A data-driven approach to viscous fluid mechanics -- the stationary case

We introduce a data-driven approach to the modelling and analysis of viscous fluid mechanics. Instead of including constitutive laws for the fluid's viscosity in the mathematical model, we suggest to directly use experimental data. Only a set of differential constraints, derived from first principles, and boundary conditions are kept of the classical PDE model and are combined with a data set. The mathematical framework builds on the recently introduced data-driven approach to solid-mechanics [KO16,CMO18]. We construct optimal data-driven solutions that are material model free in the sense that no assumptions on the rheological behaviour of the fluid are made or extrapolated from the data. The differential constraints of fluid mechanics are recast in the language of constant rank differential operators. Adapting abstract results on lower-semicontinuity and $\mathscr{A}$-quasiconvexity, we show a $\Gamma$-convergence result for the functionals arising in the data-driven fluid mechanical problem. The theory is extended to compact nonlinear perturbations, whence our results apply to both inertialess fluids and flows with finite Reynolds number. Data-driven solutions provide a new relaxed solution concept. We prove that the constructed data-driven solutions are consistent with solutions to the classical PDEs of fluid mechanics if the data sets have the form of a monotone constitutive relation.


INTRODUCTION
In this article, a new approach to the modelling and analysis of viscous fluid mechanics is introduced. The hydrostatic behaviour of an incompressible fluid at any instant in time may be described by its velocity field ∶ ↦ ( ) ∈ ℝ which induces a strain(-rate) ∶ ↦ ( ) ∈ ℝ × sym = 1 2 ∇ + ∇ , (1.1) with finite Reynolds number the force balance (1.2) has to be complemented by the inertial forces proportional to + ( ⋅ ∇) . This results (after suitable non-dimensionalisation) in the equation However, in this paper we restrict our analysis to the stationary case = 0, i.e. we study the problem ( ⋅ ∇) − div = .
Since our analysis is mainly based on variational arguments suited for stationary problems, we postpone the time-dependent case to a separate work.
1.1. The PDE-Based Approach -Constitutive Laws for Viscous Fluids. Hitherto, the modelling and analysis of a rich set of phenomena in viscous fluid mechanics relies on constitutive laws describing the relation between the strain field and the stress field . A commonly used relation is which relies on the assumption that the stress comprises two components -the hydrostatic stress id and the viscous stress 2 (| |) . Here, ∶ ↦ ( ) ∈ ℝ + denotes the viscosity of the fluid. It depends on the strain rate and measures the resistance of the fluid to deformation. Mathematically, the hydrostatic pressure ∶ ↦ ( ) ∈ ℝ is the Lagrange multiplier corresponding to the incompressibility condition div = 0. In the simplest model of a viscous fluid, the viscosity is assumed to be constant ≡ const. and the corresponding fluid is called Newtonian. In other words, the relation between the viscous forces and the local strain rate is perfectly linear, the constant viscosity being the factor of proportionality. In the case of an inertialess incompressible Newtonian fluid one obtains the well-known Stokes equations − Δ + ∇ = div = 0. Although it is reasonable in many practical applications to assume a fluid being Newtonian, real fluids that account for viscosity are in fact non-Newtonian, i.e. they feature a nonlinear relation between the stresses and the rate of strain . A widely-used constitutive relation is given by and the corresponding fluid's are called power-law fluids or Ostwald-de Waele fluids. The exponent > 0 denotes the so-called flow-behaviour exponent and 0 > 0 is the flow consistency index. In the case 0 < < 1 the fluid exhibits a shear-thinning behaviour as its viscosity decreases with increasing shear-rate, while the fluid is called shear-thickening in the case > 1. In this case the viscosity is an increasing function of the shear rate. The corresponding stationary non-Newtonian Navier-Stokes system reads ( ⋅ ∇) − div 2 (| ( )|) ( ) + ∇ = div = 0.
(1.6) For = 1 we recover a Newtonian behaviour. In practice, constitutive laws for the viscosity are derived from experimental measurements. This is done by determining the parameters inside a prescribed class of laws, for instance 0 and in the case of power-law fluids (1.5), to best approximate the measured data. A large part of the mathematical knowledge in the mechanics of viscous fluids comes from the theoretical and numerical analysis of partial differential equations such as Stokes equation and Navier-Stokes equation, that are derived using constitutive laws. Here, a lot of progress has been made by allowing for increasingly general classes of (nonlinear) viscosity laws (see for example [Lad67,MNR93,MRS05,MPS06]).
1.2. A Data-Driven Approach. Nowadays, the availability of big data and the possibility to mine them is increasing drastically. In the present work, instead of including constitutive laws in the mathematical models, we suggest to directly use experimental data in order to find the strain rate and the stress that satisfy the respective differential constraints and, at the same time, approximate the experimental data best. In order to realise this mathematically, we are inspired by the articles [KO16,CMO18], where a similar approach has first been introduced in the context of solid mechanics.
In the present paper, data sets consist of strain-stress pairs ( , ), which we think of as being extracted from an experiment. These data might be obtained by preprocessing the information coming from actual measurements of other physical quantities. We emphasise that the step of preprocessing is also necessary when deriving constitutive laws from measurements.
The motivation for replacing the classical PDE-based approach by the data-driven approach is the following. Once one accepts the fundamental assumptions (first principles) about the nature of the fluid leading to the differential constraints, the PDE-based approach generates two errors with respect to modelling the real world: First, the experimental equipment is imperfect, leading to measurement errors. Second, the fitting of a material law to the experimental data introduces a modelling error. The data-driven approach entirely skips this second step. Turning to the remaining source of errors, with perfect equipment and infinitely many measurements, we expect to recover the viscosity law of the fluid (if it exists). In reality, measurements are however restricted by • the inaccuracy of the equipment leading to a measurement error; • a limited number of data points. This comprises both 'density of measurements' (i.e. given a strain , how many data points lie in a neighbourhood of ?), as well as 'range of measurement' (how large is the range of values of that can be measured in the experiment?).
Nevertheless, if over the course of several consecutive measurement series the measurement error decreases or the density and range of data points increases, we expect the experimental data to converge to the material law. Mathematically, we give consideration to this behaviour by introducing different notions of data convergence.
In this paper, we restrict ourselves to the study of the following two settings: • data with increasing quality and an unbounded range of measurements; • data with increasing quality and a bounded but increasing range of measurements. An overview of the possible settings and where they are discussed in this paper is given in Table 1.

