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Susan Friedlander’s Contributions in Mathematical Fluid Dynamics

Alexey Cheskidov
Nathan Glatt-Holtz
Natasa Pavlovic
Roman Shvydkoy
Vlad Vicol

Communicated by Notices Associate Editor Daniela De Silva

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Susan Friedlander received her undergraduate degree in mathematics from University College London in 1967. Having been awarded one of the prestigious Kennedy Scholarships to study at the Massachusetts Institute of Technology, Friedlander moved to the US and earned her MS degree at MIT in 1970. She subsequently started her PhD studies at Princeton University, completing her doctorate thesis under the title “Spin Down in a Rotating Stratified Fluid” in 1972 with the supervision of the fluid dynamicist Louis Norberg Howard. After being a visiting member at New York University’s Courant Institute of Mathematical Sciences, Friedlander moved to the University of Illinois at Chicago, where she worked as a Professor until 2008. Since then, Friedlander is a Professor and the Director of the Center for Applied Mathematical Sciences at the University of Southern California.

Throughout her career, Friedlander has focused on the mathematical analysis of partial differential equations (PDEs) arising in fluid dynamics. While the fundamental models are several centuries old, to date fluid dynamics remains the source of some of the most fascinating and challenging problems at the intersection of mathematics and physics. Without a doubt, the phenomenon of “turbulence” is chief among them. A unifying theme in Friedlander’s research is an emphasis on problems of clear physical interest and importance.

Friedlander’s impact on the field of mathematical fluid dynamics, and on the mathematical community as a whole, extends far beyond her research contributions. Prior to 1989, she opened bridges to the fluids communities behind the iron curtain. Since the early 1990s she has served in several leadership positions at the American Mathematical Society, including as Associate Secretary. For the past 15 years Friedlander has been the Editor in Chief of the Bulletin of the AMS, and more recently Friedlander was one of the key figures in the founding of the Mathematical Council of the Americas.

Susan is an exceptional mentor. Since the early stages of our careers, the authors of this paper were fortunate enough to collaborate with Susan, benefiting from her guidance, academic generosity, and perspective on mathematics as a whole. Susan has helped shape both our careers and our views of mathematics, and we are truly thankful for her inspiration, thoughtful guidance, and limitless positive energy.

In this review celebrating Friedlander’s contributions we will focus on her work on hydrodynamic instability as it relates to the transition from laminar to turbulent flow, on dyadic models in fluid dynamics and Onsager’s conjecture analyzing the transfer of energy in turbulent flows, and on magnetohydrodynamics as it relates to large-scale motions in Earth’s fluid core.

1. Equations of Fluid Dynamics

The fundamental partial differential equations that describe the macroscopic properties of the motion of an incompressible, inviscid fluid with constant density are the Euler equations:

with the initial condition

for the unknown velocity vector field and the pressure , where , , and . Despite the fact that Leonhard Euler introduced them in 1757, many basic questions concerning Euler equations in are still unresolved. For example, it is an outstanding problem to find out if solutions of the 3D Euler equations form singularities in finite time, from smooth initial data.

The equations modeling the macroscopic properties of viscous, incompressible, homogeneous fluids were formulated by Claude-Louis Navier (1822) and Sir George Stokes (1845). The Navier-Stokes equations that they derived are written as

with the initial condition

and appropriate boundary conditions. As with the Euler equations the theory of the Navier-Stokes equations in three dimensions is far from being complete. One of the major open problems is global existence and uniqueness of smooth solutions to the Navier-Stokes equations in 3D. This is one of the Millennium Problems of the Clay Mathematics Institute.

Beyond the Navier-Stokes and Euler equations many other models animate modern research in mathematical fluid dynamics. Of particular interest in Friedlander’s work are the magnetohydrodynamics equations (cf. 3437 below) and other equations arising in geophysics.

