Hypocoercivity in algebraically constrained partial differential equations with application to Oseen equations

The long-time behavior of solutions to different versions of Oseen equations of fluid flow on the 2D torus is analyzed using the concept of hypocoercivity. The considered models are isotropic Oseen equations where the viscosity acts uniformly in all directions and anisotropic Oseen-type equations with different viscosity directions. The hypocoercivity index is determined (if it exists) and it is shown that similar to the finite dimensional case of ordinary differential equations and differential-algebraic equations it characterizes its decay behavior.


Introduction
This paper is concerned with the long-time behavior and hypocoercivity structure of (an)isotropic Oseen equations from fluid dynamics.The Oseen equations describe the flow of a viscous and incompressible fluid at low Reynolds numbers and they have the form The Oseen equations arise when one linearizes the incompressible or nearly incompressible Navier-Stokes equations describing the flow of a Newtonian fluid, = −( ⋅ ∇) − ∇ + Δ , > 0 , 0 = − div , around a prescribed vector field , that is independent of space and time, see e.g.[39].
As it also includes a (linear) convective term, it can be seen as an improvement of the flow description by the Stokes equations, see [12, §4.10], [33,Chap. 2,§11].The Oseen equations are a typical example of an operator differential-algebraic equation (DAEs) of the form with the unbounded operators = ⋅ ∇ − Δ and = div, see [23, p. 466].DAEs of the form (2) also arise when the Oseen system is semi-discretized in space e.g.via a finite element discretization.This constitutes what is often called a vertical method of lines approach, see e.g., [30,34,38].
In all described cases the equations have to be supplemented by suitable initial and boundary conditions.While in applications, see e.g.[33], the Oseen equation is typically considered on subsets of ℝ or on unbounded exterior domains, to keep the presentations and the technicalities of our analysis simple, we analyze its long-time behavior here on the torus 2 ∶= (0, 2 ) 2 , see e.g.[11].
We perform the analysis using the concept of hypocoercivity which was introduced in [43] in the study of unconstrained evolution equations (mostly partial differential equations) of the form d d = − on some Hilbert space , where the (possibly unbounded) operator − generates a uniformly exponentially stable 0 -semigroup ( − ) ≥0 .More precisely, for hypocoercive operators there exist constants > 0 and ≥ 1, such that where  is another Hilbert space, densely embedded in (ker ) ⟂ ⊂ .Often, the evolution equation d d = − is also called hypocoercive.For infinitesimal generators − , an estimate (3) with = 1 holds if (and only if) is coercive.
The long-time behavior of many systems exhibiting hypocoercivity has been studied frequently in recent years, including Fokker-Planck equations [8,9,43], kinetic equations [21,22], and reactiontransport equations of BGK-type [1,2].In these works, in particular in [1,2,9], the issue was to determine the sharp (i.e.maximal) exponential decay rate , while to determine at the same time the smallest multiplicative constant ≥ 1 is a rather recent topic, e.g.see [7].Also the short-time behavior of linear evolution equations and its link to the hypocoercivity index was recently discussed for systems of ordinary differential equations in [4,6] and for Fokker-Planck equations [10,Th. 3.6].
In this paper we consider (unbounded) operators on a Hilbert space  such that the operator is accretive, i.e. has a nonnegative self-adjoint part, and that − generates a contraction semigroup, i.e., it satisfies ‖ − ‖  ≤ 1 for all ≥ 0.
We characterize those accretive operators which are hypocoercive, i.e., those for which − generates a uniformly exponentially stable 0 -semigroup.Furthermore, based on our characterization and following the Lyapunov theory on Hilbert spaces, see e.g.[19,27,28], we will construct appropriate strict Lyapunov functionals.
The remainder of this paper is structured as follows: In §2 we review key notions from hypocoercivity for finite dimensional ODEs and DAEs, and then extend it to the Hilbert space case.In §3 we apply these techniques to analyze Oseen equations on the 2D torus where the viscosity acts uniformly in all directions.
In Section §4 we then study two anisotropic Oseen-type equations on the 2D torus, where the viscosity is different in the different space directions and the drift is either constant or space dependent.