Range of measurement Constant (unbounded) Increasing
Error Constant (no improvement) Need to deal with "bad" data Need to deal with "bad" data Decreasing Section 4.1 Section 4.2 TABLE 1. Measurement error and range of measurement.
In the case of non-increasing accuracy, measurements for a given strain rate might be located in a neighbourhood of the exact value with a certain likelihood. In this case, the set of data converges in a weak sense to some distribution, see [CHO21]. See also [RS20] for the analysis of single outliers in measurements.

Mathematical Approach for the Data-Driven Problem and Main Results.
We follow the mathematical approach proposed in [CMO18] in a solid mechanical context. To this end, we first split the stress = − id +̃ into its hydrostatic part id = − 1 tr( ) id and its viscous part̃ .
Throughout the paper we assume that the data set comprises pairs ( ,̃ ) of strain and viscous stress only. The hydrostatic pressure (i.e. the trace of ) is not included in the data set, since we allow to attain arbitrary values. This is due to the fact that the pressure does not play a role in the constitutive law for the viscosity but arises as a Lagrange multiplier corresponding to the incompressibility constraint.
Given a data set = {( ,̃ )} ∈ , consisting of pairs ( ,̃ ) of symmetric and trace-free matrices in ℝ × , we consider the functional as a measure for the distance of functions ( ,̃ ), defined on a simply connected and bounded 1 -domain Ω ⊂ ℝ , to the data set. Here, is the constraint set of fields ,̃ satisfying the prescribed differential constraints and suitable boundary conditions, and dist(⋅, ⋅) is a suitable distance function.
In the present paper, the set of differential constraints is given by (1.1) in combination with either the inertialess force balance or the stationary Navier-Stokes force balance. That is, we study both the linear constraint (1.8) as well as the nonlinear constraint set (1.9) The set of constraints is complemented by suitable boundary conditions. Typical boundary conditions in fluid mechanics are the no-slip condition = 0 on Ω (1.10) and the Navier-slip condition (1.11) Here, ≥ 0 is the inverse of the so-called slip length and denotes the outer normal to Ω. Moreover, Ω denotes the tangential bundle of Ω. The case of free slip ⋅ = 0 for ∈ Ω is included via = 0. The second condition in (1.11) expresses the non-permeability of the boundary.
Less natural is the Neumann type boundary condition = 0 on Ω. (1.12) In the linear case (1.8), we are able to handle all three types of boundary conditions (1.10), (1.11), and (1.12). In the nonlinear case (1.9), we are able to handle the physical boundary conditions (1.10) and (1.11). In some cases we allow for inhomogeneous boundary conditions, i.e. non-zero right-hand sides. Coming back to (1.7), a minimiser (or a minimising sequence) of the functional always satisfies the compatibility conditions for and̃ and is as close to the experimental data as possible. In the case in which a sequence of data sets approximates a limiting set , corresponding to a constitutive law, it is expected that the minimisers = ( ,̃ ) of the functional converge to a solution of the PDE corresponding to the constitutive law. One main contribution of the present article is to specify conditions under which this is true. We use the following notion for convergence of data sets.
The sequences and represent the relative error, while and describe the measurement range. Note that condition (i) ensures that every point in the limiting set is approximated by data points in while condition (ii) ensures that the approximates uniformly. Moreover, the notion of convergence introduced in Definition 1.1 (ii) is justified from an experimental point of view. Indeed, for a given experimental setup we expect the measurements to be precise only within a certain range, | | ≤ . For instance, in the experiment conducted by COUETTE [Cou90], the aim of which was to measure the viscosity of a fluid, the range is linked to the aspect ratio of the rotating cylinders. In the setting of this article, the absolute error is allowed to grow with the range of measurements, which extends the setting studied in [CMO18], where the absolute errors are required to converge to zero.
From a mathematical point of view, the above notion of convergence is justified by the observation that we may restrict the analysis to -equi-integrable recovery sequences in the Γ-convergence result below. Indeed, the first main result of this article is the following.
• Γ-convergence (Theorem 5.11 and Theorem 5.15): If → and the satisfy a certain growth condition, then Γ-converges to * ( , where is a suitable convex envelope of the distance function corresponding to the differential operators defining the compatibility conditions (1.1) and (1.2).
There are two main challenges in the proof of this result. One difficulty is the suitable modification of sequences of functions while preserving differential constraints and given boundary conditions. To overcome this challenge we prove the following result, which might be of independent interest.
• -equi-integrability and boundary conditions (Theorem 3.9): If a weakly convergent sequence of -functions on Ω ⊂ ℝ satisfies some differential constraint = 0 for a constant coefficient (and constant rank) differential operator , we can modify slightly in the sense of closeness in for < . The modified sequence still satisfies the differential constraint and the same boundary conditions, but is -equi-integrable (i.e. no concentrations of mass occur).
In the case of a data set given by a constitutive law, data-driven solutions provide a new solution concept. Another main result of this article proves that, in the case of monotone constitutive laws, this solution concept is compatible with the concept of weak solutions to PDEs: • Consistency (Section 6): If the data set corresponds to a monotone constitutive law, e.g. = {( , | | −1 )} in the case of power-law fluids, and if the corresponding PDE admits a solution, then for a map = ( ,̃ ) the following three statements are equivalent: (i) is a minimiser of * , i.e. a solution to the relaxed data-driven problem; (ii) * ( ) = 0, i.e. there exists a sequence ⇀ with ( ) → 0; (iii) is a solution to the corresponding PDE (i.e. to (1.6) in the nonlinear case) in the classical weak sense.
In the case of non-monotone constitutive laws, the requirement * ( ) = 0 amounts to a relaxed solution concept that might be useful for instance in order to deal with viscoelastic fluids.
1.4. Outline of the Paper. Section 2 shows how the fluid mechanical problems fit into the general theory of constant rank operators. In Section 2.1 we introduce relevant notation and recall the notion of Γ-convergence with respect to the weak topology of -spaces. In Subsection 2.1.4 we recall the generalised form of Problem (1.7), where the differential constraint ( ,̃ ) ∈ is written abstractly as = 0 and the distance function is replaced by some function ( , ). In Section 2.2 it is demonstrated that the fluid mechanical setting fits into this abstract framework.
An abstract theory for lower-semicontinuity of functionals under linear differential constraints has been developed by FONSECA & MÜLLER ( [FM99], see also [BFL00]) and we recall these results at the beginning of Section 3. The remainder of Section 3 is devoted to the modification of the corresponding arguments to fit the fluid mechanical setting of the present paper. In particular, we show the crucial Theorem 3.9, which allows us to modify sequences to be equi-integrable, while still respecting both the differential constraints and the boundary conditions. This result is used to extend relaxation results, previously obtained in [BFL00], to the situation of a semilinear differential constraint in Theorem 3.13.
For Sections 4-6 we return to the fluid mechanical setting and apply the abstract results of Section 3. In Section 4 we discuss two different notions of data convergence on a purely set-theoretic level; in particular these notions of convergence are not directly connected to the differential constraints. First, in Subsection 4.1 we introduce a form of data convergence which corresponds to fixed range of measurement (lower-left entry of Table 1) and show that this is equivalent to a suitable notion of convergence for the unconstrained functionals For results about Γ-convergence of constrained functionals of type (1.7), however, we can weaken the notion of convergence to Definition 1.1. This type of convergence is examined in Section 4.2. The reason for this convergence being of interest for Γ-convergence, is discussed already at the beginning of Section 3 in Theorem 3.6. The abstract results of Section 3 and results about distance functions to data sets of Section 4 are combined in Sections 5. In Subsection 5.1 and Subsection 5.2 we introduce the data-driven problem both for inertialess fluids and fluids with inertia. We show that, given boundary conditions and a suitable pointwise coercivity condition, the functionals in (1.7) are coercive on the phase space . Therefore, we can apply results from Section 3 to get the respective Γ-convergence result (Theorem 5.11 and Theorem 5.15).
Finally, Section 6 links the (relaxed) data-driven problem * ( ) = 0 to the partial differential equations obtained by including a constitutive law in the modelling. We show that if the data set coincides with the set obtained by a monotone constitutive law, i.e. = {( ,̃ ) ∶̃ = 2 (| |) }, then solutions to the relaxed data-driven problem are weak solutions to the classical PDE problem and vice versa.

