Paper studies periodic solutions to Navier-Stokes equations on hyperbolic manifolds.
problem Existence and uniqueness of asymptotically almost periodic solutions to Navier-Stokes equations on hyperbolic manifolds.
method Dispersive and smoothing estimates for the Stokes equation, Massera-type principle, fixed point argument.
result Existence and uniqueness of asymptotically almost periodic mild solutions in Lp(Γ(TM)) spaces. Paper compares five surface Navier-Stokes derivations and finds some are equivalent.
problem Modeling evolving fluidic surfaces using different principles and coordinate systems.
method Systematic comparison of five derivations using tangential and normal components.
result All derivations yield the same tangential surface Navier-Stokes equations.
Study on Navier-Stokes equations on non-compact manifolds, proving existence and decay of solutions.
problem Existence and asymptotic behavior of solutions to Navier-Stokes equations on non-compact manifolds.
method Used Lp−Lq-dispersive and smoothing estimates of the Stokes semigroup, fixed point arguments, and Gronwall's inequality. result Established existence and exponential decay of almost periodic and asymptotically almost periodic mild solutions.
Stokes equations help uniquely identify manifold metrics from boundary data.
problem Determining Riemannian metric from boundary Cauchy data.
method Proving uniqueness of metric from Stokes equations Cauchy data.
result Partial derivatives of all orders of the metric on the boundary are uniquely determined.
Constructs positive energy representations from Toda equations Stokes data.
problem Creating positive energy representations of affine algebras.
method Using Stokes data of tt*-Toda equations to construct representations.
result Illustrates construction with examples in conformal field theory.
What is the suitable Laplace operator on vector fields for the Navier-Stokes equation on a Riemannian manifold? In this note, by considering Nash embedding, we will try to elucidate different aspects of different Laplace operators such as de Rham-Hodge Laplacian as well as Ebin-Marsden's Laplacian. A probabilistic repr…
New equations reveal viscosity from boundary measurements.
problem Determine viscosity from boundary measurements for incompressible fluids.
method Equivalent new system of elliptic equations, Dirichlet-to-Neumann map analysis.
result Dirichlet-to-Neumann map uniquely determines viscosity and its derivatives on the boundary.
The Navier-Stokes equations on certain manifolds can perform universal computation.
problem Computational universality in viscous fluids.
method Cosymplectic geometry and harmonic 1-forms.
result Stationary Navier-Stokes solutions exhibit Turing completeness.
Paper connects Stokes phenomena to quantum groups and Poisson-Lie groups.
problem Analyse Stokes phenomena in Poisson-Lie groups and quantum groups.
method Use Ug-valued Stokes phenomena to construct quantum group U_hg and relate it to Poisson-Lie group G*.
result Show that Ug-valued Stokes phenomena can be obtained as a semiclassical limit of the KZ associator.
Paper derives explicit expression of Alekseev-Meinrenken diffeomorphism.
problem Understanding the Alekseev-Meinrenken diffeomorphism.
method Via the Stokes phenomenon of meromorphic linear systems of ODEs with Poncaré rank 1.
result Explicit expression of the Alekseev-Meinrenken diffeomorphism.
Explains quantum cohomology of Grassmannians using tt* equations.
problem Relates quantum cohomology of complex Grassmannians to projective space.
method Uses tt* equations and Lie-theoretic connections.
result Illustrates relations between tt* equations and quantum cohomology.
In "Isomonodromy aspects of the tt* equations of Cecotti and Vafa I. Stokes data" (arxiv:1209.2045) we described all smooth solutions of the two-function tt*-Toda equations in terms of asymptotic data, holomorphic data, and monodromy data. In this supplementary article we focus on the holomorphic data and its interpret…
We consider the concept of Stokes-Dirac structures in boundary control theory proposed by van der Schaft and Maschke. We introduce Poisson reduction in this context and show how Stokes-Dirac structures can be derived through symmetry reduction from a canonical Dirac structure on the unreduced phase space. In this way, …
Study well-posedness of generalized Stokes operator on cylindrical domains.
problem Analyzing the generalized Stokes operator on domains with cylindrical ends.
method Using layer potentials and developing algebra tools for limit and jump relations.
result Well-posedness results for the associated Stokes boundary value problem.
