Solenoidal Vector Research Articles

Application of Helmholtz–Hodge decomposition to the study of certain vector fields

Smooth vector fields on can be decomposed into the sum of a gradient vector field and divergence-free (solenoidal) vector field under suitable hypotheses. This is called the Helmholtz–Hodge decomposition (HHD), which has been applied to analyze the topological features of vector fields. In this study, we apply the HHD to study certain types of vector fields. In particular, we investigate the existence of strictly orthogonal HHDs, which assure an effective analysis. The first object of the study is linear vector fields. We demonstrate that a strictly orthogonal HHD for a vector field of the form F(x) = Ax can be obtained by solving an algebraic Riccati equation. Subsequently, a method to explicitly construct a Lyapunov function is established. In particular, if A is normal, there exists an easy solution to this equation. Next, we study planar vector fields. In this case, the HHD yields a complex potential, which is a generalization of the notion in hydrodynamics with the same name. We demonstrate the convenience of the complex potential formalism by analyzing vector fields given by homogeneous quadratic polynomials.

Physical topology of three-dimensional unsteady flows with spheroidal invariant surfaces.

Scope is the response of Lagrangian flow topologies of three-dimensional time-periodic flows consisting of spheroidal invariant surfaces to perturbation. Such invariant surfaces generically accommodate nonintegrable Hamiltonian dynamics and, in consequence, intrasurface topologies composed of islands and chaotic seas. Computational studies predict a response to arbitrary perturbation that is dramatically different from the classical case of toroidal invariant surfaces: said islands and chaotic seas evolve into tubes and shells, respectively, that merge into "tube-and-shell" structures consisting of two shells connected via (a) tube(s) by a mechanism termed "resonance-induced merger" (RIM). This paper provides conclusive experimental proof of RIM and advances the corresponding structures as the physical topology of realistic flows with spheroidal invariant surfaces; the underlying unperturbed state is a singular limit that exists only for ideal conditions and cannot be achieved in a physical experiment. This paper furthermore expands existing theory on certain instances of RIM to a comprehensive theory (supported by experiments) that explains all observed instances of this phenomenon. This theory reveals that RIM ensues from perturbed periodic lines via three possible scenarios: truncation of tubes by (i) manifolds of isolated periodic points emerging near elliptic lines or by either (ii) local or (iii) global segmentation of periodic lines into elliptic and hyperbolic parts. The RIM scenario for local segmentation includes a perturbation-induced change from elliptic to hyperbolic dynamics near degenerate points on entirely elliptic lines (denoted "virtual local segmentation"). This theory furthermore demonstrates that RIM indeed accomplishes tube-shell merger by exposing the existence of invariant surfaces that smoothly extend from the tubes into the chaotic shells. These phenomena set the response to perturbation-and physical topology-of flows with spheroidal invariant surfaces fundamentally apart from flows with toroidal invariant surfaces. Its entirely kinematic nature and reliance solely on continuity and solenoidality of the velocity field render the comprehensive theory and its findings universal and generically applicable for (arbitrary perturbation of) basically any incompressible flow-in fact any smooth solenoidal vector field-accommodating spheroidal invariant surfaces.

Addendum to Article “A Counterexample Related to the Regularity of the p-Stokes Problem”

We show existence of a solenoidal vector field u belonging to $$ {W}^{2,q}\left(\Omega \right)\cap {W}_0^{1,s}\left(\Omega \right), $$ s ∈ (1,∞), q ∈ (1, n), such that (δ+|Du|)p−2, p ∈ (1, 2)∪(2,∞), δ ∈ [0, 1], does not belong to the Muckenhoupt class A∞(Ω). This is an extension of our result in [1] proved for δ = 1.

On the Correct Determination of Flow of a Discontinuous Solenoidal Vector Field

We prove inequalities connecting a flow through the (n−1)-dimensional surface S of a smooth solenoidal vector field with its Lp(U)-norm (U is an n-dimensional domain that contains S ). On the basis of these inequalities, we propose a correct definition of the flow through the surface S of a discontinuous solenoidal vector field f ∈ Lp(U) (or, more precisely, of the class of vector fields that are equal almost everywhere with respect to the Lebesgue measure).