Notation
The conjugate transpose (transpose) of a matrix is denoted by ( ⊤ ).The set of Hermitian matrices in ℂ × is denoted by ℍ .Positive definiteness (semi-definiteness) of ∈ ℍ is denoted by > 0 ( ≥ 0).The unique square root of a positive semi-definite Hermitian matrix is denoted by 1∕2 and the real part of a complex number is denoted by ℜ( ).

Hypocoercivity for finite and infinite dimensional evolution equations
In this section we recall the concepts of hypocoercivity and the hypocoercivity index for (constrained) evolution equations.Our discussion will be in three steps, starting with finite dimensional cases, then infinite dimensional operator evolution equations with bounded generators, and finally some cases of unbounded generators.We begin with the class of ordinary differential equation (ODE) systems with some function ∶ [0, ∞) → ℂ and a constant matrix ∈ ℂ × .The second class are differentialalgebraic equation (DAE) systems for a pair ( , ) of constant matrices , ∈ ℂ × with = positive semi-definite.Note that if ∈ ℍ is positive definite, then it has a positive definite matrix square root 1∕2 ∈ ℍ , and by a change of variables ∶= 1∕2 and by scaling the equation by ( 1∕2 ) −1 , the DAE (5) takes the form However, if is singular then the behavior of the two systems (4)-( 5) is fundamentally different.Writing a matrix ∈ ℂ × as the sum of its Hermitian part = ( + )∕2 and skew-Hermitian part = ( − )∕2, we have the following definition.
In the following (to avoid too many indices), we often write semi-dissipative matrices in the form = − with a skew-Hermitian matrix = and a positive semi-definite Hermitian matrix = − .
For ODE systems, hypocoercivity and the hypocoercivity index are defined as follows: holds.

Remark 1.
(1) Clearly, ( 7) is equivalent to the condition for some > 0, where denotes the identity matrix.This variant will be needed in the infinite dimensional case below.Condition (8) was also used in [2].
(2) By a non-trivial result, the HC-index characterizes the short-time decay of semi-dissipative Hamiltonian ODE systems ̇ ( ) = − ( ): Its system matrix has a HC-index for some > 0.Here we used that ‖ − ‖ 2 is a real analytic function on some (small) time interval [0, 0 ), see Theorem 2.7(a) in [4].This classification can be extended to the DAE case by considering the HC-index of the dynamical part, see Proposition 6 in the Appendix A.1.
A similar equivalence result (on a subspace describing the dynamics of the system) has also been shown for DAE systems in [5], see Appendix A.1 here.
The characterization of hypocoercive matrices is related to results in control theory (as discussed in [5, Remark 2]): Remark 2 (Connection to control theory: finite dimensional setting).Consider a state-space system for constant matrices , ∈ ℂ × .A pair • For semi-dissipative matrices − , the HC-index of − is one less than the controllability index of ( , ) = ( , ).
• Moreover, for general input-output systems there is also the dual concept of observability.Conditions like (11) are most often formulated in the context of observability, but not frequently in the context of controllability.
• Whereas the controllability/observability indices have a clear interpretation in the discrete-time setting, see e.g.[16, p.171]; their interpretation in the continuous-time setting is not so clear.In particular, we are not aware of a characterization which is comparable to (9) in Remark 1.
While the two conditions (10) and (11) are clearly equivalent in the finite dimensional ODE case, only the latter one may also be used for linear operators.In fact, the above results are easily extended to evolution equations of the form with a bounded or unbounded operator on some infinite dimensional Hilbert space : Definition 3 ([32, §V.3.10]).A linear operator on a Hilbert space , with domain ( ), is said to be accretive if the numerical range of is a subset of the right half plane, that is, if ℜ⟨ , ⟩ ≥ 0 for all ∈ ( ).In this case − is said to be dissipative.And is called coercive if there exists > 0 such that ⟨ , ⟩ ≥ ‖ ‖ 2 for all ∈ ( ).
Note that in this definition we follow the convention in semigroup theory, see e.g.[24,Proposition 3.23]; whereas to be consistent with Definition 1 we would have to call such an operator semi-dissipative.
The classical Lyapunov characterization of (uniformly) exponentially stable semigroups on finitedimensional Hilbert spaces easily extends to infinite dimensional settings: Theorem 1 ([18, Theorem 4.1.3],[19]).Suppose that is the infinitesimal generator of the 0 -semigroup ( ) on the Hilbert space .Then ( ) is uniformly exponentially stable if and only if there exists a bounded positive operator ∈ () such that Equation ( 14) is called Lyapunov equation.If ( ) is uniformly exponentially stable, then the unique self-adjoint solution of ( 14) is given by Next we recall the definition of hypocoercive operators in [43], which generalizes Definition 2: Definition 4 ([43, §I.3.2]).Let be a (possibly unbounded) operator on a Hilbert space  with kernel ker .Let  be a Hilbert space, which is continuously and densely embedded in (ker ) ⟂ , endowed with a scalar product ⟨⋅, ⋅⟩  and norm ‖⋅‖  .The operator is called hypocoercive on  if − generates a uniformly exponentially stable 0 -semigroup ( − ) ≥0 on  → (ker ) ⟂ , i.e. (3) holds.
Next we shall generalize the notion hypocoercivity index to the infinite dimensional Hilbert space case.As in the finite dimensional case, we consider operators of the form = − with self-adjoint and nonnegative, and skew-adjoint.For technical reasons (related to the operator domain) we shall first discuss bounded operators and then some special situations of unbounded operators .Definition 5. Consider bounded operators , ∈ () on a Hilbert space  such that is self-adjoint and nonnegative, and is skew-adjoint, i.e., * = − .The hypocoercivity index (HC-index) of the (accretive) operator ∶= − ∈ () is defined as the smallest integer ∈ ℕ 0 (if it exists) such that for some > 0.
When defining next the hypocoercivity index for unbounded operators, we do not aim at the largest generality of equations ( 13), but rather we present a framework that covers the Oseen equations in §4 below.In our extension, the unbounded operator = − is accretive with the following assumptions:

