Investigates fluid flow perturbations using geometric theory.
problem Analyzing linear perturbations in non-equilibrium fluid flows.
method Uses second order variations of the action and Jacobi fields.
result Demonstrates numerical simulations of perturbation dynamics.
Geometric Hydrodynamics tackles open problems in fluid dynamics.
problem Open problems in fluid dynamics and invariant metrics.
method Variational settings, models for invariant metrics, Cauchy and boundary value problems.
result New constructions and recent developments in fluid dynamics.
Study shows how noise can ensure solutions to fluid dynamics equations.
problem Ensuring unique solutions to stochastic fluid dynamics equations.
method Extended existing results to linear advection of k-forms, proving existence and uniqueness of weak L^p-solutions.
result Proved existence and uniqueness of weak L^p-solutions to stochastic linear advection equation of k-forms.
Geometric framework for Newton's equations on diffeomorphism groups.
problem Modeling fluid dynamics and related systems on geometric spaces.
method Geodesic approach and infinite-dimensional information geometry.
result Unified framework for various fluid dynamics equations.
Geometric analysis on diffeomorphism groups for fluid dynamics and information geometry.
problem Geometric analysis of fluid flows and optimal mass transport.
method Review of metrics and topology on diffeomorphism groups.
result Introduction of new metrics and topology for diffeomorphism groups.
Paper develops a new fluid flow model with energy exchange through boundaries.
problem Modeling ideal fluid flow with energy exchange through boundaries.
method Port-Hamiltonian model based on Stokes-Dirac structures.
result Wide range of fluid dynamical systems can be achieved with this model.
Paper shows geometric frequency and Lagrange derivative equivalence for electric and fluid systems.
problem Understanding and classifying system operating conditions based on electric quantity waveform distortions.
method Demonstrates equivalence between geometric frequency and Lagrange derivative through numerical examples.
result Identifies components of Lagrange derivative that relate to geometric frequency and waveform distortions.
FLUID-LLM uses LLMs to predict fluid dynamics with improved accuracy.
problem Leveraging LLMs for CFD due to their pattern recognition abilities but struggles with fluid dynamics complexities.
method Combines pre-trained LLMs with spatiotemporal-aware encoding to predict unsteady fluid dynamics.
result Significant performance improvements in CFD predictions across various datasets.
The paper explores the geometric properties of fluid flows and their symmetries.
problem Understanding the geometric properties of fluid flows and their symmetries.
method Analyzing the Euler equation and its relation to geodesic flows on groupoids of multiphase diffeomorphisms.
result Generalized flows, multiphase fluids, and vortex sheets are all geodesics on certain groupoids of multiphase diffeomorphisms.
Arnold discovered geodesics in fluid dynamics.
problem Understanding fluid motion through geometric perspectives.
method Exploring Euler's equations and their connection to geodesics on diffeomorphism manifolds.
result Geodesics in the space of volume-preserving diffeomorphisms correspond to solutions of Euler's equations.
In this mostly pedagogical tutorial article a brief introduction to modern geometrical treatment of fluid dynamics and electrodynamics is provided. The main technical tool is standard theory of differential forms. In fluid dynamics, the approach is based on general theory of integral invariants (due to Poincare and Car…
Paper proves existence of conjugate points on ellipsoids but not on spheres.
problem Existence of conjugate points in incompressible Euler flows.
method Formulated a differential-geometric criterion (M-criterion) and analyzed flows on spheres and ellipsoids.
result Zonal flows on ellipsoids can satisfy M-criterion, while not on spheres.
Study Godbillon-Vey invariants in non-Lorentzian spacetimes and fluid dynamics.
problem Characterizing and measuring the local spin of spatial leaves in non-Lorentzian spacetimes.
method Relating intrinsic torsion to Godbillon-Vey class, using geometric structures to model fluid dynamics.
result Godbillon-Vey class represents an obstruction to steady flow of fluid and new conservation laws.
In this paper geometrical aspects of perfect fluid spacetime with torse-forming vector field ξare discribed and Ricci soliton in perfect fluid spacetime with torse-forming vector field ξare determined. Conditions for the Ricci soliton to be expanding, steady or shrinking are also given.
The paper studies geometric structures in perfect fluid spacetimes with specific metrics.
problem Analyzing the geometric properties of perfect fluid spacetimes with specific metrics.
method Investigates conditions for conformal Ricci-Yamabe soliton and derives Laplace equations.
result Conditions for expanding, steady, or shrinking conformal Ricci-Yamabe solitons are identified.