FUNCTIONAL ANALYTIC SETTING OF THE FLUID MECHANICAL PROBLEM
In this section we introduce an abstract functional analytic framework that offers a convenient way to reformulate the differential constraints. First, in Subsection 2.1, we recall the notion of Γ-convergence and the notion of constant rank operators. The latter requires a short reminder on some results from Fourier analysis. In Subsection 2.2 we show how the differential operators appearing in the fluid mechanical applications fit into the framework of constant rank operators.
We call × the local phase space. Recall that we assume throughout the paper that the pressure (i.e. the trace of ) is not considered as part of the data. Consequently, each data set  is a subset of × . In order to introduce a distance on × , for pairs ( ,̃ ) ∈ × , = 1, 2, we define dist ( 1 ,̃ 1 ), ( 2 ,̃ 2 ) = 1 | 1 − 2 | + 1 |̃ 1 −̃ 2 | and therewith ( 1 ,̃ 1 ), ( 2 ,̃ 2 ) = dist ( 1 ,̃ 1 ), ( 2 ,̃ 2 ) 1 max{ , } . (2.1) The function (⋅, ⋅) is defined by taking the -th, respectively the -th root of dist(⋅, ⋅), in order to guarantee that the triangle inequality is satisfied . Thus, (⋅, ⋅) defines a metric on × . Accordingly, we define the distance on the phase space by We start by proving that the distance function (⋅, ⋅), introduced in (2.1), defines a metric. In the following we embed Ω into the -dimensional torus when it is convenient. Without loss of generality we therefore assume that Ω is compactly contained in (0, 1) . In general we use as a generic constant. However, we use specific constants whenever it is convenient.
2.1.2. Γ-convergence. In this subsection we recall some well-known results on Γ-convergence that are frequently used throughout the paper. We use this notion of convergence to consider the behaviour of functionals of type (1.7) and (1.13) under convergence of the data.   (i) In metric spaces the constant sequence = possesses a Γ-limit * , namely the lowersemicontinuous hull of , given by * ( ) = inf → lim inf →∞ ( ).
(2.3) * is called the relaxation of . (ii) If each is a minimiser of and → , then is a minimiser of . (iii) One may define Γ-convergence on topological spaces, cf. [DM93]. This reproduces the definition on metric spaces when equipped with the standard topology. Weak convergence is not metrisable on Banach spaces. However, it is metrisable on bounded sets of reflexive, separable Banach spaces. Hence, if a functional satisfies a certain growth condition; i.e.
for a function ∶ [0, ∞) → ℝ with ( ) → ∞ as → ∞, we may use the metric for weak convergence defined on bounded sets of the Banach space and treat the Banach space together with the weak topology as a metric space. (iv) In topological spaces, especially in Banach spaces equipped with the weak topology, the constant sequence = does in general not possess a sequential Γ-limit, as the infimum in (2.3) does not need to be a minimum. (v) If does not satisfy the growth condition (2.4), it is possible to consider the sequential Γ-limit, given as in Definition 2.2. However, this might not exist, even if the topological Γ-limit of a sequence of functionals exists. In particular, the constant sequence might not have a sequential Γ-limit.
In the following we only consider the sequential Γ-limit of sequences in the weak topology of some Banach space (usually × ). If the functional is coercive in the sense of (2.4), then the sequential Γ-limit coincides with the topological Γ-limit.
The following lemma links Γ-convergence to uniform convergence of functionals.
Lemma 2.4 (Uniform convergence and Γ-convergence). Let be a reflexive, separable Banach space equipped with the weak topology. Suppose that , ∶ → [−∞, ∞], such that → uniformly on bounded sets of . If the sequential Γ-limit of the constant sequence exists, then also possesses a Γ-limit and Note that the sequential Γ-limit of the constant sequence exists if the functional is coercive.
which establishes both the lim sup-inequality and the lim inf -inequality.