Assumptions.
(A1) The unbounded operator with dense domain ( ) ⊂  is self-adjoint (and hence closed) and nonnegative in .The operator is bounded and skew-adjoint on .
Under these assumptions, standard arguments from semigroup theory show that − and hence also − = − + are dissipative with domain ( ) and, hence, infinitesimal generators of 0 -semigroups of contractions on , see e.g.§1.4 of [37].Moreover these semigroups are analytic, see [ For unbounded operators, the relation between the domains of the operator, its self-adjoint part and its skew-adjoint part can be subtle.Therefore, different extensions of Definition 5 are reasonable.For example, under the assumptions (A1)-(A2), each term of the sum ( 16) is well-defined on ( ).However, the following extension to ( 1∕2 ) is more convenient for the subsequent lemma: Definition 6.Let the (accretive) operator = − satisfy the Assumptions (A1), (A3).Then the hypocoercivity index (HC-index) of is defined as the smallest integer ∈ ℕ 0 (if it exists) such that for some > 0.
In both of the above settings (Definitions 5 and 6) we have the following infinite dimensional analog of Lemma 1. (B3') There exists ∈ ℕ 0 such that (17) holds for some > 0.
Moreover, the smallest possible ∈ ℕ 0 coincides in all cases (if it exists).
Proof.To prove Lemma 2 we will make use of the (equivalent) characterizations of surjective operators in [14,Theorem 2.21].To this end, we will introduce the following Hilbert spaces and operators.For ∈ ℕ 0 , the direct sum is again a Hilbert space.Define the linear operator Since 1∕2 is closed and is bounded, also the operators 1∕2 ( * ) are closed.Hence the operator is densely defined (in both cases of assumptions ( ) ∶= ( 1∕2 ) is dense in ) and closed.Its adjoint .
Next, we identify the operator and its adjoint * in our statement with the ones in [14, Theorem 2.21]: First, gives the equivalence of the conditions (B1') and (a) in [14, Theorem 2.21], where " !=" indicates the condition to be satisfied.Note that in (18) each range 1∕2 has to be evaluated on the -th component of ( * ), which may indeed be a proper superset of ( 1∕2 ). Second, Moreover, due to the assumptions, range( ) is closed if and only if range( * ) is closed.This gives the equivalence of the conditions (B2') and (c) in [14,Theorem 2.21].Third, Remark 3 (Connection to control theory: infinite dimensional setting).Consider the control system (12) where, for simplicity, , ∈ () operate on the same Hilbert space .In the infinite dimensional setting many more concepts of controllability exist, e.g.depending on the operator domains ( ), ( ) ⊂ .In [17,Theorem 3.18] and [18,Theorem 6.2.27] it is observed that for , ∈ (), system ( 12) is exactly controllable if and only if Condition (B1') in Lemma 2 holds.
After recalling the basic hypocoercivity concepts, in the next two sections we apply the techniques for the analysis of different variants of the Oseen equations.We consider the Oseen equations as a constrained PDE model on a torus.Since this allows for a modal decomposition, it reduces to an infinite system of DAEs.Depending on the detailed shape of the drift and diffusion terms in the Oseen equation, these models exhibit a wide range of hypocoercivity phenomena: it may be coercive (see §3), hypocoercive with index 1 (see §4.2), or not hypocoercive (see §4.1).

