The purpose of this work is to study well–posedness and approximation results, on weighted spaces, for some models of non–Newtonian fluids on convex polyhedral domains. To be specific we will study the following problem
We assume the domain
to be a convex polyhedron, and for a vector fieldwe denote by
its symmetric gradient. Specific assumptions for the stress tensorand for the data and will be made explicit as we study particular models.
We must make an immediate comment regarding the space dimension. While the three dimensional case is the most significant from the physical point of view, our restriction to this case is of purely technical nature. This is because many of our results heavily rely on Hölder estimates for the derivatives of the Green matrix; see section 2.4. As far as we are aware, some of these are not available in the literature in either dimension two or higher than three. As soon as these become available, our results will readily extend to these dimensions as well.
The main source of difficulty and originality in this work can be summarized as follows. First, most of the well–posedness results for non–Newtonian fluids of the form (1) are presented for domains that are at least ; see, for instance, [MR2001659, MR3582412, MR2272870, MR2338412, MR2471134]. However, this assumption is not amenable to finite element discretizations. For this reason we focus on convex polyhedra, and we are able to provide approximation results for each one of the models that we consider. Second, we allow the data to be singular. Even in the linear case, i.e., , for a constant , the study of well–posedness results on convex polyhedra is far from being trivial. Here, by singular, we mean that and where for ; see section 2 for notation. This allows to have even measure valued forcings. Finally, we may allow the constitutive relation to be degenerate, as naturally appears when considering the Smagorinski model of turbulence described in section 6. In such a setting, problem (1), once again, must be understood in suitably weighted Sobolev spaces.
The history of the analysis and approximation of classes of non–Newtonian fluids is too vast and deep to even attempt to provide a complete description here. It can, for instance, be traced back to the work of Ladyženskaja [MR0155092, MR0241832, MR0226907], and the famous model that now bears her name; see also [MR1089323]. Other classes of non–Newtonian fluids that are similar to those we considered here have been studied in [MR1602949, MR1348587]. Regarding approximation, to our knowledge, some of the first works that deal with finite element discretizations of non–Newtonian fluids are [MR1034917, MR1069652, MR1211613]. The estimates of these works were later refined and improved in [MR1213411, MR1301740]. This last reference, introduced the concept of quasi–norm error bounds, and led to further developments using Orlicz spaces and shifted N–functions that were proposed, for instance, in [MR2914267]. Similar estimates, but via different arguments, were obtained in [MR3042563].
We organize our presentation as follows. In section 2 we present notation and gather some well–known facts that shall be useful for our purposes. In particular, we present regularity results for the classical Stokes problem in convex polyhedra and suitable Hölder estimates on the associated fundamental solution. We also recall some facts about the approximation properties of finite elements in standard and weighted Sobolev spaces. Section 3 is our first original contribution. Via a duality argument we obtain an –error estimate for the discretization of the Stokes problem when the forcing is a general Radon measure. This estimate improves our previous work [DOS:19]. The fundamental solution estimates are used in section 4 to show that for every , and every , the Stokes problem is well–posed on weighted spaces . The study of non–Newtonian models begins in section 5, where we extend the well–posedness results of [MR3582412] to the case of convex polyhedra. Finally, in section 6, we study a variant of the well–known Smagorinski model of turbulence, which was originally developed in [MR3488119], and aims at reducing the well–known overdissipation effects that this model presents near walls [MR3494304].
We will mostly adhere to standard notation. For and we denote by
its average over . Spaces of vector valued functions and its elements will be indicated with boldface. Since we will mostly deal with incompressible fluids, we must indicate a way to make the pressure unique in these. To do so, for we denote by the space of functions in that have zero averages.
The relation indicates that there is nonessential constant such that . By we mean . Whenever , we indicate by its Hölder conjugate. For a cube with sides parallel to the coordinate axes we denote by the length of its sides. If is a cube, and , we denote by the cube with same center but with sidelength .
One of the tools that will allow us to deal with singular sources, and nonstandard rheologies is the use of weighted spaces and weighted norm inequalities. A weight is an almost everywhere positive function in . Let , we say that a weight is in the Muckenhoupt class if [MR1800316, MR2491902, MR1774162]
where the supremum is taken over all balls in . We call , for , the Muckenhoupt characteristic of the weight . We observe that, for , there is a certain conjugacy in the classes: if and only if . Finally, we note that for all .