Korn-Poincaré inequality.
In this subsection, we revisit a combination of Korn's inequality (i.e. the full gradient is controlled by its symmetric part) and Poincare's inequality to obtain an estimate of the form This estimate is a straightforward consequence of the -Korn inequality and the Poincaré inequality, cf. for instance [Cia10]. For the convenience of the reader we provide the proof. In the following we use the notation (2.5)
(i) There is a constant = ( , Ω), such that for any ∈ 1 (Ω; ℝ ) we have Then there is a constant = ( , Ω, ), such that for any ∈ we have Proof. (i) Recall that there is a first-order differential operator̃ with constant coefficients, such that Therefore, we can bound Using Nečas' lemma [Nec66,AG94] for functions with zero mean twice and writing ′ = ⨏ Ω ∇ d , we get To obtain an inequality featuring only the skew-symmetric part = 1 2 ′ − ( ′ ) note that by the triangle inequality The statement follows by estimating each term on the right-hand side by ‖∇ + ∇ ‖ . For the first term we combine (2.7) and (2.6) to obtain Using Poincaré's and Jensen's inequalities, the second term can be estimated by (ii) Note that the spacẽ Indeed, if (2.8) were false, then there would exist a sequence ⊂ with ‖ ‖ 1 = 1 and ‖ − ‖ 1 → 0 as → ∞. As ∈̃ is bounded and̃ is finite dimensional, there is a subsequence converging strongly to some ∈̃ . Since ‖ − ‖ 1 → 0, this implies → in 1 (Ω; ℝ ). But this is a contradiction, as is closed, ‖ ‖ 1 = 1 and ∩̃ = {0}. Part (i) in combination with (2.8) yields (ii), since = + .
2.1.4. Constant Rank Operators. In this subsection we introduce the version of constant rank operators used in this paper. To this end, we slightly adapt the notion of homogeneous constant rank operators [Mur81] since the differential operator ( ,̃ ) = (curl curl , diṽ ) appearing in the fluid mechanical application is only componentwise homogeneous. We consider a differential operator defined on functions ∶ Ω → ℝ 1 × ℝ 2 defined via where 1 and 2 are homogeneous constant coefficient differential operators of order , = 1, 2, i.e. (2.9) Recall that the Fourier symbols corresponding to the operators defined in (2.9) are given by Definition 2.6. = ( 1 , 1 ) satisfies the constant rank property if both 1 and 2 satisfy the constant rank property; that is, if for some fixed ∈ ℕ and for all ∈ ℝ ⧵ {0}.
The characteristic cone of is defined as The operator satisfies the spanning property whenever Remark 2.7. If ∈ ( ; ℝ ) can be written as If, in addition, the operators satisfy the constant rank property, then ℤ ⧵ {0} can be replaced by ℝ ⧵ {0}.

Fourier Symbols and Fourier Multipliers.
In this subsection, we recall some important facts about constant rank differential operators that are connected to the Fourier transform on the -torus . As we can consider the constraint operators 1 and 2 separately, we assume ′ ∶ ∞ (ℝ ; ℝ ) → ∞ (ℝ ; ℝ ) to be a constant coefficient differential operator of order ′ , i.e. (2.10) We call ℬ ′ a potential of ′ , whenever the corresponding Fourier symbols satisfy For such and ∶ ℝ ⧵{0} → Lin(ℝ ; ℝ ), we may define a linear operator on ∞ ( ; ℝ )∩ ( ; ℝ ), 1 ≤ < ∞, by If maps boundedly into some function space, ( ) can be defined for general ∈ ( ; ℝ ), 1 ≤ < ∞, by using density. Such an operator is called Fourier multiplier. The algebraic identity (2.11) in combination with standard Fourier multiplier theory leads to the following statements.
For weakly, but not strongly, convergent sequences on bounded sets, there are essentially two possible effects. There can be oscillations and concentrations. For weak lower-semicontinuity results, oscillations are much easier to handle than concentrations. The notion of -equi-integrability prevents concentration: be a 0-homogeneous Fourier multiplier. Then, for any 1 < < ∞ the following holds true.
Step 1: Construction of a truncated sequence. There exists > 0 and for all > 0 there exists a > 0, such that we have For > 0 consider the function ∶ ℝ → ℝ , defined by Then, for fixed > 0 and ∈ , the set { • ∶ ∈ } is bounded in ∞ ( ; ℝ ). Therefore, by (i), the set Step 2: -equi-integrability of the truncated sequence. We show that, for fixed ∈ ℕ, the set { ( • )} ∈ is -equi-integrable. Taking Step 1 into account, this follows from the fact that any bounded set ′ ⊂ 2 ( ; ℝ ) is already -equiintegrable. To prove this, assume for contradiction that there exists a bounded set ′ ⊂ 2 ( ; ℝ ) that is not -equi-integrable. Then there exist ⊂ ′ , ⊂ with | | → 0, as → ∞, and an > 0, such that By Jensen's inequality this implies which contradicts the assumption that is bounded in 2 ( ; ℝ ) and that | | → 0.
We conclude that for any > 0 there is ( ) > 0, such that, for all ∈ we have the implication (2.12) Step 3: -equi-integrability of ( ). We show that Step 2 together with -equi-integrability of implies that ( ) is -equi-integrable.
2.2. The differential operator for problems in fluid mechanics. In this section, we discuss how the fluid mechanical constraints (1.8) and (1.9) fit into the previously outlined abstract setting. We consider the two differential operators The Fourier symbol of the differential operator 1 is given by For 2 the Fourier symbol reads For a fixed ∈ ℝ ⧵ {0}, the set ker 1 [ ] × ker 2 [ ] is given as follows. Let ⊂ be defined as where ⊙ = 1 2 ( ⊗ + ⊗ ) is the symmetric tensor product. Note that is a ( −1)-dimensional subspace of . Then meaning that the space dimension of ker 1 [ ] is ( − 1) and where ̃ is defined as the unique ∈ ℝ, such that 2 [ ](̃ , ) = 0, i.e.
Remark 2.11. Due to the additional constraint div = 0, ℬ 1 is not a potential to 1 in the sense of (2.11). In particular, Proposition 2.8 cannot be applied directly. Note, however that a function ∈ 1 ( ; ℝ ) with zero average satisfies the differential constraint div = 0 if and only if = curl * for a suitable function ∈ 2, ; ℝ × skew , where curl * is the adjoint of curl; in other words curl * is a potential of div. In particular, this also means that if = 1 2 ∇ + ∇ , then there exists ∈ 2 ; ℝ × skew such that For the purpose of applying Fourier methods, we can use the symmetric gradient ℬ 1 on divergence-free matrices instead of the true potential. The suitable inverse of ℬ 1 in the Fourier space is ℬ −1 1 = curl * •B 1 , which is a Fourier multiplier of order 1 + (−2) = −1.
The potential to the differential operator 2 is not relevant in this setting. Let us remark that the condition for (̃ , ) ∈ ( ; × ℝ) and ∈ −1, ( ; ℝ ), can be rewritten in terms of̃ only, as Another strategy to tackle the linear problem from a "purely" Fourier analytic perspective would be to "forget" about the pressure by using the operator̃ 2 (̃ ) = curl • diṽ . Note that in this approach the operator curl • div acting oñ is the adjoint operator of 1 2 ∇ + ∇ • curl * which acts on . For the non-linear problem, cf. Subsection 5.2, this approach yields the equation (2.15) We believe however, that from the fluid dynamical point of view it is more instructive to include the pressure ∈ (Ω) by sticking to the more physical equation

