Divergence-free Vector Fields Research Articles

Localization of Beltrami fields: Global smooth solutions and vortex reconnection for the Navier-Stokes equations

We introduce a class of divergence-free vector fields on R3 obtained after a suitable localization of Beltrami fields. First, we use them as initial data to construct unique global smooth solutions of the three dimensional Navier-Stokes equations. The relevant fact here is that these initial data can be chosen to be large in any critical space for the Navier–Stokes problem, however they satisfy the nonlinear smallness assumption introduced in [11]. As a further application of the method, we use these vector fields to provide analytical example of vortex-reconnection for the three-dimensional Navier-Stokes equations on R3. To do so, we exploit the ideas developed in [15] but differently from this latter we cannot rely on the non-trivial homotopy of the three-dimensional torus. To overcome this obstacle we use a different topological invariant, i.e. the number of hyperbolic zeros of the vorticity field.

A decomposition theorem of surface vector fields and spectral structure of the Neumann-Poincaré operator in elasticity

We prove that the space of vector fields on the boundary of a bounded domain with the Lipschitz boundary in three dimensions is decomposed into three subspaces: elements of the first one extend to inside the domain as divergence-free and rotation-free vector fields, the second one to the outside as divergence-free and rotation-free vector fields, and the third one to both the inside and the outside as divergence-free harmonic vector fields. We then show that each subspace in the decomposition is infinite-dimensional. We also prove under a mild regularity assumption on the boundary that the decomposition is almost direct in the sense that any intersection of two subspaces is finite-dimensional. We actually prove that the dimension of intersection is bounded by the first Betti number of the boundary. In particular, if the boundary is simply connected, then the decomposition is direct. We apply this decomposition theorem to investigate spectral properties of the Neumann-Poincaré operator in elasticity, whose cubic polynomial is known to be compact. We prove that each linear factor of the cubic polynomial is compact on each subspace of decomposition separately and those subspaces characterize eigenspaces of the Neumann-Poincaré operator. We then prove all the results for three dimensions, decomposition of surface vector fields and spectral structure, are extended to higher dimensions. We also prove analogous but different results in two dimensions.

On shifting the principal eigenvalue of Dirichlet problem to infinity with non-transversal incompressible drift

We prove that it is always possible to add some divergence free drift vector field to some two dimensional spherical Dirichlet problem, such that the resulting principal eigenvalue lies above a prescribed bound. By construction those drift vector fields vanish on the boundary and their flow lines individually stay away from the boundary. The capacity of those drift vector fields to accelerate diffusivity originates from high frequency oscillation of the associated flow lines. The lower bounds for the spectrum are obtained through isoperimetric inequalities for flow invariant functions.

Error estimates of divergence-free generalized moving least squares (Div-Free GMLS) derivatives approximations in Sobolev spaces

The divergence-free generalized moving least squares (Div-Free GMLS) approximation has recently been utilized to solve some incompressible fluid flows problems. In our recent work (Mohammadi and Dehghan (2021) [28]), we have presented its formulation more precisely, and also the error estimates of derivatives have been carried out in L∞(Ω), where Ω⊂Rd is a bounded set satisfying an interior cone condition. However, the error estimates of this vector-valued approximation in Sobolev spaces are not done. So, in this paper we make the error estimates of Div-Free GMLS derivatives approximations in Lq(Ω), where 1≤q≤∞, using a stable local divergence-free polynomial reproduction property. Note that, the method is a direct approximants of exact derivatives of a divergence-free vector field, which possesses the optimal rates of convergence. This vector-valued technique can also be developed to find the numerical solution of the incompressible fluid flows problems easier than the other available mesh-dependent methods. Finally, we have shown how the proposed approximation can recover the velocity field variable of the well-known Darcy's problem in a two-dimensional space.

Existence of global symmetries of divergence-free fields with first integrals

The relationship between symmetry fields and first integrals of divergence-free vector fields is explored in three dimensions in light of its relevance to plasma physics and magnetic confinement fusion. A Noether-type theorem is known: for each such symmetry, there corresponds a first integral. The extent to which the converse is true is investigated. In doing so, a reformulation of this Noether-type theorem is found for which the converse holds on what is called the toroidal region. Some consequences of the methods presented are quick proofs of the existence of flux coordinates for magnetic fields in high generality, without needing to assume a symmetry such as in the cases of magneto-hydrostatics or quasi-symmetry.

Rigidity Results On Generalized m-Quasi Einstein Manifolds with Associated Affine Killing Vector Field.

