The paper connects eigenvalue problems for various operators and establishes inequalities and asymptotic formulas for heat traces.
problem Eigenvalue problems and heat trace asymptotics for different operators.
method Establishes connections and inequalities for eigenvalues and heat traces.
result Eigenvalue inequalities and three-term asymptotic formulas for heat traces of various operators.
Proves invertibility of layer potentials for generalized Stokes operators on smooth domains.
problem Invertibility of layer potentials for generalized Stokes operators on smooth domains.
method Developed algebra toolkit to handle layer operators' limit and jump relations; proved Fredholm property and invertibility.
result Proves invertibility of layer potentials for generalized Stokes operators on smooth domains.
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.
The author presents the generalized Stokes theorem for R-linear forms on Lie algebroids (which can be non-local). We apply the Stokes formula on forms to prove that two homotopic homomorphisms of Lie algebroids implies the existence of a chain operator joining their pullback operators.
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…
For a projective algebraic variety V with isolated singularities, endowed with a metric induced from an embedding, we consider the analysis of the natural partial differential operators on the regular part of V. We show that, in the complex case, the Laplacians of the de Rham and Dolbeault complexes are discrete op…
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.
For a given bounded domain Ω⊂Rn with smooth boundary, we explicitly calculate the first two coefficients of the asymptotic expansion of the heat trace associated with the Stokes operator as t→0+. These coefficients (i.e., heat invariants) provide precise information for the volume of the domain $…
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.
Numerical experiments support conjecture about opers and nonabelian Hodge.
problem Testing predictions of Gaiotto-Moore-Neitzke and Gaiotto conjectures.
method Numerical experiments on polynomial holomorphic differentials.
result Supports conjectural formulas for Stokes data and Hitchin metric tensor.
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…
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. In this paper we prove that Dirac operators on non-compact complete orbifolds which are sufficiently regular at infinity, admit a unique extension. Additonally, we prove a generalized orbifold Stokes'/Divergence theorem.
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. Unified framework for Arnold-type invariants via dual complexes and finite-difference structures.
problem Study of Arnold-type invariants of immersed curves and surfaces.
method Framework on dual complexes, locally normalized maps, finite-difference structures, and Shumakovitch-type identities.
result Unified evaluation of Arnold-type invariants St(1) and St(2) on dual skeleta. A conservative discretization of incompressible Navier-Stokes equations is developed based on discrete exterior calculus (DEC). A distinguishing feature of our method is the use of an algebraic discretization of the interior product operator and a combinatorial discretization of the wedge product. The governing equatio…
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.
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…
Geometric theory of integration developed in SDG.
problem Combining infinitesimal contributions to finite quantities.
method Analysis of differential forms and new operators in SDG.
result Discovery of new types of differential forms and operators.
Deep neural network approximates flow averages for rough walls in multiscale simulations.
problem Approximating flow averages in rough-wall Stokes flow simulations.
method Fourier neural operator for local averages, parameterized by local wall geometry.
result Stable and accurate HMM solution with reduced micro problem solving cost.
Lean 4 formalizes Stokes' theorem for smooth singular cubes.
problem Formalizing Stokes' theorem for singular cubes in arbitrary dimensions.
method Using true differential-form pullback via Frechet derivative, bridging to mathlib4's extDeriv.
result d^2=0 for singular cubical chains, chain-level Stokes extended.
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. 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.
A kinematic method selects the deformation Laplacian for fluid dynamics on Riemannian manifolds.
problem Ambiguity in viscous operator choice for Navier-Stokes equations on Riemannian manifolds.
method Kinematic construction of strain rate from Lie-dragged vectors, excluding Hodge Laplacian due to antisymmetric part.
result Kinematic selection uniquely identifies the deformation Laplacian, resolving analytical obstructions.