EXISTENCE OF MINIMISERS -WEAK LOWER-SEMICONTINUITY AND COERCIVITY
It is the structure of the differential constraints, with constant rank operators of different order, the quasilinear perturbation of the otherwise linear constraints, the boundary conditions, and the natural location of and̃ in different spaces, that necessitates the following Section 3, where all these challenges are adressed in an abstract setting.
3.1. -Quasiconvexity. In order to study weak lower-semicontinuity results, we first introduce the notion of -quasiconvexity for a constant rank operator = ( 1 , 2 ) as defined in the previous section.

Definition 3.1. A (measurable and locally bounded) function
it holds that For ∈ (ℝ 1 × ℝ 2 ) we define the -quasiconvex envelope of as is convex.
Note that the -quasiconvex envelope of a continuous function is the largest -quasiconvex function smaller than [FM99]. Moreover, a function is -quasiconvex if and only if = .
Proposition 3.2 (Properties of -quasiconvex functions). Let = ( 1 , 2 ) be a differential operator satisfying the constant rank property and the spanning property and let ∶ ℝ 1 × ℝ 2 → ℝ. Then the following holds true.
We do not provide the proof of Proposition 3.3 here, since it is largely analogous to the proof of [FM99, Theorem 3.6], which is based on a suitable notion of equi-integrable sequences. In the ( , )-setting, the right notion of equi-integrability is the following.
The key insight for Proposition 3.3 is that it suffices to consider ( , )-equi-integrable sequences. This is the content of the following proposition which is again a straightforward adaption of the -setting.
Proof. (i) The main idea of the proof is to show that a suitable version of Proposition 3.5 holds, namely that sequences ⊂ as in (H1) already satisfy (i). To this end, let ⊂ be bounded, and let ⊂ be a Due to (H4) and the ( , )-equi-integrablility of the first term tends to 0. In order to estimate the second term, let > 0 be a constant such that ‖ ‖ , ‖ ‖ ≤ . Then, using (H2), for any > 0 we obtain The first integral on the right-hand side of this inequality converges to 0 as → ∞, since − → 0 in measure by (H1). Moreover, since the sequence is ( , )-equi-integrable, the second integral can be bounded by a constant with → 0 as → ∞. Consequently, (iii) If the sequential Γ-limit of exists (we denote it by * ), then for all ∈ the following holds true.
The lim inf -inequality for is ensured by (ii), i.e. if ⇀ in , then as̄ ⇀ in . On the other hand, the lim sup-inequality follows from (i): the recovery sequence (or at least a suitable subsequence) can be modified to an equi-integrable recovery sequence . By (i), we find that * ( ) ≥ lim sup This completes the proof.
The main challenge in applying Theorem 3.6 to the case in which is a set given by differential constraints and boundary conditions is to verify Hypothesis (H1). In Section 4 we check the conditions (H2)-(H4) on the integrand . To verify (H1), for a given sequence we need to construct a suitable ( , )-equi-integrable modification that conserves both the differential constraints and the boundary conditions. For this purpose we need the following two auxiliary results.
which is the required uniform convergence.
The following result is due to [FM99,Lemma 2.15]. It allows to construct ( , )-equi-integrable modified sequences. However, in general these modified sequences fail to conserve the constraints.
Proposition 3.8. Let be a bounded sequence in (Ω; ℝ ). Then there exists a -equi-integrable sequencẽ with the following properties: The following theorem allows to obtain modified sequences that continue to satisfy both differential constraints and boundary conditions.