We study a non-trivial generalized $m$-quasi Einstein manifold $M$ with finite $m$ and associated divergence-free affine Killing vector field, and show that $M$ reduces to an $m$-quasi Einstein manifold. In addition, if $M$ is complete, then it splits as the product of a line and an $(n-1)$-dimensional negatively Einstein manifold. Finally, we show that the same result holds for a complete non-trivial $m$-quasi Einstein manifold $M$ with finite $m$ and associated affine Killing vector field.

Exploring Finite-Time Singularities and Onsager’s Conjecture with Endpoint Regularity in the Periodic Navier Stokes Equations

It has recently been proposed by the author of the present work that the periodic NS equations (PNS) with high energy assumption can breakdown in finite time but with sufficient low energy scaling the equations may not exhibit finite time blowup.This article gives a general model using specific periodic special functions, that is degenerate elliptic Weierstrass P functions whose presence in the governing equations through the forcing terms simplify the PNS equations at the centers of cells of the 3-Torus. Satisfying a divergence free vector field and periodic boundary conditions respectively with a general spatio-temporal forcing term which is smooth and spatially periodic, the existence of solutions which blowup in finite time for PNS can occur starting with the first derivative and higher with respect to time. P. Isett (2016) has shown that the conservation of energy fails for the 3D incompressible Euler flows with Hlder regularity below 1/3.(Onsager’s second conjecture) The endpoint regularity in Onsager’s conjecture is addressed, and it is found that conservation of energy occurs when the Hlder regularity is exactly 1/3. The endpoint regularity problem has important connections with turbulence theory. Finally very recent developed new governing equations of fluid mechanics are proposed to have no finite time singularities.

A sharp interface immersed edge-based smoothed finite element method with extended fictitious domain scheme

This paper proposes a versatile and robust immersed edge-based smoothed finite element method with the mass conservation algorithm (IESFEM/Mass) to solve partitioned fluid–structure interaction (FSI). A gradient smoothing technique was used to solve the system governing equations, which can improve the calculated capability of the linear triangular elements in two phases. Based on the quadratic sharp interface representation of immersed boundary, an extended fictitious domain constructed by a least squares method approximately corrected the residual flux error. The compatibility for boundary conditions on moving interfaces was satisfied, thus eliminating spurious oscillations. The results from all numerical examples were consistent with those from the existing experiments and published numerical solutions. Furthermore, the present divergence-free vector field had a faster-converged rate in the flow velocity, pressure, and FSI force. Even if in distorted meshes, the proposed algorithm maintained a stable accuracy improvement. The aerodynamics of one- and two-winged flapping motions in insect flight has been investigated through the IESFEM/Mass. It can be seen that the wing–wake interaction mechanism is a vital factor affecting the lift. The applicability of the present method in the biological FSI scenario was also well-demonstrated.

Lyapunov exponents and entropy for divergence-free Lipschitz vector fields

Let mathfrak {X}^{0,1}_nu (M) be the subset of divergence-free Lipschitz vector fields defined on a closed Riemannian manifold M endowed with the Lipschitz topology Vert ,{cdot },Vert _{0,1} where nu is the volume measure. Let mathfrak {X}^{0,1}_{nu ,ell }(M)subset mathfrak {X}^{0,1}_nu (M) be the subset of vector fields satisfying the ell -property, a property that implies C^1-regularity nu -almost everywhere. We prove that there exists a residual subset with respect to Vert ,{cdot },Vert _{0,1} such that Pesin’s entropy formula holds, i.e. for any the metric entropy equals the integral, with respect to nu , of the sum of the positive Lyapunov exponents.

Adaptive partition of unity interpolation method with moving patches

The adaptive partition of unity interpolation method, introduced by Aiton and Driscoll, using Chebyshev local interpolants, is explored for interpolating functions with sharp gradients representing two-medium problems. For functions that evolve under vector fields, the partition of unity patches (covers) can be shifted and resized to follow the changing dynamics of local profiles. The method is tested for selected 1D and 2D two-medium problems with linear divergence-free vector fields. In those cases, the volume fraction in each patch contributing to volume conservation throughout the domain can be kept in high accuracy down to machine precisions. Applications that could benefit from the method include volume tracking and multiphase flow modeling.

Sharp Uncertainty Principle inequality for solenoidal fields

This paper solves the L2 version of Maz'ya's open problem [1, Section 3.9] on the sharp uncertainty principle inequality∫RN|∇u|2dx∫RN|u|2|x|2dx≥CN(∫RN|u|2dx)2 for solenoidal (namely divergence-free) vector fields u=u(x) on RN. The best value of the constant for N≥3 turns out to be CN=14(N2−4(N−3)+2)2 which exceeds the original value N2/4 for unconstrained fields. Moreover, we show the attainability of CN and specify the profiles of the extremal solenoidal fields: for N≥4, the extremals are proportional to a poloidal field that is axisymmetric and unique up to the axis of symmetry; for N=3, there exist extremal toroidal fields, in addition to extremal poloidal fields; for N=2, the extremal fields are all toroidal.