Isotropic Oseen equation on the 2D torus
As first model we consider the time-dependent, incompressible Oseen equation of fluid dynamics with isotropic viscosity on the 2D torus 2 ∶= (0, 2 ) 2 , for the vector-valued velocity field = ( , ) and the scalar pressure = ( , ) in the space variable ∈ 2 and the time variable ≥ 0. The constant > 0 denotes the viscosity coefficient and ∈ ℝ 2 is the constant drift velocity.Note that since in (19) diffusion acts uniformly in all directions, we call the Oseen model ( 19) isotropic.
For (19) we assume periodic boundary conditions in both and .Hence, this model actually could be simplified right away: Taking the divergence of the first equation in (19) yields Δ (⋅, ) = 0 and hence (⋅, ) is constant in .It also shows that the vector-valued transport-diffusion equation preserves the incompressibility if the initial condition satisfies div (0) = 0, which is assumed in the sequel.Since it is known that in this case the normal component of has a periodic extension [20, §IX.1.2+3],[26, §II.5], it follows that, for any initial condition and hence (19) has a unique smooth solution for > 0, and its explicit Fourier representation can be obtained from the first equation of the Fourier expansion (22) below.
Here, in order to pave the way for more general applications, we proceed differently and employ negative hypocoercivity of matrix pencils in semi-dissipative Hamiltonian DAEs.So we ignore this possible simplification and rather follow our discussion from [5, §3].The following analysis is an extension of §4.1 in [5], which considered the Stokes equation.
The modal functions ( ), ∈ ℤ 2 correspond to the function ( ) in §2 and [5], since = [ 1 , 2 ] ⊤ is used here for the spatial variable.Following the notation from §2, we decompose as In order to determine the hypocoercivity index of ( 23) we employ the unitary transformation of ( 23) to staircase form via the Algorithm in [5, Lemma 5], which we recall as Lemma 6 in Appendix A.1.
When modifying (19) into an anisotropic Oseen equation with viscosity only in the 2 -direction, the dynamics becomes more interesting: Depending on the prescribed convection field , the dynamics may be hypocoercive or not.This is the topic of the next section.

Anisotropic Oseen-type equations on the 2D torus
In [15,36] the 3D Navier-Stokes equation with anisotropic viscosity was studied, where the vertical viscosity was reduced or even zero.In analogy, we shall study now Oseen-type models with anisotropic viscosity on the torus (similar to [42]).We consider these equations as a useful simplified mathematical model to illustrate (hypo)coercivity, rather than considering them for their physical interest (similar to [26,40,41]).For simplicity of the presentation we again confine ourselves here to the 2D case, i.e. to 2 ∶= (0, 2 ) 2 : The model is subject to periodic boundary conditions; it has prescribed transport with a drift velocity vector ( ) ∈ ℝ 2 which may depend on ∈ 2 , and diffusion only in 2 (hence, we call the model (32) anisotropic).This is strictly speaking not an Oseen equation anymore, since the derivation of Oseen needs that is constant in space.We analyze this model in two variants: For a constant convection field ∈ ℝ 2 , the modes still decouple but the generator of the evolution is (depending on ) either coercive or not even hypocoercive.If the convection field is non-constant in space, e.g. if = [sin( 2 ), 0] ⊤ , then the spatial modes are coupled and the generator of the (infinite dimensional) problem becomes hypocoercive.