Optimal Sobolev regularity for the Stokes equations on a 2D wedge domain

In this note we prove that the solution of the stationary and the instationary Stokes equations subject to perfect slip boundary conditions on a 2D wedge domain admits optimal regularity in the $$L^p$$ -setting, in particular it is $$W^{2,p}$$ in space. This improves known results in the literature to a large extend. For instance, in Maier and Saal (Discrete Contain Dyn Syst Ser S 7(5):1045–1063, 2014) [Theorem 1.1 and Corollary 3] it is proved that the Laplace and the Stokes operator in the underlying setting have maximal regularity. In that result the range of p admitting $$W^{2,p}$$ regularity, however, is restricted to the interval $$1<p<1+\delta $$ for small $$\delta >0$$ , depending on the opening angle of the wedge. This note gives a detailed answer to the question, whether the optimal Sobolev regularity extends to the full range $$1<p<\infty $$ . We will show that for the Laplacian this does only hold on a suitable subspace, but, depending on the opening angle of the wedge domain, not for every $$p\in (1,\infty )$$ on the entire $$L^p$$ -space. On the other hand, for the Stokes operator in the space of solenoidal fields $$L^p_\sigma $$ we obtain optimal Sobolev regularity for the full range $$1<p<\infty $$ and for all opening angles less than $$\pi $$ . Roughly speaking, this relies on the fact that an existing “bad” part of $$L^p$$ for the Laplacian is complemented to the space of solenoidal vector fields.

Resolvent Kernels of Self-Adjoint Extensions of the Laplace Operator on the Subspace of Solenoidal Vector Functions

The Laplace operator on the subspace of solenoidal vector functions of three variables vanishing at the origin together with first derivatives is a symmetric operator with deficiency indices (3). Krein’s theory allows one to derive an expression for the resolvent kernel of a self-adjoint extension of the operator in question as a sum of the Green’s function of the vector Laplace operator and some additional kernel of finite rank.

The Scalar Products of the Regular Analytic Vectors of the Laplace Operator in the Solenoidal Subspace

The Laplace operator on the subspace of solenoidal vector functions of three variables vanishing together with their first derivatives at selected points xn, n = 1, . . .,N, is a symmetric operator with deficiency indices (3N, 3N). The calculation of the scalar products of its regular analytic vectors is the key step in the construction of the resolvents of its self-adjoint extensions via Krein’s formula.

Hydrodynamics and Electromagnetism: Differential—Geometric Aspects and Analogies

The well-known evolution equations of a solenoidal vector field with integral curves frozen into a continuous medium are presented in an invariant form in the four-dimensional spacetime. A fundamental 1-form (4-potential) is introduced, and the problem of variation of the action (integral of the 4-potential along smooth curves) is considered. The extremals of the action in the class of curves with fixed endpoints are described, and the conservation laws generated by symmetry groups are found. Under the assumption that the electric and magnetic fields are orthogonal, Maxwell’s equations are represented as evolution equations of a solenoidal vector field. The role of the velocity field is played by the normalized Poynting vector field.

Equations of Motion in the Theory of Relativistic Vector Fields

Within the framework of the theory of relativistic vector fields, the covariant expressions are presented for the equations of motion of the matter and the field. These expressions can be written either in terms of the field tensors, that is, the fields’ strengths and solenoidal vectors, or in terms the four-potentials, that is, the fields’ scalar and vector potentials. This state of things is due to the fact that the Lagrange function initially implied the complementarity of description in terms of the strengths and the field potentials. It is shown that the equation for the fields, obtained by taking the covariant derivative in the equation for the metric, has a deeper meaning than the ordinary equation of motion of the matter, found with the help of the principle of least action. In particular, the above-mentioned equation for the fields leads to the generalized Poynting theorem, and after integration over the volume it allows us to introduce for consideration the integral vector as a measure of the energy and the fields’ energy fluxes, associated with a system of particles and fields.

