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Mixing in Incompressible Flows: Transport, Dissipation, and Their Interplay

Michele Coti Zelati
Gianluca Crippa
Gautam Iyer
Anna L. Mazzucato

Communicated by Notices Associate Editor Daniela De Silva

1. Introduction

Mixing in fluid flows is a ubiquitous phenomenon and arises in many situations ranging from everyday occurrences, such as mixing of cream in coffee, to fundamental physical processes, such as circulation in the oceans and the atmosphere. From a theoretical point of view, mixing has been studied since the late nineteenth century in different contexts, including dynamical systems, homogenization, control, hydrodynamic stability and turbulence theory. Although certain aspects of these theories still elude us, significant progress has allowed to provide a rigorous mathematical description of some fundamental mixing mechanisms. In this survey, we address mixing from the point of view of partial differential equations, motivated by applications that arise in fluid dynamics. A prototypical example is the movement of small, light tracer particles (in laboratory experiments, these are often tiny glass beads) in a liquid. One can visually see the particles “mix,” and our interest is to quantify this phenomenon mathematically and formulate rigorous results in this context.

When the diffusive effects are negligible, the evolution of the density of tracer particles is governed by the transport equation

Here is a scalar representing the density of tracer particles, and  denotes the velocity of the ambient fluid. We will always assume that the ambient fluid is incompressible, which mathematically translates to the requirement that  is divergence free. Moreover, we will only study situations where  is a passive scalar (or passively advected scalar)—that is, the effect of tracer particles on the flow is negligible and the evolution of  does not influence the velocity field . One example where passive advection arises in nature is when light, chemically nonreactant particles are carried by a large fluid body (e.g., plankton blooms in the ocean). Examples of active scalars (i.e., tracers for which their effect on the flow cannot be neglected) are quantities such as salinity and temperature in geophysical contexts.

Our interest is to study mixing away from boundaries, and hence we will study 1.1 with periodic boundary conditions. For simplicity, and clarity of exposition, we fix the dimension . The spatial domain is henceforth the torus , which we normalize to have unit sidelength. We mention, however, that most of the results we state can be extended to higher dimensions without too much difficulty. We supplement 1.1 with an initial condition at time . If the velocity is sufficiently regular (Lipschitz continuous in space uniformly in time, to be precise), solutions to equation 1.1 can be expressed explicitly in terms of the time-dependent flow map of the velocity field , which is obtained by solving the system of ordinary differential equations:

Now, a direct calculation shows that the unique solution to 1.1 with initial condition is given by the formula

This is known as the method of characteristics.

The condition is equivalent to imposing that for each time , , the map is area preserving. Since  is divergence free, integrating 1.1 in space shows that the total mass

is a conserved quantity (i.e., remains constant in time). Thus replacing  with  if necessary, there is no loss of generality in assuming (and hence ) has zero mean. We stipulate from now on that . Physically, now represents the deviation from the mean of the density of tracer particles.

Mixing, informally speaking, is the process by which uneven initial configurations transform into a spatially uniform one. In our setting (since the flow is area preserving), the area of regions of relatively higher (or lower) concentration is preserved. That is, for any , the area of the sublevel sets  and the superlevel sets  are both constant in time. Thus if initially the set where  is positive occupies half the torus, then for all time the set where  is positive must also occupy half the torus. The process of mixing will transform  in such a way that the set  will be stretched into many long thin filaments (that still occupy a total area of half), and are interspersed with filaments of the set  in such a manner that averages at any fixed scale become small (see Figure 1, below).

Figure 1.

Example of mixing. The red and blue level sets in both figures have exactly the same area. For the left figure, averages on scales comparable to of the period are of order . On the right, however, the sets are stretched and interspersed in such a manner that averages at the same scale are much smaller.