2. Instabilities

The late 1970s and early 1980s were marked by the discovery of a new type of instability in incompressible fluids—the so-called shortwave or broad-band instability. Such instabilities occur when a fluid rushes through a pipe leading to the formation of elliptical vortices near the walls. Although such vortices themselves are two dimensional, and in fact stable under two-dimensional perturbations, they are manifestly unstable when perturbed in the direction of their axes of rotation. Moreover, the frequency of unstable modes corresponding to the same exponential rate ,

corresponds to a range of values instead of being uniquely determined by a dispersive relation . Hence, the term “broad-band.” Grounded in numerous physical works by Orszag, Patera, Bayly, Pierrehumbert, Craik, Criminale, and others, these novel instabilities lacked a rigorous foundation presenting a unique challenge for the mathematical community in the early 1980s.

2.1. The fast dynamo problem

A similar type of instability appears in the kinematic dynamo problem. This problem seeks to describe persistent growth of a magnetic field transported by a given velocity field of electrically conducting fluid in the limit of vanishing magnetic resistivity. Specifically, for satisfying the system

the dynamo is called fast if one has

where is the exponential type of the -semigroup generated by 78. Similar to the fluid problem such instabilities are expected to be of highly oscillatory nature as the corresponding spectral problem would require an increasing range of frequencies as .

The groundbreaking works FV91FV92VF93 marked the beginning of a productive collaboration of Friedlander with Misha Vishik, who developed a novel approach to the fast dynamo problem Vis89. This proved to be a universal tool to tackle a range of instability questions in fluids, geophysics, and magnetohydrodynamics. The approach is based on studying shortwave asymptotic expansions of the corresponding evolution semigroup or the associated Green’s function. Here the general methodology is to reduce the evolution of an infinite-dimensional system to the leading order “core” dynamics. Remarkably in many cases the reduced dynamics is governed by a finite-dimensional system of ODEs.

In the context of the fast dynamo problem 79, the Green’s function of the evolution operator can be represented in Lagrangian coordinates

as the Fourier integral operator

where the symbol has an asymptotic expansion

Here the principal symbol which plays the determining role in exponential growth of the dynamics is obtained by

the tangent push-forward transport map.

The technical analysis of the asymptotic series 10 is rather involved. Ultimately it connects the limiting exponential rate of as over the energy space to that of the inviscid problem, and hence to the ODE 11. At the same time, the asymptotic behavior of 11 is well known. For steady states it is simply determined by the largest Lyapunov-Oseledets exponent of the underlying velocity field :

(or otherwise the exponent of the corresponding cocycle family), which is positive if and only if the flow-map exhibits exponential stretching of its trajectories. The main result of FV91 reads as follows.

Theorem 2.1.

If the system 78 has a fast dynamo 9, then necessarily , and in fact

Thus, a necessary condition for a fast dynamo is the presence of an instability in the underlying conducting fluid itself.

2.2. The geometric optics method

As we already described above the asymptotic methods developed by Friedlander and Vishik in attacking the fast dynamo problem proved to be applicable to a wide range of problems arising in fluid dynamics. Indeed, the works VF93FV92 laid the foundation to what is now called the geometric optics approach to shortwave instabilities, a particular case of which is the elliptic instability we mentioned in the beginning of this section.

To describe the method in more detail let us consider the example of the classical incompressible Euler system linearized around a given steady state :

where is the linear perturbation of and is the perturbed pressure, which plays the role of projecting the right side of 12 onto the space of divergent-free fields. We consider 12 with periodic boundary conditions, , or the whole space , . The method seeks to find effective dynamics of a localized oscillatory wave written in the form of a geometric optics ansatz

If initially is localized near position , and the frequency of initial oscillation is , i.e., , we obtain a new wave at time approximately localized near the Lagrangian particle . Plugging this ansatz into the Euler system 12 one reads off the leading order evolution of the amplitude and phase in the Lagrangian coordinates of the underlying field :

where is the frequency covector. To write the system in closed form we replace the transport of by the transport of the frequency vector, and the resulting system reads

supplemented by initial conditions , , , and the incompressibility constraint . This is a so-called bicharacteristic-amplitude system (BAS for short). From a dynamical viewpoint the first two equations represent a bicharacteristic flow on the tangent bundle of the fluid domain , denoted , and the last amplitude equation represents evolution of a vector on the fiber bundle over with fibers given by orthogonal planes . Thus, is a cocycle family of maps over the flow .