Let and . On weighted spaces the following inf–sup condition holds
see [DOS:19, Lemma 6.1]. This estimate will become useful in the sequel.
Let be an interior point of and . Define
The weight provided that . Notice that there is a neighborhood of where has no degeneracies or singularities. This observation motivates us to define a restricted class of Muckenhoupt weights [MR1601373, Definition 2.5].
[class ] Let be a Lipschitz domain, and . We say that belongs to if there is an open set , and such that:
In [OS:17infsup] it was shown that, provided the weight belongs to this class, the Stokes problem is well–posed on weighted spaces and Lipschitz domains. One of the highlights of this work is that we, in a sense, remove this restriction on the weight.
2.2 Maximal operators
For , the Hardy–Littlewood maximal operator is defined by [MR1800316, Chapter 7]
where the supremum is taken over all cubes containing . One of the main properties of the Muckenhoupt classes introduced above is that, for , the maximal operator is continuous on [MR1800316, Theorem 7.3].
We will also make use of the sharp maximal operator, which is defined, for , by [MR1800316, Chapter 6, Section 2]
The supremum is taken over all cubes that contain . It is important to notice that, when bounding the sharp maximal operator it suffices to bound the difference between and any constant . In fact,
2.3 The Stokes problem
Here we collect some facts on the Stokes problem that are well–known and will be used repeatedly. In other words, we set the constitutive relation , where . Problem (1) thus becomes
Problem (8) has a unique solution provided and , with a corresponding estimate. We also have the following regularity result.
[regularity] Let be a convex polyhedron. If and , then the solution to (8) is such that
See [MR977489], [MR1301452], and [MR2987056, Corollary 1.8].
Problem (8) is also well–posed in spaces.
[well–posedness in ] Let and be a convex polyhedron. If and , then problem (8) has a unique solution that satisfies the estimate
Evidently, we only need to comment on the case . See the first item in Section 5.5 of [MR2321139] for a proof of this result when . Using the equivalent characterization of well–posedness via inf–sup conditions, one can deduce well–posedness for .
[equivalence] In the literature, the Stokes problem is usually presented with the term replaced by . Using the elementary identity
2.4 The Green matrix
We introduce the Green matrix for problem (8) as follows [KMR, MR2808700]. Let be such that
then we represent the entries of as
where the pairs are distributional solutions of
Here, denotes the canonical basis of and the Dirac distribution. For uniqueness, we also require that
The existence and uniqueness of the Green matrix follows from [KMR, Theorem 11.4.1]. Note that for and . The importance of this matrix lies in the fact that it provides a representation formula for the solution of (8). In particular, we have that [KMR, Section 11.5]
where denotes a suitable duality pairing. This representation shall become useful in the sequel.
A useful property of the Green matrix is the estimates given below.
[Hölder estimates] Let be a convex polyhedron. There is a that depends only on the domain, such that for any multiindices , the Green matrix satisfies
for with and
for with , whenever and . Here, and
Reference [MR2808700] provides the claimed estimates for the Green’s matrix for the case where the operator acting on the –components of the matrix is the Laplacian. It suffices to proceed with the change of variables described in Remark 2.3 and observe that the –components of the Green’s matrix have the same differentiability properties as derivatives of the –components.
2.5 Finite elements
Many of the results we wish to discuss involve error estimates for finite element schemes. For this reason we assume that we have at hand, for each , finite dimensional spaces that satisfy, for all and , the compatibility condition
Moreover, we require that these spaces have approximation properties that are usually verified by finite element spaces over quasiuniform meshes of size . In particular, we require the existence of a stable operator that preserves the space and satisfies the error estimates
and, for all and ,
We finally comment that, since the continuous inf–sup condition (3) holds, (10) is equivalent to the existence of a so–called Fortin operator [Guermond-Ern, Lemma 4.19], that is an operator that preserves the divergence, i.e.,
and is stable. This immediately implies that possesses quasioptimal approximation properties, i.e., for all ,
Within the unweighted setting, examples of such pairs are very well–known in the literature [MR851383, CiarletBook, Guermond-Ern]. For extensions to the weighted case, we refer the reader to [DOS:19, Section 6]. An operator satisfying (11) and (12) has been constructed in [NOS3]. Usually these spaces consists of piecewise polynomials subject to a quasiuniform, in the sense of [MR851383, CiarletBook, Guermond-Ern], triangulation of size of the domain . Since we assume to be a convex polyhedron, this domain can be triangulated exactly.