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 submanifolds with boundaries in Heisenberg groups using Stokes' Theorem.
problem Understanding submanifolds with boundaries in Heisenberg groups.
method Introduce and study CH1-regular submanifolds with boundaries, prove Stokes' Theorem for them. result Prove a version of Stokes' Theorem for submanifolds with boundaries in Heisenberg groups.
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.
We compute Stokes matrices and monodromy for the quantum cohomology of projective spaces. We prove that the Stokes' matrix of the quantum cohomology coincides with the Gram matrix in the theory of derived categories of coherent sheaves.
Algorithm describes Fourier transform of Stokes data at infinity.
problem Understanding the Fourier transform of Stokes data at infinity.
method Topological description and algorithmic approach using recent results and language of Stokes local systems.
result Explicit isomorphisms between wild character varieties are induced.
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.
Categorifies Stokes coefficients in Chern-Simons theory models.
problem Stokes phenomenon in Chern-Simons theory around flat connections.
method Finite-dimensional model for analytically continued Chern-Simons theory, categorification of Stokes coefficients.
result Stokes coefficients can be promoted to graded vector spaces.
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 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.
This work is thought as an operative guide to discrete exterior calculus (DEC), but at the same time with a rigorous exposition. We present a version of (DEC) on cubic cell, defining it for discrete manifolds. An example of how it works, it is done on the discrete torus, where usual Gauss and Stokes theorems are recove…
Stokes theorem holds for Lipschitz forms on a smooth manifold.
The paper validates Stokes' theorem for differential subcomplexes in positively graded Lie groups.
problem Validating Stokes' theorem for differential subcomplexes in positively graded Lie groups.
method Introducing geometric conditions and spectral complexes to recover Stokes' theorem on locally smooth intrinsic graphs.
result Stokes' theorem holds for Rumin complex and new spectral complexes on Carnot groups.
In this paper, we show the existence of real-analytic stationary Navier-Stokes flows with isotropic streamlines in all latitudes in some simply-connected flow region on a rotating round sphere. We also exclude the possibility of having a Poiseuille's flow profile to be one of these stationary Navier-Stokes flows with i…
We present type θ Stokes' theorem for type θ k-chains which extends the fundamental theorem of calculus in higher dimensions.
Study submanifolds with boundary in Heisenberg groups, proving Stokes' Theorem.
problem Understanding submanifolds with boundary in sub-Riemannian Heisenberg groups.
method Introduced examples and proved Stokes' Theorem involving Rumin's differential forms.
result Stokes' Theorem for submanifolds with boundary in Heisenberg groups.
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.
Improved lower bounds for poly-Laplacian eigenvalues in arbitrary dimensions.
problem Lower bounds for higher eigenvalues of the poly-Laplacian operator.
method Sharp inequalities and eigenvalue bounds in low and arbitrary dimensions.
result Improved lower bounds for eigenvalues of the poly-Laplacian in arbitrary dimensions.
By proving graph theoretical versions of Green-Stokes, Gauss-Bonnet and Poincare-Hopf, core ideas of undergraduate mathematics can be illustrated in a simple graph theoretical setting. In this pedagogical exposition we present the main proofs on a single page and add illustrations. While discrete Stokes is is old, the …
The Gauss formula is extended to various Laplacians on submanifolds.
problem Deriving formulas for Laplacians on submanifolds.
method Extending the Gauss formula to different types of Laplacians.
result Formulas for various Laplacians on submanifolds.
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.
Stokes-Dirac structures are infinite-dimensional Dirac structures defined in terms of differential forms on a smooth manifold with boundary. These Dirac structures lay down a geometric framework for the formulation of Hamiltonian systems with a nonzero boundary energy flow. Simplicial triangulation of the underlaying m…
We consider infinite dimensional port-Hamiltonian systems. Based on a power balance relation we introduce the port-Hamiltonian system representation where we pay attention to two different scenarios, namely the non-differential operator case and the differential operator case regarding the structural mapping, the dissi…
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.