Poloidal-toroidal decomposition of solenoidal vector fields in the ball

Poloidal-toroidal decomposition of solenoidal vector fields in the ball

Equations of Motion in the Theory of Relativistic Vector Fields

Within the framework of the theory of relativistic vector fields, the covariant expressions are presented for the equations of motion of the matter and the field. These expressions can be written either in terms of the field tensors, that is, the fields’ strengths and solenoidal vectors, or in terms the four-potentials, that is, the fields’ scalar and vector potentials. This state of things is due to the fact that the Lagrange function initially implied the complementarity of description in terms of the strengths and the field potentials. It is shown that the equation for the fields, obtained by taking the covariant derivative in the equation for the metric, has a deeper meaning than the ordinary equation of motion of the matter, found with the help of the principle of least action. In particular, the above-mentioned equation for the fields leads to the generalized Poynting theorem, and after integration over the volume it allows us to introduce for consideration the integral vector as a measure of the energy and the fields’ energy fluxes, associated with a system of particles and fields.

Poloidal-Toroidal Decomposition of Solenoidal Vector Fields in the Ball

Under study is the polynomial orthogonal basis system of vector fields in the ball which corresponds to the Helmholtz decomposition and is divided into the three parts: potential, harmonic, and solenoidal. It is shown that the decomposition of a solenoidal vector field with respect to this basis is a poloidal-toroidal decomposition (the Mie representation). In this case, the toroidal potentials are Zernike polynomials, whereas the poloidal potentials are generalized Zernike polynomials. The polynomial system of toroidal and poloidal vector fields in a ball can be used for solving practical problems, in particular, to represent the geomagnetic field in the Earth’s core.

The 3D Navier–Stokes Equations: Invariants, Local and Global Solutions

In this article, I consider local solutions of the 3D Navier–Stokes equations and its properties such as an existence of global and smooth solution, uniform boundedness. The basic role is assigned to a special invariant class of solenoidal vector fields and three parameters that are invariant with respect to the scaling procedure. Since in spaces of even dimensions the scaling procedure is a conformal mapping on the Heisenberg group, then an application of invariant parameters can be considered as the application of conformal invariants. It gives the possibility to prove the sufficient and necessary conditions for existence of a global regular solution. This is the main result and one among some new statements. With some compliments, the rest improves well-known classical results.

Weak solutions for the Oseen system in 2D and when the given velocity is not sufficiently regular

The aim of this work is twofold: proving the existence of solution (u,π)∈H1(Ω)×L2(Ω) in bounded domains of R2 and the whole plane for the Oseen problem (O) for solenoidal vector fields v in L2(Ω), and analyzing the same problem in bounded domains of Rn for n=2,3 when h=0, g=0 and the solenoidal field v belongs to Ls(Ω) for s<n.

Simulation of the electromagnetic wall response during Vertical Displacement Events (VDE) in ITER tokamak

The key basis for tokamak plasma disruption modeling is to understand how currents flow to the plasma facing surfaces during plasma disruption events. In ITER tokamak, the occurrence of a limited number of major disruptions will definitively damage the chamber with no possibility to restore the device. In the current exchange plasma-wall-plasma, according to the Helmholtz decomposition theorem, our surface current density in the conducting shell - the unknown of our problem - being a vector field twice continuously differentiable in 3D, has been splited into two components: an irrotational (curl-free) vector field and a solenoidal (divergence-free) vector field. Developing a weak formulation form and minimizing the correspondent energy functionals in a Finite Element approach, we have obtained the space and time distribution of the surface currents. We verified successfully our numerical simulation with an analytical solution with pure homogeneous Neumann B.C. and satisfying the necessary existence condition. By considering the iron core presence in JET tokamak, we have split the magnetization currents - the unknowns in some integral equations - into two components, the first producing a magnetic field in the iron region only and the second producing a magnetic field in the vacuum, obtaining thus a better evaluation of the influence of the iron core on the plasma equilibrium. To reduce the influence of the singularities appearing during the surface currents determination in multiply connected domains (L-shaped domains) we have used a conformal transformation method.

