Fluid Dynamics, Computer Science, and Geometry – Barcelona 2024

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After the June 2023 workshop organised in Royal Institution, London on “Navier-Stokes regularity, fluid computing, and machine learning”, which hosted the guest lecture by Prof. Eva Miranda, who was awarded the Hardy lecturer title, Eva, Daniel, and Ángel organised a first of its own kind workshop in Barcelona, in September 2024. Spanning almost a week, the workshop hosted plenary talks, exploratory sessions, and other social-academic activities and witnessed up to hundred participants who were experts/enthusiasts of three different fields — computer science, geometry, and functional analysis — and still their works were centred around the regularity issue of the Euler/Navier-Stokes equations (NSE).

Let us state NSE in the following format:

\frac{\partial u_{i}}{\partial t} + \sum_{j=1}^{n} u_{j} \frac{\partial u_{i}}{\partial x_{j}} = \nu \Delta u_{i} - \frac{\partial p}{\partial x_{i}} + f_{i}(x,t) \quad (x \in \mathbb{R}^{n}, t \geq 0) \quad (1a)
\textrm{div} \, u = \sum_{i=1}^{n}\frac{\partial u_{i}}{\partial x_{i}}=0 \quad \quad (x \in \mathbb{R}^{n}, t \geq 0)) \quad (1b)

While it is certain that we would expect certain proceedings from the workshop soon, I decided to summarise, in this post, three key lectures that were helpful to me in certain way. The notes from the workshop can be found here.

In the first lecture, Robert Ghrist discussed his work on the theory of sheaf cohomology (inspired by Hodge theory and with an aim to find combinatorial graph Laplacian but for the sheaves of lattices) for cellular sheaves defined in the category of lattices. The motivation for this construction is that the problems in topics such as opinion dynamics, distributed optimization, network flow and coding problems, signal processing, are in certain sense, data science problems, and vaguely speaking, it is possible to reformulate the structure of a highly dense data in a local-to-global setting. Upon the introduction of sheaf theory in this setting, the local data could be stored in stalks and the local contraints in the restriction maps. Ultimately, the cohomology of the sheaf collects the global information in its global sections, thereby formalizing the local-to-global structure.

Given a cell complex {\textrm{X}}, let us define a sheaf {\mathcal{F}: \textrm{F}_{\textrm{C}}(\textrm{X}) \rightarrow \textrm{Ltc}}, such that the restriction maps for the cells are the connections, and 0-cochains {C^{0}(\textrm{X};\mathcal{F})} are choices of data on the vertex stalks. The Laplacian, analogous to Hodge Laplacian for sheaves of vector spaces, also called Tarski Laplacian, is given by

\displaystyle \left( L \mathbf{x} \right)_v = \bigwedge_{e \in \delta v} \left( \mathcal{F}_{v \trianglelefteq e} \right) \cdot \left( \bigwedge_{w \in \delta e} \left( \mathcal{F}_{w \trianglelefteq e} \cdot \left( x_w \right) \right) \right) \ \ \ \ \

where {\mathcal{F}_{\sigma \triangleleft \tau} = \left( \left( \mathcal{F}_{\sigma \trianglelefteq \tau} \right) \cdot , \left( \mathcal{F}_{\sigma \trianglelefteq \tau} \right) \cdot \right) } is the canonical form of a connection. The Laplacian operator models the propagation of information via the sheaf restriction maps. Further construction on this details down various kind of cohomologies that are typically defined on the cellular sheaves, namely: Grandis cohomology; Tarski cohomology; and Hodge theory, such that

\displaystyle GH \subset TH \subset HH \ \ \ \ \

On a side note, Ghrist also conducted a survey on the opinions of the workshop’s participants surrounding the Navier-Stokes and Euler equations’ singularity in finite-time. The outcomes of the survey are here; with the sentiment shifting more towards finite-time singularity than it was in 2007.