Theorem 3.9 (Equi-integrable sequences & boundary values).
Suppose that ∶ ∞ (ℝ ; ℝ ) → ∞ (ℝ ; ℝ ) is a homogeneous differential operator of order , satisfying the constant rank property and that ℬ is a potential of in the sense of (2.11). Let Ω ⊂ ℝ be an open and bounded set with Lipschitz boundary. Let ⇀ 0 in (Ω; ℝ ) and → 0 in − (Ω; ℝ ). Then there exists a sequence ⊂ ℬ (Ω; ℝ ℎ ) such that the following holds true: The main difficulty in the proof compared to the statement without boundary values in [FM99] is to obtain the compact support.
Step 3: Upper Bound on ‖ℬ , − ‖ . First, we note that, by definition, , is compactly supported in Ω for any ∈ ℕ, as is compactly supported in Ω. Moreover, it holds
3.3. Relaxation. If the function is not -quasiconvex, the functional in (3.4) fails to be weakly lowersemicontinuous. Hence, we cannot ensure existence of minimisers just by using the direct method in the calculus of variations. However, when studying the data-driven problem, it is enough to consider approximate minimisers, i.e. minimising sequences with ( ) converging to the infimum of , and their weak limits * . In the following, we define a functional * such that it is the relaxation of . Thus, any weak limit * of a minimising sequence is a minimiser of * and, vice versa, any minimiser of * is a weak limit of approximate minimisers.
3.3.1. Relaxation under a linear differential constraint. We recall the definition of from (3.4). For simplicity, we use for the quasiconvex envelope of a function ∶ Ω × ℝ 1 × ℝ 2 → ℝ the short-hand notation Note that by Proposition 3.3 the functional * given by * ( ) ∶= is weakly lower-semicontinuous in (Ω; ℝ 1 ) × (Ω; ℝ 2 ). That * is indeed the relaxation of is a consequence of the following (linear) result [BFL00].
We say that a functional is coercive on , provided there is a uniform bound on the (Ω; ℝ 1 ) × (Ω; ℝ 2 )-norm of . By taking a diagonal sequence of we may conclude the existence of a recovery sequence satisfying Coercivity as defined in (3.18) is classically obtained by assuming that This strong pointwise coercivity condition is however not suitable for our setting. The distance function to a set only satisfies (3.19) if the set is bounded. Instead, we use a weaker coercivity condition of the type (3.20) In general, 1 ⋅ 2 does not have a good pointwise bound. Nevertheless, in the fluid mechanical setting, appropriate boundary conditions allow us to bound the integral ∫ Ω 1 ⋅ 2 d , cf. Section 5.

3.3.2.
Relaxation under a semi-linear differential constraint. As above, let Ω ⊂ ℝ be an open and bounded domain with Lipschitz boundary. Instead of considering a linear differential constraint, e.g.
Remark 3.15. The statement of Theorem 3.13 is taylored towards its application for fluid dynamics, cf. Subsection 5.2. Observe that in the proof of Theorem 3.13, a main step was to solve the differential equation together with suitable boundary conditions. This equation is solved by the observation, that (Θ( 1, ) − Θ( 1 )) already satisfies the boundary conditions. If we generalise the setting to other non-linearities, we need more assumptions on the non-linearity. For example, consider a constraint like . Then weak-strong continuity is not enough, as one also needs to solve the analogue of (3.22) with suitable boundary conditions. If for example, 2 = div, then a further condition is as follows: Whenever 1 and ′ 1 satisfy spt( 1 − ′ 1 ) ⊂⊂ Ω, then ∫ ( 1 ) − ( ′ 1 ) d = 0 (such that the divergence-equation is solvable, cf. [Bog79]).

CONVERGENCE OF DATA SETS
In this section, we define two different notions of data convergence, i.e. we define a suitable topology on closed subsets of × . We show that these notions are equivalent to convergence of the unconstrained functionals in (1.13). In particular, these notions of data convergence are independent of the underlying differential constraint. Recall that we assume that the data consist of pairs of strain and the viscous part̃ of the stress; the pressure is not part of the data.  We consider the functionals defined on by Theorem 4.2. Let , be closed, nonempty subsets of × . The following statements are equivalent: (ii) For all ∈ it holds that lim →∞ ( ) = ( ) and this convergence is uniform on bounded subsets of .
Proof. '(i) ⇒ (ii)'. Suppose without loss of generality that 0 ∈ . Otherwise we translate the underlying space which at most changes , by a bounded factor. Let ∈ , with ∫ Ω dist( , 0) d ≤ . We assume without loss of generality that ≥ . Then for ∈ ℕ we may estimate where ( ) ∈ is a point in such that ( ( ), ( )) = ( ( ), ). Note that, as 0 ∈ and due to the uniform approximation property, we obtain a pointwise bound on , i.e. ( ( ), 0) ≤ 2 ( ( ), 0) for large enough. Therefore, for some > 0 we get Note that ∫ Ω ( , ) d is bounded from above (for large enough) by 2 ∫ Ω ( , 0) d ≤ 2 as 0 ∈ and 0 is approximated uniformly by elements of . Therefore, for any > 0 we may choose and 0 ∈ ℕ such that for all > 0 we have Consequently, there exists ( , ) → 0, such that for all ∈ with ∫ Ω dist( , 0) d ≤ it holds that ( ) ≤ ( ) + ( , ). (4.1) For the lower bound on ( ) we can do the same calculation using fine instead of uniform approximation and find that for any ∈ with ∫ Ω dist( , 0) d ≤ we have We argue as for the lower bound, to obtaiñ ( , ) → 0, such that for all ∈ with ∫ dist( , 0) d ≤ Therefore, the convergence ( ) → ( ) is uniform on bounded subsets of .

If
is not a fine approximation of , the argument is similar. Then there exists > 0 and a subsequence ∈ , such that, Again, assume that 0 ∈ . We may assume that there exists a sequence ′ → 0 with ′ ∈ , otherwise for ≡ 0, it holds that Let Σ be a subset of Ω with measure |Ω|(1 + dist( , 0)) −1 and define As argued before, ∫ Ω dist( , ) d is bounded uniformly by |Ω| and for ∈ ℕ we find that But, for the distance to we have Therefore, the convergence ( ) → ( ) cannot be uniform on bounded subsets of .
The definition of this type of convergence is motivated by Lemma 2.4. In particular, we have as a consequence that if ⟶ , then the sequential Γ-limit of and of the constant sequence coincide, i.e

Proof. '(i) ⇒ (ii)':
The proof is similar to the proof of Theorem 4.2. We only prove that fine and uniform approximation imply that, for a ( , )-equi-integrable subset ⊂ , we have The converse inequality follows similarly. For simplicity assume that 0 ∈ and that ≥ . For some fixed > 0 we estimate (4.4) We now estimate both integrals on the right-hand side from below and start with the second term. The set ⊂ is ( , )-equi-integrable. Hence, there is an increasing function ∶ ℝ + → ℝ + such that The set is bounded. Thus, defining we find that the measure of {dist( , 0) > } is bounded by −1 . Consequently, we obtain We turn to the first term in (4.4). If dist( ( ), 0) ≤ , we may find some ( ) ∈ with dist( ( ), 0) ≤ (2 + 2 ) , and dist( ( ), ) = dist( ( ), ( )).
Due to uniform approximation for all ( ), we can estimate for large enough Together with (4.5) this implies Choosing ( ) and large enough, then for any there is , such that which establishes (4.3).