Locally-symplectic neural networks for learning volume-preserving dynamics

We propose locally-symplectic neural networks LocSympNets for learning the flow of phase volume-preserving dynamics. The construction of LocSympNets stems from the theorem of the local Hamiltonian description of the divergence-free vector field and the splitting methods based on symplectic integrators. Symplectic gradient modules of the recently proposed symplecticity-preserving neural networks SympNets are used to construct invertible locally-symplectic modules, which compositions result in volume-preserving neural networks LocSympNets. To further preserve properties of the flow of a dynamical system LocSympNets are extended to symmetric locally-symplectic neural networks SymLocSympNets, such that the inverse of SymLocSympNets is equal to the feed-forward propagation of SymLocSympNets with the negative time step, which is a general property of the flow of a dynamical system. LocSympNets and SymLocSympNets are studied numerically considering learning linear and nonlinear volume-preserving dynamics. In particular, we demonstrate learning of linear traveling wave solutions to the semi-discretized advection equation, periodic trajectories of the Euler equations of the motion of a free rigid body, and quasi-periodic solutions of the charged particle motion in an electromagnetic field. LocSympNets and SymLocSympNets can learn linear and nonlinear dynamics to a high degree of accuracy even when random noise is added to the training data. In all numerical experiments, SymLocSympNets have produced smaller errors in long-time predictions compared to the LocSympNets. When learning a single trajectory of the rigid body dynamics locally-symplectic neural networks can learn both quadratic invariants of the system with absolute relative errors below 1%. In addition, SymLocSympNets produce qualitatively good long-time predictions, when the learning of the whole system from randomly sampled data is considered. LocSympNets and SymLocSympNets can produce accurate short-time predictions of quasi-periodic solutions, which is illustrated in the example of the charged particle motion in an electromagnetic field.

Spectral analysis and domain truncation for Maxwell's equations

We analyse how the spectrum of the anisotropic Maxwell system with bounded conductivity σ on a Lipschitz domain Ω is approximated by domain truncation. First we prove a new non-convex enclosure for the spectrum of the Maxwell system, with weak assumptions on the geometry of Ω and none on the behaviour of the coefficients at infinity. We also establish a simple criterion for non-accumulation of eigenvalues at iR as well as resolvent estimates. For asymptotically constant coefficients, we describe the essential spectrum and show that spectral pollution may occur only in the essential numerical range We(L∞)⊂R of the quadratic pencil L∞(ω)=μ∞−1curl2−ω2ε∞, acting on divergence-free vector fields. Further, every isolated spectral point of the Maxwell system lying outside We(L∞) and outside the part of the essential spectrum on iR is approximated by spectral points of the Maxwell system on the truncated domains. Our analysis is based on two new abstract results on the (limiting) essential spectrum of polynomial pencils and triangular block operator matrices, which are of general interest. We believe our strategy of proof could be used to establish domain truncation spectral exactness for more general classes of non-self-adjoint differential operators and systems with non-constant coefficients.

Helicity, linking and the distribution of null-homologous periodic orbits for Anosov flows* *2022. This work is licensed under a CC BY license. Solly Coles was supported by the UK Engineering and Physical Sciences Research Council.

This paper concerns connections between dynamical systems, knots and helicity of vector fields. For a divergence-free vector field on a closed three-manifold that generates an Anosov flow, we show that the helicity of the vector field may be recovered as the limit of appropriately weighted averages of linking numbers of periodic orbits, regarded as knots. This complements a classical result of Arnold and Vogel that, when the manifold is a real homology three-sphere, the helicity may be obtained as the limit of the normalised linking numbers of typical pairs of long trajectories. We also obtain results on the asymptotic distribution of weighted averages of null-homologous periodic orbits.

On the advection-diffusion equation with rough coefficients: Weak solutions and vanishing viscosity

We deal with the vanishing viscosity scheme for the transport/continuity equation ∂tu+div(ub)=0 drifted by a divergence-free vector field b. Under general Sobolev assumptions on b, we show the convergence of such scheme to the unique Lagrangian solution of the transport equation. Our proof is based on the use of stochastic flows and yields quantitative rates of convergence. This offers a completely general selection criterion for the transport equation (even beyond the distributional regime) which compensates the wild non-uniqueness phenomenon for solutions with low integrability arising from convex integration constructions, as shown in recent works [8,28–30], and rules out the possibility of anomalous dissipation.