A theory for the evolution of a metric g driven by the equations of three-dimensional continuum mechanics is developed. This metric in turn allows for the local existence of an evolving three-dimensional Riemannian manifold immersed in the six-dimensional Euclidean space. The Nash-Kuiper theorem is then applied to th…
In the present report, by using the Stokes-Helmholtz decomposition theorem the 3-dimensional Navier-Stokes equation (NSE) is uncoupled and transformed into a scalar equation for the velocity potential when the flow field is toroidal. The dynamics of the velocity potential is independent of the vector potential. The red…
We describe all smooth solutions of the two-function tt*-Toda equations (a version of the tt* equations, or equations for harmonic maps into SL(n,R)/SO(n)) in terms of (i) asymptotic data, (ii) holomorphic data, and (iii) monodromy data. This allows us to find all solutions with integral Stokes data. These include solu…
Quantum dilogarithm function proven from a linear difference equation.
problem Proving Faddeev's quantum dilogarithm from a linear difference equation.
method Proved Faddeev's quantum dilogarithm using Borel summation of a formal power series solution of a linear difference equation.
result Borel summation of a formal power series solution produces Faddeev's quantum dilogarithm.
We propose a Lie-theoretic definition of the tt*-Toda equations for any complex simple Lie algebra g, based on the concept of topological-antitopological fusion which was introduced by Cecotti and Vafa. Our main result concerns the Stokes data of a certain meromorphic connection, whose isomonodromic deform…
We show that non-uniqueness of the Leray-Hopf solutions of the Navier--Stokes equation on the hyperbolic plane observed in arXiv:1006.2819 is a consequence of the Hodge decomposition. We show that this phenomenon does not occur on the hyperbolic spaces of higher dimension. We also describe the corresponding general Ham…
Model uses Navier-Stokes equations to assess liquidity and systemic risk.
problem Traditional models fail to capture real market fluctuations and extreme events.
method Develops and validates a mathematical model based on Navier-Stokes equations, incorporating 13 macroeconomic and financial parameters.
result Model effectively describes liquidity dynamics, systemic risk, and extreme scenarios.
Paper variates Navier-Stokes-Fourier system for thermodynamic consistency.
problem Modeling compressible fluid dynamics with thermodynamic constraints.
method Variational discretization with discrete exterior calculus.
result Derives a nonholonomic variational integrator for NSF system.
Developed MF-PINNs to solve coupled Stokes-Darcy equations more accurately.
problem Solving coupled Stokes-Darcy equations with varying physical constants.
method Combining VP and SV forms with adjusted weights in MF-PINNs.
result Improved accuracy of streamline and pressure fields in numerical experiments.
Complex Chern-Simons theory reveals peacock patterns in perturbative series.
problem Understanding the structure of partition functions in complex Chern-Simons theory.
method Analyzing the partition function as a holomorphic function and using resurgence theory.
result Perturbative series are resurgent, with trans-series involving non-perturbative variables.
Analyzes tt*-structures from ADE-type Stokes data.
problem Classifying tt*-structures over C∗. method Isomonodromic deformations with upper unitriangular real Stokes matrices.
result Establishes a direct analytic realization of the ADE classification. Framework predicts Navier-Stokes solutions on 2D domains using graph neural networks.
problem Predicting stationary Navier-Stokes solutions in non-parametrized 2D geometries.
method Graph-based multi-fidelity learning framework combining reduced-order models, Transformers, and Mamba architectures.
result Mamba architecture reduces computational cost while maintaining performance.
In the first part of the paper, we solve the boundary and monodromy problems for the isomonodromy equation of the n×n meromorphic linear system of ordinary differential equations with Poncaré rank 1. In particular, we derive an explicit expression of the Stokes matrices of the linear system, via the boundary …
We prove the analyticity in time for solutions of two parabolic equations in the whole space, without any decaying or vanishing conditions. One of them involves solutions to the heat equation of exponential growth of order 2 on $\M$. Here $\M$ is Rd or a complete noncompact manifold with Ricci curvature bounded f…
In this paper, a new algorithm based on differential geometry viewpoint to solve the 3D rotating Navier-Stokes equations with complex Boundary is proposed, which is called Bi-parallel algorithm. For xample, it can be applied to passage flow between two blades in impeller and circulation flow through aircrafts with comp…
Global stability proved for Navier-Stokes equations on hyperbolic space.
problem Stability of the Navier-Stokes equations on hyperbolic space.
method Proved global stability with exponential decay rate for small initial data.
result Exponential decay rate of $μλ_\Def^{(3)}$ for Navier-Stokes equations on hyperbolic space.
The paper extends stability theorem for Navier-Stokes equations to negatively curved manifolds.
problem Stability of the three-dimensional Navier-Stokes equations on negatively curved manifolds.
method Analysis of the deformation Laplacian, overcoming obstacles with curvature pinching and spectral gap.
result Global mild solution with exponential decay for small data on negatively curved manifolds.