Analysis of Vlasov plasma dynamics using matched pair Lie-Poisson formulation.
problem Understanding the dynamics of Vlasov plasma and its kinetic moments.
method Hamiltonian (Lie-Poisson) analysis and matched pair decomposition.
result Observation of mutual interactions between subdynamics in Vlasov plasma.
Introduces GFC for learning complex dynamical systems with geometric constraints.
problem Challenges in accurately modeling and predicting complex dynamical systems with geometric constraints.
method Geometric Contact Flows (GFC) using Riemannian and Contact geometry as inductive biases.
result Ensemble of contactomorphisms adapt the latent contact Hamiltonian model to target dynamics while preserving desirable properties.
The Skew Mean Curvature Flow(SMCF) is a Schrödinger-type geometric flow canonically defined on a co-dimension two submanifold, which generalizes the famous vortex filament equation in fluid dynamics. In this paper, we prove the local existence and uniqueness of general dimensional SMCF in Euclidean spaces.
New method efficiently simulates fluid flows across various conditions.
problem High computational cost in simulating fluid flows.
method Parameter-conditioned sequential generative modeling of neural networks.
result Trained models simulate fluid flows at orders of magnitude faster than traditional methods.
A machine learning model captures non-Newtonian fluid dynamics from molecular details.
problem Creating accurate non-Newtonian fluid models from molecular data.
method Developed a machine learning framework that maps micro-scale polymer configurations to macro-scale fluid dynamics, preserving molecular fidelity.
result The deep non-Newtonian model (DeePN2) accurately predicts fluid behavior without empirical closures. Certain solutions of a sextic sigma-model Lagrangian reminiscent of Skyrme model correspond to perfect fluids with stiff matter equation of state. We analyse from a differential geometric perspective this correspondence extended to general barotropic fluids.
Study reveals how to determine area and curvature from fluid flow resonances.
problem Determining geometric properties from fluid flow data.
method Asymptotic expansion of heat kernel and Steklov spectral invariants.
result Area and total mean curvature can be inferred from Steklov eigenvalues.
Geometrical aspects of a perfect fluid spacetime are described in terms of different curvature tensors and η-Ricci and η-Einstein solitons in a perfect fluid spacetime are determined. Conditions for the Ricci soliton to be steady, expanding or shrinking are also given. In a particular case when the potential vector…
Novel deep learning approach for fast, differentiable fluid simulations.
problem Challenges in solving incompressible fluid dynamics equations efficiently.
method Physics-constrained training approach for convolutional neural networks.
result Trained models can handle various fluid phenomena and offer fast simulations.
Paper extends port-Hamiltonian model to include internal energy for compressible and incompressible flow.
problem Modeling fluid flow dynamics with internal energy and constraints.
method Derived port-Hamiltonian model using interconnection maps and added internal energy and constraint forces.
result Model accurately represents both compressible and incompressible fluid flow.
Study of Riemann solitons and η-hyperbolic Ricci solitons on Bochner-flat Lorentzian Kähler spacetime manifolds.
problem Analyzing soliton behaviors on Bochner-flat Lorentzian Kähler spacetime manifolds.
method Deriving explicit formulas for soliton parameters and analyzing their behaviors.
result Criteria for shrinking, steady, and expanding behaviors of solitons.
Hybrid model combines neural networks and fluid dynamics for efficient, generalized simulations.
problem Inefficient and poor generalization of deep learning approximations of fluid dynamics.
method Combines graph neural networks with a differentiable PDE solver inside a neural network.
result Hybrid model generalizes well to new scenarios and outperforms both neural network and traditional methods.
These are the proceedings of the workshop "Math in the Black Forest", which brought together researchers in shape analysis to discuss promising new directions. Shape analysis is an inter-disciplinary area of research with theoretical foundations in infinite-dimensional Riemannian geometry, geometric statistics, and geo…
Toolbox for stochastic Euler equations using Ebin-Marsden theory.
problem Applying geometric framework to stochastic PDEs.
method Combining infinite-dimensional geometry and stochastic analysis.
result Local well-posedness of maximal solutions for incompressible Euler equation with noise.
Fluid approximations have seen great success in approximating the macro-scale behaviour of Markov systems with a large number of discrete states. However, these methods rely on the continuous-time Markov chain (CTMC) having a particular population structure which suggests a natural continuous state-space endowed with a…
The study identifies conjugate and cut points in ideal fluid motion configurations.
problem Understanding stability and re-convergence of fluid configurations.
method Existence and non-existence of conjugate points in specific fluid configurations, using geometric and physical analysis.
result Existence of conjugate points in Kolmogorov flows and non-existence in Arnold steady states.