The asymptotic expansion of the Euler semigroup , defined by 1213, is dominated by the -cocycle playing the role of the principal symbol

where is the Leray projection onto the divergence-free fields, ,

is the leading order pseudodifferential operator, and is a similar operator of order . For the high frequency waves the frequency localization of pseudodifferential operator leads precisely to the ansatz 14 which becomes justified a posteriori.

The shortwave instabilities can now be studied by looking into the Lyapunov spectrum of the BAS whose maximal exponent is given by

The main result of VF93 establishes a direct relationship between the growth rate of the BAS and the growth rate of the Euler dynamics in the energy space.

Theorem 2.2.

Let denote the exponential rate of the semigroup in . Then

The high frequency asymptotic relationship between and makes it possible to relate shortwave instabilities to the essential spectrum of the semigroup. The BAS was found to be fully descriptive of the essential spectrum in later works. For particular flows, however, Theorem 2.2 proved to be extremely versatile in many different situations. For example, for the aforementioned elliptic vortices, locally given by , the growth of the BAS becomes a Floquet problem over time-periodic elliptic trajectories. This case was also studied in works of Lifschitz and Hameiri around the same time. The amplitudes become unstable in directions pointing off of the -plane, which is consistent with the empirical observations of Orszag, Patera, and others. A systematic study of the BAS and various dynamic scenarios leading to instabilities was performed in FV92. First, in 2D, the quantity is conserved. Hence, is related precisely to the exponential stretching of the underlying field . Here the cotangent cocycle associated with the -equation has the same Lyapunov spectrum as that of the tangent cocycle 11. Thus, in this case. In particular, all parallel shear flows are shortwave stable. In 3D, the analogue of this law is conservation of the volume

for any pair of amplitudes over the same frequency . Hence, in 3D we obtain

In any case, exponential stretching makes the flow spectrally unstable. On the other hand, some integrable flows , where , on nondegenerate level tori of the Bernoulli function are found to be stable, namely . Geometric instability criteria for vortex rings without swirl were provided as well.

In several subsequent works (see FSV97FS05 and references therein), Friedlander expanded the geometric optics method to a range of models appearing in geophysics and magnetohydrodynamics. In all these cases the underlying bicharacteristic flow remains the same but the amplitude equation changes according to a simple recipe—it captures the principal symbol of the linearization:

Thus, for the surface quasigeostrophic equation describing evolution of a potential temperature on a horizontal surface the -equation reads

where is the underlying steady temperature. In this case the essential spectrum is neutral, . For density stratified fluids we have

Here is the gravitational potential. The kinematic dynamo falls under the same scheme and yields 11. Camassa-Holm (Euler-) gives

and for inviscid systems of nonrelativistic superconductivity we have

Numerous other applications of the theory were found to non-Newtonian fluids also. The reach of the method proved to be truly astonishing.

2.3. From linear to nonlinear instability

Justification of the linearization procedure for inviscid fluids remains a very challenging problem to this day. Providing an explicit bound on the “bad” essential part of the spectrum given by exponent is a helpful tool to prove a range to sufficient conditions for the analogue of the Lyapunov theory—going from linear to nonlinear instability FSV97. In several subsequent works Friedlander established several pioneering results in this direction. First, in 2D if there is a point spectrum (exact eigenvalue) beyond , i.e.,

then the underlying steady flow is unstable in the energy norm VF03. Construction of flows with oscillatory laminar regions that fulfill this condition have been provided in Friedlander’s works with Yudovich. In the region outside of the essential spectrum, in fact one can also construct unstable invariant manifolds by analogy with finite-dimensional theory and dissipative systems as was done later in works of Lin and Zeng. Next, for the linearized Navier-Stokes system

both in 2D and in 3D, those dominant eigenvalues reappear for small viscosities in a strong spectral limit: for any eigenvalue of the linearized Euler system with and sufficiently small the Navier-Stokes system gains point spectrum in a vicinity of with the same multiplicity, and moreover the Riesz projection corresponding to those spectral subspaces near tends to that of the Euler equation in the uniform operator topology. This result proves to be particularly interesting in view of the fact that the nonlinear Navier-Stokes system inherits instability in (and in fact any for ) from the linearization, a classical result of Yudovich. Thus, any steady flow in 3D that has inviscid spectrum beyond becomes nonlinearly unstable in the vanishing viscosity sense.