This was followed by Kai Cieliebak’s work on the stationary solution to the inviscid Navier-Stokes equations (Euler equations). The main result proved in the lecture is that if the velocity field of the stationary solution has no zeroes and a real analytic Bernoulli function, then it is possible to perform rescaling to the Reeb vector field, which does not hold true if the real analyticity hypothesis breaks down. Stationary Euler equations are given by  \nabla_X X = - \nabla p L_{X} \mu = 0 where {(X,p)} are the time-independent velocity field and pressure variables. Additionally, it is possible to rewrite Eq. 1(a) as {\textrm{curl} \, X \times X = - \nabla h}, where {h:=p+|X|^2 / 2} is the Bernoulli function. The key result proved is as follows

Theorem 1 Let {X} be a solution of the stationary Euler equations Eq. 1 (a,b) with respect to a metric {g} and volume form {\mu} on a closed, oriented 3-manifold {M}. Suppose that {X} has no zeroes and its Bernoulli function is real analytic. Then X also solves equation Eq. 1(a) with respect to a different metric {\hat{g}} such that the corresponding Bernoulli function is constant.

The key corollary of this result is as follows

Corollary 2 A vector field X as in Theorem (1) can be rescaled by a positive function to the Reeb vector field of some stable Hamiltonian structure.

In fact, there is a counterexample proven next to these results, which establishes the following result

Proposition 3 There exists a smooth, nowhere vanishing vector field {X} solving the stationary Euler equations Eq. 1(a,b) on some closed, oriented 3-manifold {M} which cannot be rescaled by a positive function to the Reeb vector field of a stable Hamiltonian structure.

This means that it is sometimes possible to have vector fields with certain properties which do not stay in the analytic sets, and thus any analysis on the Euler equations, would need a generic technique (irrespective of the categories of sets that one is dealing with). Further details of the techniques used to prove these results are given in Cieliebak & Volkov 2015.

The next lecture after this was by Cornelia Vizman on the dual pairs involved in the group of volume preserving diffeomorphisms, and their occurence in the Euler equations. As Joe Monaghan puts it, “A fluid moves to get out of its own way as efficiently as possible”. Given a Riemannian manifold {(M,g)} with Levi-Civita connection {\nabla} and {\mu} volume form, the theorem by Vladimir Arnold in 1966 states that the geodesic equation on the group {G=\textrm{Diff}_\textrm{vol}(M)} of volume preserving diffeomorphisms of a Riemannain manifold {(M,g)} for the right variant {L^2} metric is given by

\displaystyle \langle u,v \rangle = \int_{M} g(u,v) \mu \quad \textrm{such that}, u,v \in \mathfrak{g} = \mathfrak{X}_{\textrm{vol}}(M) and is equivalent to the Euler equation for ideal fluid flow with velocity {u} and pressure {p}. The vorticity 2-form {\xi \in d \Omega^1 (M)} satisfies the Helmholtz equation {\partial_t \xi = - L_{u} \xi}. The smooth dual of {\mathfrak{g}=\mathfrak{X}_{\textrm{vol}(M)}} is given by {\mathfrak{g}^{*}_{\textrm{reg}}=\Omega^{1}(M)/d\Omega^{0}(M)} with the non-degenerate pairing \displaystyle ([\gamma],u)= \int_{M}\gamma(u) \mu such that if {H^{1}_{\textrm{dR}}(M)=0}, then the space of vorticities is given by {d \Omega^1(M)=\mathfrak{g}^{*}_{\textrm{reg}}} and the vorticity 2-form is confined to a coadjoint orbit.

Defining an ideal fluid dual pair goes back to an article by Marsden and Weinstein. As noted above, the dual of the Lie algebra is given by the space of vorticities, and the Kelvin’s circulation can be reformulated as the preservation of coadjoint orbits. Therefore, the Clebsch variables are just momentum maps. There are certainly more structures — point vortices, vortex patches, and vortex filaments which form coadjoint orbits and their symplectic structures are defined on {\mathbb{R}^1}, and understood using a special kind of Clebsch variables, but further details on this are available in the work by Marsden and Weinstein.

P.S.: At the time of the writing, eastern parts of Spain (Valencia) experienced heavy floods because of climate change aggravated cold drop. The post dedicates to the ongoing efforts to tackling and dealing with the aftermath of the disaster.

Additional note: My friend and colleague, Adnan Mahmud, kindly attended the conference. During his stay at Barcelona, which he was visiting for the first time, he wrote a poem, available on his website. The title is, “Obhāgār Desh”, and its translation, transliteration, recitation and melody, is available there.

Published by Saksham

Ph.D. graduate in fluid dynamics from the University of Cambridge

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