃CP(p,Ω)>0:∥u∥p≤CP∥∇u∥p∀u∈W01,p(Ω);

2. the Korn inequality

   (2.2) ∃CK(p,Ω)>0:∥∇u∥p≤CK∥Du∥p∀u∈W01,p(Ω),

where Du denotes the symmetric part of the matrix of derivatives ∇u.

For 1 ≤ p<3 we have, as a combination of (2.1)-(2.2), also the Sobolev-type inequality

where p∗:=3p3−p .

When working with incompressible fluids it is natural to incorporate the divergence-free constraint directly in the definition of the function spaces. These spaces are built upon completing the space of solenoidal smooth functions with compact support (here denoted by C0,σ∞(Ω) in an appropriate topology. For 1 < p<∞, we define

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).

Since the range of the allowed p ∈ (1, 2) will play a relevant role, we first recall that the restriction p > 6/5 is quite natural for the problem, at least for what concerns existence of weak solutions. In the case ν₀=0, the weak formulation of (1.3) is in fact: find u∈W0,σ1,p(Ω) such that

To properly define the quadratic term, one needs (at least) that u∈Lloc2(Ω), which follows, for instance, if u ∈ W^1,p(Ω) for p≥65 . The basic a priori estimate obtained testing with u itself shows that, if f ∈ W^−1,p′(Ω), then ∥Du∥_p ≤ C. Next, Korn inequality and the embedding W01,p(Ω)↪L2(Ω) (which holds for p=6/5 in three space-dimensions) give that u⊗u ∈ L¹(Ω). This restriction on p is intrinsic to the problem, due to the growth of the convective term. The limiting case p=6/5 is excluded from the existence theorem due to some technical compactness arguments which are used in the proof.

The following existence result holds true.

3 On the double-phase problem

In this section, we consider the problem (2.7) with δ_p=δ_q=0 and we observe that–on one hand–the “good” estimates (lower bound) for the stress tensor of the difference are valid if 1 < p<2, 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 define 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 [16]. Taking the difference of two solutions u_i ∈ W0,σ1,q(Ω) corresponding to the same force f ∈ W^−1,q′(Ω), one gets for the difference U the following estimates:

obtained by using Hölder, Korn, and Sobolev inequalities, and also (2.10). The calculations are justified 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 u₁, 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 different 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:

Before proving the theorem, we observe that the significant case is when the “q-phase” is such that δ_q=0. Also the possible degeneracy δ_p=0 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 fluid 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-modified 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 first significant difference comes into play if q is smaller or larger than the space dimension, since this will imply estimates involving the Sobolev exponent q∗=3q3−q or, alternatively, estimates in L^∞(Ω).

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 first have a result which follows by integrating by parts:

Proof. The proof of the first equality is simply obtained by interchanging the dummy variable in the double summation

and using that Du₁ is, by definition, 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,u₁ in spaces in which C0,σ∞(Ω) is dense, provided that the integrals are well-defined (as an example Lemma 3.1 is valid for u,u1∈W0,σ1,q(Ω) , with q=9/5

The nonlinear term is now estimated with the following inequalities:

Proof. We observe that the integral ∫ Ω ( U ⋅ ∇ ) u 1 ⋅ U d x is finite by (2.9) since all functions belong to W0,σ1,q(Ω), hence all calculations we perform are completely justified.

We start from the case q < 3 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>0, we can freely multiply and divide by δ+Du1+|DU|2−p2≠0 . We use the Hölder inequality which is valid if

and Sobolev inequalities to get

The proof in the case ? q > 3 follows more or less the same lines and if the following inequality is satisfied

we can write

We can now give the proof of the main result of this paper.

Proof of Theorem 3.1. We write the difference between the two weak solutions of (2.7) and we use U=u1−u2∈ W0,σ1,q(Ω) as test function to get

Since the operator sq,0(⋅) is monotone, we get

and the right-hand side is finite due to the fact that q > 2

We now estimate the left-hand side by Proposition 2.1 and, for u_i at least in W0,σ1,p(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

by the Minkowski inequality the following holds:

and, if δp∈0,δ0 for some δ0>0 , then

Hence, in the case of the problem with a non-degenerate stress tensor Sp,δp , that is if δp>0 , 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 ∇u1q≤ϵ0 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 0 1 , 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