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Mathematically, this process is known as weak convergence. More precisely, we say  becomes mixed as  if  converges weakly to  in . That is, for every  test function  we have

The standard notation for this convergence is to write

while we denote strong  convergence by

which means as . As the name suggests, strong convergence implies weak convergence, but the converse is generally false. In fact, in our situation where is divergence free, for all  we have . Thus while many of our examples exhibit mixing (i.e., weak convergence of  to ), they will not have strong convergence of  to , unless  is identically . We remark that all norms are conserved by a volume-preserving map, but is a natural choice. Indeed, readers familiar with ergodic theory will recognize that weak convergence in is equivalent to the concept of strong mixing in dynamical systems.

Even though weak convergence is a natural way to study mixing, the disadvantage is that it does not a priori give a quantifiable rate. To explain further, if  converges to strongly in , then at time  the quantity  is a measure of how close  is to its (strong) limit. If  converges to weakly, then  may not contain any useful information about the convergence. (Indeed, for solutions to 1.1 is independent of .) It turns out, however, that in our situation, weak  convergence to  is equivalent to strong convergence to  in any negative Sobolev space. Now the norm in these negative Sobolev spaces (which we will define shortly) can be used as a measure of how “well mixed” the distribution is (see for instance Thi12).

It is easiest to define the negative Sobolev norms using the Fourier series. Given an (integrable) function  on the torus, we define its Fourier coefficients by

For mean-zero functions the -th Fourier coefficient, , vanishes. Now, for any  we define the homogeneous Sobolev norm of index  by

Note that for  the norm puts more weight on higher frequencies. Thus functions that have a smaller fraction of their Fourier mass in the high frequencies will be “less oscillatory” and have a smaller  norm. For , however, the norm puts less weight on higher frequencies. Thus functions that have a larger fraction of their Fourier mass in the high frequencies will be “very oscillatory,” and have a smaller  norm. This is consistent with what we expect from “mixed” distributions. Moreover, the result mentioned previously guarantees that mixing of  is equivalent to

Thus, for any , the quantity  can be used as a measure of how “mixed” the distribution is at time .

For this reason negative Sobolev norms are often referred to as “mix-norms.” Choosing is particularly convenient, as the ratio of the norm to the norm scales like a length, and can be interpreted as a characteristic scale of the tracer field. Since the norm is preserved by equation 1.1, we can identify the norm with the mixing scale of the scalar field . We mention that there is also a related notion of mixing scale, which is more geometric in nature (see for instance Bre03), but when studying evolution equations the mix-norms described above are easier to work with.

Practically, in order to mix a given initial configuration to a certain degree, one has to expend energy by “stirring the fluid.” A natural, physically meaningful question is to bound the mixing efficiency LTD11. That is, given a certain “cost” associated with stirring the ambient fluid, what is the most efficient way to mix a given initial configuration? Two cost functions that are particularly interesting are the energy  (which is proportional to the actual kinetic energy of the fluid, assuming the fluid is homogeneous), or the enstrophy  (which is proportional to the fluid’s viscous energy dissipation rate). Hence our question about mixing efficiency can now be formulated as follows: what are optimal bounds on in terms of the fluid’s energy or enstrophy?

Before answering this question, we note that solutions to 1.1 can be given a classical meaning if the velocity field is Lipschitz. However, the natural constraints on the velocity described above do not require  to be Lipschitz. Moreover, one intuitively expects efficient mixing flows to be “turbulent,” and many established turbulence models have velocity fields that are only Hölder continuous at every point. Thus, in many natural situations arising in the study of fluids, one has to study 1.1 when the advecting velocity field is not Lipschitz. Seminal work of DiPerna and Lions in ’89, and the important extension of Ambrosio in ’04, addresses this situation, and shows that certain “renormalized” solutions to 1.1 are unique, provided .

Returning to the question of mixing efficiency formulated above, one can use direct energy estimates to show that if the fluid is energy constrained (i.e., if , for some constant ), then  can decrease at most linearly as a function of time. An elegant slice-and-dice construction of Bressan Bre03, using piecewise-constant shear flows in orthogonal directions in a self-similar fashion, provides an example where this bound is indeed attained (see LLN12). In particular, this provides an example where for some  and an incompressible, finite-energy velocity field , we have  weakly in  as . That is, the fluid mixes the initial configuration “perfectly” in finite time.