Given , with , we define its Stokes projection as the pair that satisfies
We also recall that, under the given assumptions on the finite element spaces, the Stokes projection is stable on weighted spaces.
[weighted stability estimate] Let be a convex polyhedron. If then, the finite element Stokes projection, defined in (13), is stable in , in the sense that
where the hidden constant is independent of , and . The proof follows after small modifications to [DOS:19, Theorem 4.1]; see Appendix A for details.
3 An error estimate in
In this section we discuss error estimates for discretizations of (8) in the case and , the space of vector valued, bounded Radon measures. In doing so, we shall extend the results of [MR812624] to the Stokes problem, and slightly improve the error estimate of [DOS:19, Corollary 5.4].
Let . Since , we have that . Therefore,
Invoking Theorem 2.3, we have that problem (8) is well–posed for such data. If, in addition, we assume that , then we also have that . As a consequence, it makes sense to provide an error estimate in . Our main result in this direction is the following.
with a hidden constant independent of , , , and . For let be the solution of
for all and . Since is convex, owing to Proposition 2.3 we have that . This, in particular, implies that if denotes its Stokes projection, we have
where we used the interpolation error estimate (11), a basic inverse inequality, the invariance property of , the stability of in , an interpolation estimate for in , and the regularity results of Proposition 2.3.
Consider now problem (14) for , and set . This immediately yields
Observe that, since corresponds to the Stokes projection of and the pair , we have
On the other hand, since , and , then we have that for every . We can thus consider as a test function in the weak version of (8) with to arrive at
4 Well–posedness of the Stokes problem on weighted spaces
Let us now consider the Stokes problem (8) with , , and . We begin by recalling that, for general Lipschitz domains, there is such that, if , and belongs to the restricted class , then this problem is well–posed; see [OS:17infsup, Theorem 17]. On the other hand, if is , then [MR3582412, Lemma 3.2] shows well–posedness for all and all . Here we will show a result that, in a sense, is intermediate between these two. We remove the restriction on the integrability index and the boundary behavior of the weight, thus showing well–posedness for all and all , but at the expense of requiring that is a convex polyhedron.
The main tool that we shall use is the representation of the velocity given in (9) and the Hölder estimates of the Green matrix described in Theorem 2.4. We will follow the ideas of [DDO:17], and extend the results therein to the Stokes problem.
We begin by noting that, by density, it suffices to assume that , so that from (9) we can write
We begin with a simplified version of the Bogovskiĭ [MR553920] decomposition of a function with integral zero; see also [MR3198867]. Since, in our setting, the proof of this result is so simple, we include it for completeness.
[decomposition] Let , , , and be such that . Then, there are , , such that
where the hidden constant is independent of and . To simplify notation let us set and . Notice also that
so that .
Let now be such that , on and on . Set
Note that the functions , for , have the requisite support property. In addition,
The functions , however, do not integrate to zero. Thus, we correct them as follows. Define
Then, we have that and, moreover,
we are able to obtain the estimates
we obtain with a constant that only depends on .
Note now that the function
satisfies , , and, since ,
Finally, using (18), we have
where the hidden constant only depends on .
This concludes the proof.
With this decomposition at hand we can obtain an a priori estimate on the oscillation of the gradient of , much as in [DDO:17, Lemma 2.4] and [MR1800316, Lemma 7.9].
[oscillation estimate] Let be a convex polyhedron, , , , and . Let be the velocity component of the solution of (8) with . Then, for any and , we have that
where the hidden constant is independent of , , and . Let be a cube with center in such that , where . Extend and to zero outside and decompose , with , and with , , as in Lemma 4 but with replaced by . Let now be the velocity component of the solution to (8) with data . It suffices then to bound the oscillation of for all . Fix and set . With this notation, for , we have .
To estimate we follow (7) and bound the average of the difference between and any constant. Thus, we have
We bound each of the terms separately.
First, by Hölder’s inequality, for any , we have
Next, the unweighted Korn’s inequality, the unweighted estimate given in Theorem 2.3, and the fact that and vanish outside yield