3. Dyadic Models

One way to gain an understanding of certain aspects of the equations of fluid motion is to introduce toy models which share properties with the actual equations, but are simpler to analyze. During the last two decades, the work of Friedlander has shaped studies of so-called dyadic models of the fluid equations, which simulate the energy cascade through dyadic frequency shells.⁠Footnote1 Specifically, a th dyadic shell refers to a region where the Fourier frequency lies in the annular domain . The fluid velocity in the th dyadic shell is modeled with a single representative, . In these models, the nonlinearity of the Euler equations is simplified so that only local interactions between neighboring scales are considered. However, simplifications of the nonlinear term vary, and as a consequence the models differ. Some of the first examples of models of this type were derived by Desnyanskiy and Novikov in the context of oceanography, and by Gledzer, whose model was subsequently generalized by Ohkitani and Yamada (and is now known as the GOY model).

The dyadic models that Friedlander explored are designed to share with the actual equations of fluid motion the scaling of the nonlinear term in 3D (which we motivate in the next subsection) and the following properties:

A skew-symmetry property of the nonlinear term,

Conservation of energy for the classical solutions to the Euler equations,

which is a consequence of 18 and the divergence-free condition, as can be seen by pairing the Euler equation 1 with in the sense.

Decay of energy for classical solutions to the Navier-Stokes equations 46,

Broadly speaking, dyadic models provide a framework for studying specific aspects of turbulence theory, while being mathematically accessible. Moreover, in some instances these models motivated results on actual equations of fluid motion, as was the case in e.g. CCFS08.

3.1. Introducing a dyadic model

Let us now recall a version of a dyadic model from CFP07. This model was inspired by a wavelets model introduced in KP02 as a tool to help guide a partial regularity result for actual Navier-Stokes equations with hyperdissipation. We start by briefly revisiting the wavelets model.

First, we recall some terminology from KP02. A cube in is called a dyadic cube if its sidelength is an integer power of , , and the corners of the cube are on the lattice . Let denote the set of dyadic cubes in . Let denote the subset of dyadic cubes having sidelength . Then the parent of , denoted by , is introduced as the unique dyadic cube in which contains . On the other hand, one defines , the th-order grandchildren of , to be the set of those cubes in which are contained in .

The modeling starts by replacing a vector-valued function by a scalar-valued one. An orthonormal family of wavelets is denoted by , with the wavelet associated to the spatial dyadic cube . Then can be represented as

Note that due to spatial localization of ,

On the other hand

Having in mind 21 and 22, a cascade-down operator is defined through its th coefficient as follows:

with the scaling that reflects the upper bound on the nonlinear term implied by 2122. Similarly, a cascade-up operator is defined as the adjoint of via

Then the cascade operator is introduced as

Having defined the Laplacian as , one introduces the following model equations:

• Dyadic Euler equation:

• Dyadic Navier-Stokes equation:

By construction of the cascade operators, we have which implies the skew-symmetry property of the operator ,

A simple consequence of 23 is conservation of energy for the dyadic Euler equations and decay of energy for the dyadic Navier-Stokes equations, at least at the formal level (for sufficiently regular solutions).

The above dyadic models are special cases of the following infinite system of coupled ordinary differential equations, which was studied by Friedlander:

for , where , is a positive parameter related to intermittency, and represents the total energy in the frequencies of order . The force is such that and for all , so that the energy is pumped on low modes.