On the other hand, if one imposes an enstrophy constraint (i.e., a restriction on the growth of ), or more generally a restriction on the growth of , then the DiPerna-Lions theory guarantees finite-time perfect mixing cannot occur. Indeed, equation 1.1 is time reversible, and so finite-time perfect mixing would provide one nontrivial solution to 1.1 with initial data . Since  is clearly another solution, we have nonuniqueness for weak solutions to 1.1, which is not allowed by the DiPerna-Lions theory when . So one can not have finite-time perfect mixing in this case.

Quantitatively, one can use the regularity of DiPerna-Lions flows CDL08 to obtain explicit exponential lower bounds on the mix norm. Namely, one can prove IKX14

for every , and some constants that depend on  and . (We remark that, for , an elementary proof of the lower bound 1.3 follows from Gronwall’s inequality and the method of characteristics. For , however, the proof is more involved and requires some tools from geometric measure theory.)

Interestingly, whether or not 1.3 holds for  is an open question. Indeed, the proof of the needed regularity estimates for the flow in CDL08 relies on boundedness of a maximal function, which fails for . The bound 1.3 for is related to a conjecture of Bressan Bre03 on the cost of rearranging a set, which is still an open question.

For optimality, there are now several constructions of velocity fields that show 1.3 is sharp. These constructions produce enstrophy-constrained velocity fields for which  decays exponentially in time. A construction in ACM19 does this by starting with initial data which is supported in a strip and finds a Lipschitz velocity field that pushes it along a space filling curve. Constructions in BCZG23BBPS21MHSW22 produce regular velocity fields for which

for every initial data . Such flows are called exponentially mixing, and we revisit this in more detail in Section 2.

In addition to optimal mixing, there are three other themes discussed in this article. We briefly introduce these themes here, and elaborate on them in subsequent sections.

(A.)

Loss of regularity. When  is regular, classical theory guarantees regularity of the initial data  is propagated by the equation 1.1. However, when  is irregular (e.g., when  with ), it may happen that all regularity of the initial data  is immediately lost. Not surprisingly, this loss is intrinsically related to mixing. Indeed, the process of mixing generates high frequencies, making the solution more irregular. When  is not Lipschitz one can arrange rapid enough growth of high frequencies to ensure that all Sobolev regularity of the initial data is immediately lost. We describe this construction in detail in Section 3.

(B.)

Enhanced dissipation. In several physically relevant situations, both diffusion and transport are simultaneously present. The nature of diffusion is to rapidly dampen high frequencies. Since mixing generates high frequencies, the combined effect of mixing and diffusion will lead to energy decay of solutions that is an order of magnitude faster than when diffusion acts alone. This phenomenon is known as enhanced dissipation and is described in Section 4.

(C.)

Anomalous dissipation. Even under the enhanced dissipation mentioned above, for which the energy decay is much faster due to the combined effect of diffusion and transport, the energy decay rate vanishes with the diffusivity. In some sense, this outcome is expected, as solutions to 1.1 (formally) conserve energy. For certain (irregular) flows, however, it is possible for the energy decay rate in the presence of small diffusivity to stay uniformly positive, a phenomenon known as anomalous dissipation. It implies, in particular, that the vanishing-diffusivity limit can produce dissipative solutions of 1.1, that is, weak solutions for which the energy decreases with time. We discuss anomalous dissipation in Section 5.

2. Optimally Mixing Flows

In this section, our primary focus centers around understanding the concept of shearing as one of the central mechanisms of mixing, and how this mechanism gives rise to flows that mix optimally.

2.1. Shear flows

Shear flows are the simplest example of incompressible flows on . Their streamlines (lines tangent to the direction of the velocity vector) are parallel to each other and the velocity takes the form . The corresponding transport equation is

the solution of which can be computed explicitly via the method of characteristics.