'(ii) ⇒ (i)':
This implication is a consequence of the same counterexamples as in Theorem 4.2. Indeed, suppose that the sets do not uniformly approximate on bounded sets. Then there exist > 0, > 0 and a sequence ⊂ , such that dist( , 0) ≤ and dist( , By the same construction as in the proof of Theorem 4.2, that is we obtain a sequence, such that ( ) = 0 and ( ) ≥ |Ω| with uniformly bounded in ∞ (Ω; × ) and hence is also ( , )-equi-integrable. For fine approximation the argument is again very similar.

THE DATA-DRIVEN PROBLEM IN FLUID MECHANICS
In this section we apply the theory developed in the previous sections to the setting of fluid mechanics. We thus specialise to an explicit set of constraints consisting of differential constraints and boundary conditions. In Subsection 5.1 we consider the case of inertialess fluids, leading to a set of linear differential constraints. In Subsection 5.2 we consider nonlinear differential constraints. In both cases we work with the following boundary conditions defined on three mutually disjoint and relatively open parts of the boundary Γ , Γ , Γ ⊂ Ω that satisfy and have 1 -boundary as subsets of the manifold Ω. We consider ( ,̃ ) ∈ (Ω; ) × (Ω; ) with an associated velocity field ∶ Ω → ℝ , where = 1 2 ∇ + ∇ and a pressure field ∶ Ω → ℝ, such that and = − id +̃ satisfy the following boundary conditions. for ∈ 1−1∕ (Γ ) and ℎ ∈ −1∕ (Γ ; ℝ ). Here, ≥ 0 is the inverse slip-length and Ω is the orthogonal projection to the tangent space. Note that the second equation can equivalently be cast as (5.1)
(i) The boundary conditions for can be understood as conditions for in a suitable weak formulation. For instance, if Γ = Ω, then (D) is equivalent to the following condition on . For any ∈ 1 (Ω; ) with div = 0 we have However, since an that is contained in the constraint set automatically admits a corresponding (see (linD) below and following explanation), we write the conditions directly for . A similar remark applies to the appearance of .
(iv) For simplicity we assume in the following that either Γ = Ω or Γ ≠ ∅. This allows us to control ‖ ‖ 1 in terms of ‖ ‖ and the boundary data via the Korn-Poincaré inequality, cf. Lemma 2.5. If Γ ≠ ∅, while Γ = ∅, it becomes tedious to specify under which conditions this control can still be obtained. See Lemma 5.2 and Remark 5.3 below.
In order to obtain a Korn-Poincaré type inequality, has to be uniquely determined by the above boundary conditions

Proof. (i):
The assertion follows from the fact that if ∇ 1 + ∇ 1 = ∇ 2 + ∇ 2 , then 1 − 2 = + for some ∈ ℝ × skew and ∈ ℝ . Condition (5.4) then implies that = 0 and = 0. Since ℝ × skew is one-dimensional, we can explicitly set It follows that the only sets not satisfying (5.4) are such that Γ is a subset of concentric circles. Moreover, if Γ ≠ ∅, then (5.4) is automatically satisfied. In dimension = 3, the situation is similar. Indeed, if Γ ≠ ∅, then (5.4) is satisfied. If Γ = ∅, then, if Γ is a subset of the boundary of a domain that is rotationally symmetric around a certain axis, (5.4) is not satisfied.
Remark 5.4. Uniqueness of is only important for fluids with inertia. For inertialess fluids, only appears in the constraints through boundary conditions. Therefore, even if = 1 2 (∇ 1 + ∇ 1 ) = 1 2 (∇ 2 + ∇ 2 ) for 1 ≠ 2 enjoying the same boundary conditions, it does not matter for the system of equations whether we take 1 or 2 . In contrast, for fluids with inertia, the contribution ( ⋅ ∇) in the differential constraints causes the choice of to be important. Therefore, in the linear setting, even if the prescribed boundary conditions (D), (R) and (N) allow to choose different ∈ 1 (Ω; ℝ ), for example if Γ = Ω, we may project onto a subspace that does not allow multiple solutions to Consequently, we can apply Lemma 2.5 in this situation.
5.1.1. Coercivity. In this subsection we verify coercivity of the functionals and .
Remark 5.6. In Section 4 we examine data convergence without the differential constraints, in particular we study the unconstrained functional . In general, we do not expect a coercivity statement of the type In the following we prove that coercivity follows in the presence of the differential constraints together with suitable boundary conditions, i.e. it holds that ≥ 2 ∕( + 1). Suppose that ∶ × → ℝ is ( , )-coercive and has ( , )-growth. Then there are Inserting (5.15) into (5.12) and using the result together with (5.13), (5.14), (5.16), and (5.17) in (5.10) yields where we used Young's inequality in the last step and the constants depend on , Ω, , , , ℎ, ℎ .
Thus, we may assume that 2 = 0 since this only shifts 2 in (5.6). Then is ( , )-homogeneous, i.e. ( ,̃ ) ∈ ⇒ ( , ∕ ̃ ) ∈ for all > 0. This in turn implies that the distance function is ( , )-homogeneous, i.e. where we use that the right-hand side is smaller than 0 on in the complement of , while it is smaller than in . This and (5.20) show that the distance function dist is ( , )-coercive.
Proof. The hypotheses of Theorem 3.6 are all satisfied with = dist(⋅, ), = dist(⋅, ) and = lin . Indeed, (H1) is Corollary 3.10, (H4) is the assumption ⟶ and (H2) is satisfied by distance functions of sets, such that , ∩ (0, ) ≠ ∅ for some > 0. This in turn follows from nonemptyness and ⟶ . Condition (H3) follows from the fact that the functions in our setting are distance functions, hence even locally Lipschitz continuous. Finally, the set = lin is weakly closed because for a bounded sequence = ( ,̃ ) ⊂ the pressure satisfies, after suitable renormalisation, and is thus also bounded. Since the differential constraints (linD) are linear, it is possible to take the limit for a subsequence. Therefore, Theorem 3.6 implies that Γ-converges to the Γ-limit of , which is given by * due to Proposition 3.11.
Remark 5.12. Theorem 4.5 establishes equivalence between data convergence and uniform convergence of towards if there is no differential constraint = 0. It is not clear whether such an equivalence holds for the constrained functionals and . Indeed, in an abstract degenerate setting, e.g. ker [ ] = {0} for all ∈ ℝ ⧵ {0}, so that only constant functions are in ker , it is easy to see that the equivalence does not hold. Indeed, uniform approximation for bounded/equi-integrable functions in the constraint set is equivalent to pointwise uniform approximation on bounded sets. That is, there are → ∞ and̃ → 0, such that for all ∈ with dist( , 0) ≤ dist( , ) ≤̃ . This is considerably weaker than the notions of convergence introduced in Definition 4.1 and Definition 4.3. A similar notion holds for fine approximation. Nevertheless, from a physical viewpoint, the pointwise data convergence ⟶ is a reasonable assumption and we are thus not interested in a complete characterisation of convergence for the constrained functionals.