Oseen-type equation with constant drift velocity
Let us first consider (32) as an evolution equation on the space of divergence-free vector fields.In the case of a constant drift velocity vector ∈ ℝ 2 , we can argue like in §3 to find (⋅, ) = const, and we again normalize the pressure as ≡ 0.Then, the (linear) generator of (32) takes the shape − = − ⋅ ∇ + 2 2 and it acts identically on the two components 1 and 2 .Proceeding as in §3 and using the analog of the modal evolution equation (22) shows that, for any initial condition (0) ∈ (div 0, 2 ), Equation (32) with ∈ ℝ 2 has a unique mild solution for > 0. Hence, the operator − generates a 0 -semigroup on  ∶= (div 0, 2 ), see also [24,37].But due to the degenerate, anisotropic diffusion and the lack of hypocoercivity (see Proposition 1 below), solutions are in general not smooth here.This is illustrated by an example, see (36) below.
In order to check the hypocoercivity of as characterized in (3), we determine the kernel of in : (32).
We continue to analyze the anisotropic Oseen-type model (32) with ∈ ℝ 2 and 1 ≠ 0. Following the characterization of hypocoercivity in (3) and Lemma 3, we also introduce the Hilbert space endowed with the 2 -inner product.Since the condition ∫ 2 d = 0 is preserved under the flow, − also generates a 0 -semigroup on , see e.g.[37,24].But related to its long-time behavior we have the following result: Proof.First we introduce the maximal domain of in : For future reference we also give their characterization in Fourier space (cf.( 21) and [26, §II.5]): To investigate the coercivity of , we compute for ∈ ( ): where we have used that div = 0.The last integral does not involve 1 2 .Hence, ∶= [0, sin( 1 )] ⊤ is a counterexample to coercivity on  since ⟨ , ⟩  = 0.
This lack of hypocoercivity in the model (32) with constant ∈ ℝ 2 and 1 ≠ 0 can be understood quite easily: It includes drift and diffusion in the 2 -direction, but the uniform transport in the 1 -direction does not entail mixing between the different vertical layers.If the flow field has only a vertical component 2 , possibly different in each vertical layer (as shown in (36)), then the flow field gets transported in the 1 -direction and remains incompressible.
The lack of hypocoercivity can also be verified by considering (32) as a constrained partial differential equation (PDAE), and bringing its modal representation into staircase form, see Example 1 in Appendix A.1.

Oseen-type equation with non-constant drift velocity
As second example we consider the anisotropic equation (32) with a non-constant drift field In order to simplify the detailed hypocoercivity analysis below, we choose 1 ( 2 ) = sin( 2 ); in this case only neighboring 2 -modes are coupled.We conjecture that choosing another non-constant 2 -periodic drift field 1 would lead to an analogous result, but it would need a more cumbersome analysis.
In contrast to §4.1, this model does not preserve vertical layers of the flow field but rather mixes them, hence giving rise to hypocoercivity.While the pressure could have been eliminated from the beginning in the model with constant drift term, this is not possible here.Hence, (37) with 1 ( 2 ) has to be considered as a (true) constrained PDE.