If the initial datum only depends on , then the solution remains constant for all times. Otherwise, a hint of creation of small scales is given by the growth of linearly in time. To deduce a quantitative mixing estimate, one can take a partial Fourier transform in of 2.1: denoting , with , the Fourier coefficients of , 2.1 becomes

Since , mixing follows by estimating oscillatory integrals of the form

A duality argument and an application of the stationary-phase lemma entails a (sharp) mixing estimate.

Theorem 2.1.

Assume that , for some integer , and its derivatives up to order do not vanish simultaneously: , for all . Then there exists a positive constant such that

for all initial data with vanishing -average.

The mixing rate is solely determined by how degenerate the critical points of are, and Theorem 2.1 tells us that the flatter the critical points, the slower the (universal) mixing rate. In the case of simple or nondegenerate critical points, for which the second derivative does not vanish, such as for the Kolmogorov flow , we have .

We observe that if  in 2.2 then  is constant in time; hence unless , no mixing can occur. Therefore, it is essential to assume that the initial condition has vanishing average in the variable, which is equivalent to imposing that . This requirement excludes functions that are constant on streamlines, i.e., eigenfunctions (with eigenvalue 0) of the transport operator, which do not enjoy any mixing.

2.2. General two-dimensional flows

Although regular shear flows can achieve algebraic mixing rates, we could be inclined to think that their simple structure constitutes an obstruction to faster mixing. It turns out that if is an autonomous, nonconstant Hamiltonian function on , of class , generating an incompressible velocity field , then the mixing rate of is at best , and can be even slower depending on the structure of , see BCZM22. This result can be interpreted as follows: despite the fact that could have hyperbolic points, at which the flow map displays exponential stretching and compression, shearing is the main mixing mechanism in 2d. This can be deduced from the existence of an invariant domain for the Lagrangian flow on which is bounded away from zero, which in turn implies that there exists a well-defined, regular and invertible change of coordinates , where the interval is determined by the invariant set. An example is a simple cellular flow, where . The level sets of the Hamiltonian are given in Figure 2, and clearly show shearing away from the separatrices.

Figure 2.

Level sets of the Hamiltonian .

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Proposition 2.2.

Let . There exists an invariant open set such that for any with , the corresponding solution of 1.1 satisfies

for some and all .

The set of coordinates are the so-called action-angle coordinates, and reduce the transport operator to the much simpler from , where , which is a function, is the period of the closed orbit . The analogy with 2.1 is then apparent, and estimate 2.3 is derived from the explicit solution, obtained via the method of characteristics. Thanks to interpolation, the growth 2.3 is a lower bound on the mix-norm of , hence proving that is a lower bound on the mixing rate of .

2.3. Exponentially mixing flows

Obtaining a faster mixing rate necessarily involves nonautonomous velocity fields. A widely used exponential mixer, especially in numerical simulations, is due to Pierrehumbert Pie94, and consists of randomly alternating shear flows on . The beauty of this example is its simplicity: at discrete time steps , it alternates the horizontal shear and the vertical shear . Here, is a sequence of independent uniformly distributed random variables so the phases are randomly shifted. While widely believed to be exponentially mixing, the first proof of this fact appeared only recently in BCZG23.

Theorem 2.3.

There exists a random constant (with good bounds on its moments) and such that we have 1.4, almost surely.

By taking a realization of the above velocity field, this result settles the question of the existence of a smooth exponential mixer on , although it does not produce a time-periodic velocity field.

The proof of Theorem 2.3 relies on tools from random dynamical system theory and adopts a Lagrangian approach to the problem: this approach involves proving the positivity of the top Lyapunov exponent of the flow map via Furstenberg’s criterion and a Harris theorem.

A related example has been produced in MHSW22ELM23, constructed by alternating two piecewise linear shear flows. This example is fully deterministic and produces a time-periodic, Lipschitz velocity field. The important feature of this flow is that it generates a uniformly hyperbolic map on .