Fluids with Inertia.
In this subsection we consider the system of differential constraints, corresponding to a fluid with inertia Regarding the boundary conditions, we make the following assumptions throughout this subsection: (B1) Γ = ∅, i.e. there are only no-slip and Navier-type boundary conditions; (B2) Γ ≠ ∅; (B3) One of the following two statements is true (B3a) > 2; (B3b) = 0 and = 0.
Proof. The proof is very similar to the proof of Theorem 5.11. Indeed, as the constraint set nl is weakly closed by Lemma 5.14, the only difficulty, given ∈ nl , is to find a recovery sequence lying in nl . This is achieved in Theorem 3.13.

CONSISTENCY OF DATA-DRIVEN SOLUTIONS AND PDE SOLUTIONS IN THE CASE OF MATERIAL LAW DATA
In this section we consider data that are given by a constitutive law, i.e.
= 2 (| |) , ∈ , for a viscosity ∶ ℝ → ℝ. We compare the solutions obtained by the classical PDE approach to minimisers of the data-driven functional. As before, we assume Γ = ∅ and call a pair ( ,̃ ) ∈ (Ω; ) × (Ω; ) a weak solution to the stationary Navier-Stokes equation, if there is ∈ 1 (Ω; ℝ ) and a pressure ∈ (Ω), such that ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ = 1 2 ∇ + ∇ , ∈ Ω div = 0, ∈ Ω ( ⋅ ∇) − div(2 (| |) ) + ∇ = , ∈ Ω ( ), ( ), ∈ Ω, (6.1) where (6.1) 3 has to be satisfied in −1 (Ω; ℝ ). Note that the system (6.1) is equivalent to ∈ Ω ( ), ( ), ∈ Ω. (6.2) We may interpret the convergence of data sets discussed in Section 4 as an increase of the accuracy of measurement. If a constitutive law exists, then the limit of data sets should represent this law. Since we assume that the set is given by a constitutive law ↦̃ ( ), we consider data sets = {( ,̃ ) ∶̃ =̃ ( )}. (6.3) For typical constitutive laws, a solution to the induced partial differential equation (6.2) exists and it is natural to ask whether (approximate) solutions to the data-driven problem with converge to a solution of (6.2). It turns out that this is true if the constitutive relation is monotone. Indeed, assume that ( ,̃ ) ∈ nl , i.e. that the differential constraints Consequently, a minimiser of satisfying ( ) = 0 is a solution to the partial differential equation. Conversely, given a constitutive law̃ and a weak solution to the partial differential equation (6.2), we may construct the set as in (6.3) and observe that any solution to the partial differential equation (6.2) is also a minimiser of . If the data set is a limit of measurement data sets , it is not clear a priori whether a sequence of (approximate) minimisers of converges weakly to a solution to the partial differential equation because we can only infer * ( ) = 0 and not ( ) = 0. This is addressed in the following proposition, which directly follows from the relaxation statement Theorem 5.15. Proposition 6.1. Let > 3 ∕( + 2) and let ↦̃ ( ) be a given constitutive law. Moreover, assume that the corresponding data set is given by (6.3), such that the distance function dist(⋅, ⋅) is ( , )-coercive. If the partial differential equation then any such approximate solution * is already a solution to the partial differential equation (6.2).
In the following we characterise some constitutive laws satisfying (6.4). To this end, we study the set We call a set ⊂ × -( , )-quasiconvex if = ( , ) .
Remark 6.7. Starting from the constitutive law ↦̃ ( ), there are two choices for ℳ . We may define ℳ as in (6.8) or only take the set introduced in (6.8). For the -quasiconvex hull this does not make a difference, i.e. Indeed, (6.9) can be verified by calculating the Λ -convex hull of the set (that is, we successively take convex combinations along Λ ). The Λ -convex hull is a subset of the -quasiconvex hull. Therefore, it suffices to show that the Λ -convex hull of contains ℳ . This in turn follows from the fact that ker 2 [ ] = {̃ ∈ ∶̃ = 0} + ℝ( ⊗ ) ⟹ Λ 2 = . Using this observation, the Λ -convex hull of {(0,̃ ) ∶ |̃ | = } ⊂ is the convex hull 0 . Consequently, the Λ -convex hull and therefore also the -quasiconvex hull of contain ℳ .