Analytic framework and well-posedness:
For the analysis of this case we apply the divergence to the evolution equation (37a), leading to the following condition on the pressure: 1 2 , on 2 , periodic boundary conditions for , where the last condition was added to ensure uniqueness of .For a given inhomogeneity ∈ ( 2 ( 2 )) 2 , Equation ( 38) has a unique weak solution in the homogeneous Sobolev space ̇ 1 ( 2 ), which we denote by = [ ].A standard elliptic estimate shows The generator of the flow (37) with 1 ( 2 ) = sin( 2 ) takes the form Note that div = 0 and [ ] from (38) imply that div( ) = 0.The assertion ∫ d = 0 follows trivially from ∫ d = 0.As before, we choose the Hilbert spaces and the maximal domain of is ( With this framework we have the following result: (37).Then: (1) ker = ℝ 2 , i.e. the constant-in-flows; and the 2 -orthogonal complement of ker is .
(2) The operator is not coercive on . Proof.
For the density of ( ) in  we first consider Truncating the Fourier representation of any ∈  (in analogy to (34)) one easily finds that  is 2dense in , and hence also ( ) in .
(2) For the second statement we easily compute which is defined at least on  and also dissipative.Hence, Corollary 1.4.4 (to the Lumer-Phillips Theorem) from [37] implies that − is the infinitesimal generator of a 0 -semigroup of contractions on .
Next we shall illustrate the short-time behavior of ‖ − ‖ (in the spirit of ( 9)) by analyzing the Taylor expansion of the norm for a single trajectory at = 0. Using the same initial condition as in the proof of Lemma 4, i.e. (0) = [0, sin( 1 )] ⊤ which is a linear combination of the modes ±1 0 , we compute: Hence we obtain This implies the following lower bound on the propagator norm: The first non-constant term in the Taylor expansion of ‖ ( )‖ 2  has exponent = 3.In Proposition 3 below, we show that the modal expansion of (37) with 1 ( 2 ) = sin( 2 ), has hypocoercivity index = 1.Although the analog of the result (9) has not yet been established for unbounded generators − , we note that the exponent = 3 in the Taylor series of ‖ ( )‖ 2  satisfies again = 2 + 1.

Exponential decay of the dynamical part:
Using DAE concepts we separated the dynamical part from the algebraic constraint.The following proposition studies the hypocoercivity of the dynamical part.More precisely, with the above setup we can now establish the exponential decay of 1 2 , i.e., the modal decomposition of the vorticity rot (in fact in ̇ −1 ( 2 ), see Remark 6 for details): (37).Then, for each 1 ∈ ℤ ⧵ {0}, the modal dynamics (55) is hypocoercive in the sense of (3) in 2 (ℤ; ℂ).Moreover, its HC-index = 1.
We remark that determining the hypocoercivity index of − 1 ∶= ( ̂ − ̂ 1 ) 2,2 directly via the coercivity condition in Definition 6 would be much more tedious, since both the matrix − 1 and the corresponding left-hand-side of (17) (with = 1) are 5-diagonal, see Appendix B.1.
In order to derive a hypocoercivity estimate (3) for the evolution equation (55) we proceed similarly to §4.3 in [1] (there for a linear transport-reaction equation in 1D) and construct a strict Lyapunov functional In the following, we will use the canonical unit vectors ( ) ∈ℤ as orthonormal basis for 2 (ℤ), which are defined for , ∈ ℤ as We construct (an ansatz for) a strict Lyapunov functional using [5, Algorithm 3], see Appendix A.2, starting with Π 0 = , ̃ 0 = − ̂ 1 2,2 , and Here, by abuse of notation, sequences (such as 0 ) are interpreted as infinite column vectors and 0 ⊤ 0 is the outer product of two (infinite) vectors such that and . In the next iteration, Hence, the orthogonal projection Due to Π 2 ∶= Π2 Π 1 = 0, the iteration terminates after one iteration in agreement with = 1.Finally, the ansatz for the weight matrix 1 is chosen as for some 1 > 0 to be determined.We remark that all blank elements in the matrix in (62) are zero.In the finite-dimensional setting, for sufficiently small 1 > 0, the squared weighted norm ‖ ⋅ ‖ 2 1 yields a strict Lyapunov functional, see [3].In this infinite dimensional example, we shall verify this statement directly.The infinite matrix Finally, the coefficients 1 , 1 ≠ 0 should be chosen such that the Lyapunov matrix inequalities (LMIs) Proof.We first prove that for divergence-free flow fields ∈ , and that is a bijection from the divergence-free subspace of 2 (ℤ; ℂ 2 ), i.e., for ⋅ = 0, to 2 (ℤ); see also the remark on the leading 2 × 2-subblock of 1 after (53).To this end we first note that the right hand side of (69) is related to the vorticity of , since the modal Given a divergence-free ( 1 ,⋅) ∈ 2 (ℤ; ℂ 2 ) with 1 ≠ 0 we compute with (69): where we used ⋅ = 0 twice.For the other direction, given a 1 2 ∈ 2 (ℤ) with 1 ≠ 0 we define the divergence-free flow field which is compatible with (69).Then we have and this proves the isometry and bijectivity.Finally, we sum up the (square of the) modal inequalities (65) for 1 ≠ 0 and the inequalities (43) for 1 = 0 but 2 ≠ 0. This yields the claimed decay estimate (67) for ( ) − ∞ with using (66).The decay of ‖∇ ( )‖ ( 2 ( 2 )) 2 then follows from the estimate (39).Remark 6.The modal isometry (68) can be extended to physical space by combining all modes 1 2 .Recalling from (69) the dependence of 2 = 2 [ ] on , let 2 ∶= 2 ∈ℤ 2 ⧵{0} .For scalar functions on 2 we define the following homogeneous Sobolev space via the Fourier decomposition of : Then we have the following relation for divergence-free flow fields on 2 with vanishing average, i.e. ∫ 2 ( ) d = 0: The space ̇ −1 ( 2 ) [for rot = − 2 1 + 1 2 ] is isometrically isomorphic to Then, the associated solution due to (65), where < min ≤ 1∕ √ 2, 1,min = 1,min ( ) is defined in (97), and ‖ ‖ denotes the operator norm of a bounded operator ∈ ( (ℤ)).This finishes the proof of Proposition 4.
Altogether, for consistent initial data, solutions of system (44) (and resp.( 42)) converge to the constant equilibrium with a uniform exponential rate.