In general, constructing exponentially mixing flows on has proven to be quite a challenge, and only recently there have been tremendous developments. Besides the two works described above, that constitute the latest works in the field, we mention the deterministic constructions of ACM19EZ19, the latter building upon prior results of Yao and Zlatos, and the beautiful work on velocity fields generated by stochastically forced Navier-Stokes equations of BBPS21.

3. Loss of Regularity

One of the effects of mixing is the creation of striation in the scalar field. Quantitatively, this effect corresponds to growth of derivatives of , which can also be seen from the well-known interpolation inequality:

In fact, recalling that the norm of is conserved by the flow of , if the negative Sobolev norms of decay to zero at some time , at that same time the positive Sobolev norms must blow up. However, we note that growth of derivatives is a local phenomenon that can occur in the absence of mixing, which is a global phenomenon.

One can ask whether the growth of Sobolev norms can lead to loss of regularity for solutions of the transport equation 1.1, when the velocity field is not sufficiently smooth. The Cauchy-Lipschitz theory implies that, if is Lipschitz uniformly in time, then the flow of is also Lipschitz continuous, although its Lipschitz constant can grow exponentially fast in time. Hence, at least some regularity of the initial data is preserved in time. When the gradient of is not bounded, but it is still in some space with , the direct estimates from the Cauchy-Lipschitz theory do not apply. Therefore, it is natural to investigate what, if any, regularity of the initial data is preserved under advection by .

We present two examples to show that no Sobolev regularity is preserved in general: the first where the flow is mixing and all Sobolev regularity, including fractional regularity, is lost instantaneously; the second where the flow is not mixing and we are able to show at least that the (and any higher) norm blows up instantaneously. The second construction applies to (almost) all initial conditions in , though the resulting velocity field still depends on the initial condition . In both examples, the simple key idea is to utilize the linearity of the transport equation to construct a weak solutions by adding infinitely many suitably rescaled copies of a base flow and a base solution. The rescaling pushes energy to higher and higher frequencies or small scales, leading to an accelerated growth of the derivatives, which ultimately results in an instantaneous blow-up (see CEIM22 and references therein).

The first result is the following:

Theorem 3.1.

There exists a bounded velocity field such that , , uniformly in time and a smooth, compactly supported function , such that both and the unique bounded weak solution with initial data are compactly supported in space and smooth outside a point in , but does not belong to for any and .

This result implies lack of continuity of the flow map in Sobolev spaces and can be shown to be a generic phenomenon in the sense of Baire’s Category Theorem.

We sketch the proof of Theorem 3.1. Utilizing a suitable exponentially mixing flow on the torus, it is possible to construct a smooth, bounded, divergence-free vector field with , , uniformly in time and a smooth solution of the transport equation with velocity field , both supported on the unit square in the plane, such that all positive Sobolev norms , , grow exponentially fast in time. For each , we define velocity fields and functions on squares of sidelength by rescaling:

for some sequences , , and to be chosen, up to some rigid motions, which do not change the norms and which we suppress for ease of notation. The squares can be taken pairwise disjoint. Then, by setting

we have that is a weak solution of 1.1 with velocity field . Lastly, we pick , , and in such a way that the squares converge to a point, the only point where and are not smooth, the norms and are controlled, while the norm .

The second result is the following:

Theorem 3.2.

Given any nonconstant function , there exists a bounded, compactly supported, divergence-free velocity field , with for any uniformly in time, and smooth outside a point in , such that the unique weak solution of 1.1 in with initial data does not belong to (even locally) for any .

In fact, a stronger statement is true. The velocity field is in all Sobolev spaces that are not embedded in the Lipschitz space (namely, for all , ).

The main steps in the proof of Theorem 3.2 are as follows. The first step follows by a direct calculation on functions on the torus . Given any nonconstant periodic function , applying either a sine or cosine shear flow parallel to one of the coordinate axes must increase the  norm of  by a constant factor at time . Hence, at time , where