Sharp second order uncertainty principles

We study sharp second order inequalities of Caffarelli-Kohn-Nirenberg type in the euclidian space RN, where N denotes the dimension. This analysis is equivalent to the study of uncertainty principles for special classes of vector fields. In particular, we show that when switching from scalar fields u:Rn→C to vector fields of the form u→:=∇U (U being a scalar field) the best constant in the Heisenberg Uncertainty Principle (HUP) increases from N24 to (N+2)24, and the optimal constant in the Hydrogen Uncertainty Principle (HyUP) improves from (N−1)24 to (N+1)24. As a consequence of our results we answer to the open question of Maz'ya [21, Section 3.9] in the case N=2 regarding the HUP for divergence free vector fields.

Local divergence-free polynomial interpolation on MAC grids

Divergence-free vector fields play an important role in many types of problems, including the incompressible Navier-Stokes equations and the equations for magnetohydrodynamics. In the discrete setting, these fields are often obtained by projection, resulting in a discrete approximation of the continuous field that is discretely divergence-free. For many applications, such as tracing particles, this discrete field must then be extended to the entire region using interpolation. This interpolated field is continuous and differentiable (almost everywhere), but in general it will not be divergence-free. In this paper, we construct approximation schemes with the property that discretely divergence-free data interpolates to an analytically divergence-free vector field. Our focus is on data stored in a MAC grid layout that is divergence free under the second order central difference stencil, a case that is common in projection methods for the Navier-Stokes equations. While existing schemes with this property are known, they tend to be global (the interpolated value at a point depends on data stored on the grid far from that point) or discontinuous. We construct C0 and C1 continuous approximation schemes for 2D and 3D that are local and satisfy the divergence-free property. We also construct interpolating versions of the schemes that reproduce the MAC data at face centers. All eight schemes are explicit piecewise polynomials over small stencils.

Continuity Equation and Characteristic Flow for Scalar Hencky Plasticity

AbstractWe investigate uniqueness issues for a continuity equation arising out of the simplest model for plasticity, Hencky plasticity. The associated system is of the form where 𝜇 is a nonnegative measure and 𝜎 a two‐dimensional divergence‐free unit vector field. After establishing the Sobolev regularity of that field, we provide a precise description of all possible geometries of the characteristic flow, as well as of the associated solutions. © 2022 Courant Institute of Mathematics and Wiley Periodicals LLC.

Solution properties of the incompressible Euler system with rough path advection

The present paper aims to establish the local well-posedness of Euler's fluid equations on geometric rough paths. In particular, we consider the Euler equations for the incompressible flow of an ideal fluid whose Lagrangian transport velocity possesses an additional rough-in-time, divergence-free vector field. In recent work, we have demonstrated that this system can be derived from Clebsch and Hamilton-Pontryagin variational principles that possess a perturbative geometric rough path Lie-advection constraint. In this paper, we prove the local well-posedness of the system in L2-Sobolev spaces Hm with integer regularity m≥⌊d/2⌋+2 and establish a Beale-Kato-Majda (BKM) blow-up criterion in terms of the Lt1Lx∞-norm of the vorticity. In dimension two, we show that the Lp-norms of the vorticity are conserved, which yields global well-posedness and a Wong-Zakai approximation theorem for the stochastic version of the equation.

Rectifiability of divergence-free fields along invariant 2-tori

We find conditions under which the restriction of a divergence-free vector field B to an invariant toroidal surface S is rectifiable; namely constant in a suitable global coordinate system. The main results are similar in conclusion to Arnold’s Structure Theorems but require weaker assumptions than the commutation [B,nabla times B] = 0. Relaxing the need for a first integral of B (also known as a flux function), we assume the existence of a solution u: S rightarrow {mathbb {R}} to the cohomological equation Bvert _S(u) = partial _n B on a toroidal surface S mutually invariant to B and nabla times B. The right hand side partial _n B is a normal surface derivative available to vector fields tangent to S. In this situation, we show that the field B on S is either identically zero or nowhere zero with Bvert _S/Vert BVert ^2 vert _S being rectifiable. We are calling the latter the semi-rectifiability of B (with proportionality Vert BVert ^2 vert _S). The nowhere zero property relies on Bers’ results in pseudo-analytic function theory about a generalised Laplace-Beltrami equation arising from Witten cohomology deformation. With the use of de Rham cohomology, we also point out a Diophantine integral condition where one can conclude that Bvert _S itself is rectifiable. The rectifiability and semi-rectifiability of Bvert _S is fundamental to the so-called magnetic coordinates, which are central to the theory of magnetically confined plasmas.