Conclusions
After extending the notion of hypocoercivity index to evolution equations in (infinite dimensional) Hilbert spaces, we have performed the analysis of the long-time decay behavior of three variants of isotropic and anisotropic Oseen-type equations from fluid dynamics (for simplicity on a 2D torus).Due to the torus setting we used DAE theory in Fourier space to classify the hypocoercivity index.These equations are either coercive, hypocoercive with index 1, or even not hypocoercive (the latter showing exponential convergence only to a traveling wave solution).

A.1 Hypocoercivity in linear semi-dissipative DAEs
Here, we recall the basic theory of hypocoercivity for finite-dimensional linear semi-dissipative Hamiltonian DAEs [5]: Definition 4]).A matrix pencil − is called negative hypocoercive if the pencil is regular, of DAE-index at most two and the finite eigenvalues of the pencil − have negative real part.

A.2 Review of Algorithm 3 from [5]
The purpose of Algorithm 3 from [5] is to construct an ansatz for strict Lyapunov functionals for semidissipative Hamiltonian ODEs (4) with negative hypocoercive matrix ∈ ℂ × .In [3], explicit restrictions on (relative to other parameters) were derived such that a suitable choice of turns the ansatz in Step 10 of Algorithm 3 into a strict Lyapunov functional.
Thus, we choose Hermitian part is positive definite, and it is called hypocoercive if the spectrum of lies in the open right half plane.Let , ∈ ℂ × satisfy = ≥ 0 and = − .The hypocoercivity index (HC-index) of the matrix = − is defined as the smallest integer ∈ ℕ 0 (if it exists) such that ∑ =0 ( ) > 0

Lemma 2 .
Let the operators , ∈ () on a Hilbert space  satisfy either the assumptions in Definition 5 (if = − is bounded) or the assumptions in Definition 6 (if is unbounded).Then the following three conditions are equivalent: (B1') There exists ∈ ℕ 0 such that span ⋃ This gives the equivalence of the conditions (B3') and (b) in[14, Theorem 2.21].The three equivalences established in Theorem [14, Theorem 2.21] thus imply the three equivalences of Lemma 2.
2 .(The lines indicate the partitioning of the block matrices ̃ and ̃ in the previous step.Note that the positive semi-definiteness of the Hermitian matrix implies the 0 structure