We consider a numerical approach for the incompressible surface Navier-Stokes equation. The approach is based on the covariant form and uses discrete exterior calculus (DEC) in space and a semi-implicit discretization in time. The discretization is described in detail and related to finite difference schemes on stagger…
We prove generic fibre of Painlevé moduli spaces are Weinstein handlebodies.
problem Understanding the generic fibre of Painlevé moduli spaces.
method Attach Weinstein handles to Stokes Legendrian.
result Generic fibre of Painlevé moduli spaces are Weinstein handlebodies.
Proves resurgent nature of a series solution to deformed Painlevé I equation.
problem Analyzing the resurgent nature of a series solution to the deformed Painlevé I equation.
method Proves resurgent nature through formal ℏ-power series solution and Borel summability. result Borel transform defines a global multivalued holomorphic function on a Fermat quintic surface.
Summary of main work 1999-2012
Study infinite-dimensional Toda manifold at irregular singularity, revealing non-uniqueness of formal solutions.
problem Non-uniqueness of formal solutions to the Dubrovin equation at irregular singularity.
method Revisited canonical coordinates, formal solutions analysis, Borel resummation, Stokes matrices computation.
result Infinite-dimensional Stokes matrices computed from resummed formal solutions.
New framework uses dynamics to justify Gaussian process for turbulent flows.
problem Lack of rigorous justification for Gaussian process priors in turbulent flows.
method Introduces a dynamics-informed Gaussian process framework based on quasi-Gaussianity.
result Provides a principled, long-time dynamical justified GP prior for turbulent flows.
Novel neural operator predicts complex spatiotemporal dynamics from partial observations.
problem Capturing complex operator dynamics in infinite-dimensional function spaces.
method Integrates Koopman operator theory with deep neural networks to approximate nonlinear operators between Banach spaces.
result BNO achieves robust zero-shot super-resolution in unsteady flow prediction and outperforms conventional methods.
Trains neural networks to efficiently solve Navier-Stokes equations across parameter space.
problem Efficiently solving Navier-Stokes equations in parameter space.
method Physics-informed neural networks, active learning algorithm.
result Neural networks can accurately interpolate and aggregate solutions to physical problems.
Using numerical simulations of the axisymmetric Navier-Stokes equations with swirl on a no-slip flat boundary, Hsu-Notsu-Yoneda [J. Fluid Mech. 2016] observed the creation of a high-vorticity region on the boundary near the axis of symmetry. In this paper, using a differential geometric approach, we prove that such flo…
In Part I (arXiv:1209.2045) we computed the Stokes data, though not the "connection matrix", for the smooth solutions of the tt*-Toda equations whose existence we established by p.d.e. methods. Here we give an alternative proof of the existence of some of these solutions by solving a Riemann-Hilbert problem. In the pro…
We consider finite energy and L2 differential forms associated with strongly local regular Dirichlet forms on compact connected topologically one-dimensional spaces. We introduce notions of local exactness and local harmonicity and prove the Hodge decomposition, which in our context says that the orthogonal compleme…
Analyzes how quadratic differential trajectories change with variation, proving a wall-crossing formula.
problem Analyzing how the number of trajectories of quadratic differentials changes with variation.
method Proves an analytic wall-crossing formula using Fock-Goncharov coordinates and characterizes birational automorphisms.
result Characterizes certain birational automorphisms and computes Stokes automorphisms.
The physics informed neural network (PINN) is evolving as a viable method to solve partial differential equations. In the recent past PINNs have been successfully tested and validated to find solutions to both linear and non-linear partial differential equations (PDEs). However, the literature lacks detailed investigat…
Stokes' theorem's boundary maximizes entropy.
problem Characterizing the boundary of a manifold using entropy.
method Maximizing entropy for codimension-1 submanifolds satisfying Stokes' theorem.
result The boundary of a manifold maximizes the entropy functional.
Several intrinsic topological ways to encode connections on vector bundles on smooth complex algebraic curves will be described. In particular the notion of {\em Stokes decompositions} will be formalised, as a convenient intermediate category between the Stokes filtrations and the Stokes local systems/wild monodromy re…
Proves existence and uniqueness of solutions for A_n tt*-Toda equations.
problem Existence and uniqueness of solutions for A_n tt*-Toda equations.
method Proof of existence and uniqueness for any n, new treatment of asymptotic data.
result Existence and uniqueness of global solutions for any n.
The paper develops a new approach to solve vector-valued PDEs on manifolds with minimal regularity.
problem Well-posedness and Lp-based Sobolev regularity of vector-valued PDEs on compact manifolds. method Develops a parametrization-free variational approach using classical results in reflexive Banach spaces.
result Establishes higher-order Wm,p regularity for vector-valued PDEs on manifolds of minimal regularity.