Identifies most probable flows for Kunita SDEs in fluid dynamics.
problem Modeling stochastic processes with Eulerian noise and deterministic drifts.
method Equipping the domain with a Riemannian metric from the noise, solving the resulting PDEs.
result Most probable flows differ from deterministic flows, especially under noise.
In this note we survey some recent results for the Euler equations in compressible and incompressible fluid dynamics. The main point of all these theorems is the surprising fact that a suitable variant of Gromov's h-principle holds in several cases.
SPH-ParVI uses fluid dynamics to sample unknown densities efficiently.
problem Sampling partially known densities or using gradients in probabilistic models.
method Smoothed Particle Hydrodynamics (SPH) for modeling fluid dynamics to approximate target densities.
result SPH-ParVI provides fast, flexible, scalable, and deterministic sampling for Bayesian inference and generative models.
The paper explores how a geometric flow can turn a black hole into a traversable wormhole.
problem The study investigates how a static, spherically symmetric black hole can be transformed into a traversable wormhole.
method The approach involves analyzing almost η-Ricci-Yamabe solitons and their geometric coupling with the Hawking temperature. result The geometric flow successfully transforms the black hole into a traversable wormhole, opening the throat and preserving the exact cosmological spacetime.
In this article we study the induced geodesic distance of fractional order Sobolev metrics on the groups of (volume preserving) diffeomorphisms and symplectomorphisms. The interest in these geometries is fueled by the observation that they allow for a geometric interpretation for prominent partial differential equation…
The study identifies unique fluid flow patterns.
problem Understanding incompressible fluid flows with straight streamlines.
method Local differential geometry of line congruences to integrate Euler equations.
result Only specific fluid flows are possible with straight streamlines.
Theory of point vortices extended to closed surfaces.
problem Extending point vortex dynamics to closed surfaces.
method Unified theory of point vortex dynamics on the plane, sphere, and closed surfaces.
result Comprehensive guide to point vortex dynamics on closed surfaces with genus zero and vanishing total vorticity.
Researchers identify surfaces with special fluid flow fields.
problem Understanding fluid flows on curved surfaces.
method Defined and analyzed hydrodynamic Killing vector fields (HKVF) on surfaces.
result Any connected, orientable surface with HKVF is conformally equivalent to one of 14 canonical Riemann surfaces.
FLUID uses flows to unify filtering and smoothing for complex systems.
problem Bayesian filtering and smoothing for high-dimensional nonlinear systems.
method FLUID encodes observation histories into a fixed summary statistic, using flows for filtering and smoothing.
result FLUID provides accurate approximations of filtering and smoothing distributions.
New approach tackles biological fluid adaptivity in AI.
problem Weak performance of AI in dynamic environments.
method Derives strategies from biological fluid adaptivity.
result Equips AI with fluid adaptivity similar to biology.
Paper presents a new approach to continuum mechanics using port-Hamiltonian framework.
problem Geometric formulation of solid and fluid mechanics.
method Port-Hamiltonian framework, Dirac structures, Hamiltonian reduction theory.
result Systematic derivation of port-Hamiltonian models for solid and fluid mechanics.
Study the exponential map on surfaces using fluid dynamics.
problem Exponential map of volume-preserving diffeomorphisms on closed surfaces.
method Fluid dynamical proof of Ebin--Misiołek--Preston theorem and extension of Shnirelman's rigidity result.
result Exponential map is a nonlinear Fredholm mapping of index zero and Fredholm quasiregular.
Over the past few years, we developed a mathematically rigorous method to study the dynamical processes associated to nonlinear Forchheimer flows for slightly compressible fluids. We have proved the existence of a geometric transformation which relates constant mean curvature surfaces and time-invariant pressure distri…
Study on static perfect fluid space-time geometry and boundary estimates.
problem Investigate the geometry and boundary properties of static perfect fluid space-time.
method Used generalized Reilly's formula to establish geometric inequalities and boundary estimates.
result Obtained new boundary estimates involving the Brown-York mass and first eigenvalue of the Jacobi operator.
New minimal surfaces found from vortex crystals.
problem Minimal surfaces and vortex crystals.
method Gluing helicoids into minimal surfaces.
result New minimal surfaces and vortex crystals discovered.
Paper applies fluid dynamics to stock market behavior.
problem Understanding stock market dynamics using physical principles.
method Uses Stokes law to model stock market as fluid system.
result Stock market dynamics can be explained by physical properties.
We are concerned with underlying connections between fluids, elasticity, isometric embedding of Riemannian manifolds, and the existence of wrinkled solutions of the associated nonlinear partial differential equations. In this paper, we develop such connections for the case of two spatial dimensions, and demonstrate tha…