As we have seen above, the model preserves many features of the fluid equations, while the nonlinearity is simplified by considering only local interactions between scales. Moreover, the choice of the constant ensures that the nonlinearity in the dyadic model obeys the same -based estimates as Euler (see 19) and Navier-Stokes equations (see 20). Thanks to these -based estimates and a certain monotonicity present in the model, a finite time blow-up was exhibited for the inviscid dyadic model KP05, as well as the viscous dyadic model with some “small” degrees of dissipation KP05 or large values of Che08. For instance, solutions blow up when , in which case the dyadic model scales as -dimensional Navier-Stokes equations. Such a monotonicity property resembles monotonicity of certain quantities present in so-called “cooperative” systems (see for example the work of Palais and the work of Bernoff and Bertozzi where singularities in a modified Kuramoto-Sivashinsky equation were identified). Finite time blow-up in the inviscid case was sharpened by Kiselev and Zlatoš. A three-dimensional vector model for the incompressible Euler equations was introduced in FP04, which is similar in some features to a discretized approximate model constructed by Dinaburg and Sinai for the Navier-Stokes equations in Fourier space. It was shown in FP04 that for special initial data the evolution equations of the divergence-free vector model reduced to the scalar dyadic Euler system and finite time blow-up occurs in this model for the three-dimensional incompressible Euler equations. This was a brief snapshot of results for dyadic models around 2005, when Susan Friedlander initiated the study of phenomena related to turbulence at the level of dyadic models.

3.2. Onsager’s and Kolmogorov’s conjectures

Up to now we discussed conservation of energy at a formal level, i.e., for sufficiently regular solutions to Euler equations 1. However one might wonder about the minimal level of regularity of a solution to the Euler equation that guarantees conservation of energy. In fact this seemingly naive question is connected with the statistical theories of turbulence developed by Kolmogorov (1941) and Onsager (1949). In their seminal works, it is suggested that an appropriate mathematical description of three-dimensional turbulent flow is given by weak solutions of the Euler equations which are not regular enough to conserve energy. Onsager conjectured that for the velocity Hölder exponent the energy is conserved⁠Footnote2 A very rough motivation for the appearance of the Hölder exponent is “sharing” of one derivative among three copies of velocity in the skew-symmetry relation 18, which has a crucial role in producing energy equalities 19 and 20. and that this ceases to be true for . This latter phenomenon is now called turbulent or anomalous dissipation. Kolmogorov’s theory predicts that in a fully developed turbulent flow the energy spectrum in the inertial range is given by

where is the average of the energy dissipation rate.

While the rigidity part of Onsager’s conjecture (namely the regime corresponding to the conservation of energy) has been well understood due to works of Eyink and Constantin-E-Titi—prior to works on dyadic models—the flexibility part represented a challenge for a long period of time. Thanks to advances in the method of convex integration due to De Lellis-Székelyhidi, the flexibility part of the Onsager conjecture for the Euler equation has been very recently settled by Isett, and by Buckmaster-De Lellis-Székelyhidi-Vicol for dissipative solutions. However, the state of the puzzle regarding the flexible part of the conjecture was completely open back in 2007. In that context, the dyadic model 24 provided a mathematical laboratory for addressing the phenomena predicted by Onsager and Kolmogorov.

More precisely, in CFP07CFP10 Friedlander et al. showed that the inviscid () dyadic model possesses a unique fixed point , whose energy spectrum , which is just on the borderline of the Sobolev space , where the Sobolev space is equipped with the norm

This showed that all the solutions of the dyadic Euler model stop satisfying energy equality at some time (which resolved Onsager’s conjecture in the negative direction), and the long-time behavior is exactly as predicted by Kolmogorov’s theory of turbulence, but with extreme (or what we now call fully intermittent) energy spectrum. In fact, as it was observed in CF09, the dyadic model 24 covers the whole intermittency range , where the intermittency dimension is connected to the parameter as (see Subsection 3.3). Then

Theorem 3.1 (CFP07CFP10).

The following hold for the dyadic system 24 in the inviscid case :

(1)

Every regular solution (defined to be a solution with bounded norm) satisfies the energy equality.

(2)

There exists a unique fixed point to 24, which is a global attractor. The fixed point is not in . In fact, it lies exactly in the space (defined in 31 below), which takes into account intermittency.

(3)

The energy spectrum of the fixed point

where is the energy input rate, that corresponds to the anomalous energy dissipation rate.

(4)

Every solution blows up in finite time in