Global well-posedness of 3-D inhomogeneous Navier–Stokes system with initial velocity being a small perturbation of 2-D solenoidal vector field

Motivated by [21], we consider the global wellposedness to the 3-D incompressible inhomogeneous Navier–Stokes equations with large horizontal velocity. In particular, we proved that when the initial density is close enough to a positive constant, then given divergence free initial velocity field of the type (v0h,0)(xh)+(w0h,w03)(xh,x3), we shall prove the global wellposedness of (1.1). The main difficulty here lies in the fact that we will have to obtain the L1(R+;Lip(R3)) estimate for convection velocity in the transport equation of (1.1). Toward this and due to the strong anisotropic properties of the approximate solutions, we will have to work in the framework of anisotropic Littlewood–Paley theory here.

Sharp decay estimates in a bioconvection model with quadratic degradation in bounded domains

The paper studies large time behaviour of solutions to the Keller–Segel system with quadratic degradation in a liquid environment, as given byunder Neumann boundary conditions in a bounded domain Ω ⊂ ℝn, where n ≥ 1 is arbitrary. It is shown that whenever U : Ω × (0,∞) → ℝn is a bounded and sufficiently regular solenoidal vector field any non-trivial global bounded solution of (⋆) approaches the trivial equilibrium at a rate that, with respect to the norm in either of the spaces L1(Ω) and L∞(Ω), can be controlled from above and below by appropriate multiples of 1/(t + 1). This underlines that, even up to this quantitative level of accuracy, the large time behaviour in (⋆) is essentially independent not only of the particular fluid flow, but also of any effect originating from chemotactic cross-diffusion. The latter is in contrast to the corresponding Cauchy problem, for which known results show that in the n = 2 case the presence of chemotaxis can significantly enhance biomixing by reducing the respective spatial L1 norms of solutions.

On Isomorphism for the Space of Solenoidal Vector Fields and Its Application to the Incompressible Flows

We consider the space of solenoidal vector fields in an unbounded domain $\Omega\subset \mathbb{R}^n$ whose boundary is given as a Lipschitz graph. It is shown that the space of solenoidal vector fields is isomorphic to the $n-1$ product space of the space of scalar functions in some natural topology such as $L^2(\Omega)$. As an application, we introduce a systematic reduction of the equations describing the motion of incompressible flows. This gives a new perspective of the derivation of Ukai's solution formula for the Stokes equations in the half space and provides a key step for the generalization of Ukai's approach to the Stokes semigroup in the case of the curved boundary.

Gradients estimation from random points with volumetric tensor in turbulence

We present an estimation method of fully-resolved/coarse-grained gradients from randomly distributed points in turbulence. The method is based on a linear approximation of spatial gradients expressed with the volumetric tensor, which is a 3×3 matrix determined by a geometric distribution of the points. The coarse grained gradient can be considered as a low pass filtered gradient, whose cutoff is estimated with the eigenvalues of the volumetric tensor. The present method, the volumetric tensor approximation, is tested for velocity and passive scalar gradients in incompressible planar jet and mixing layer. Comparison with a finite difference approximation on a Cartesian grid shows that the volumetric tensor approximation computes the coarse grained gradients fairly well at a moderate computational cost under various conditions of spatial distributions of points. We also show that imposing the solenoidal condition improves the accuracy of the present method for solenoidal vectors, such as a velocity vector in incompressible flows, especially when the number of the points is not large. The volumetric tensor approximation with 4 points poorly estimates the gradient because of anisotropic distribution of the points. Increasing the number of points from 4 significantly improves the accuracy. Although the coarse grained gradient changes with the cutoff length, the volumetric tensor approximation yields the coarse grained gradient whose magnitude is close to the one obtained by the finite difference. We also show that the velocity gradient estimated with the present method well captures the turbulence characteristics such as local flow topology, amplification of enstrophy and strain, and energy transfer across scales.

Calculating the inductive electric field in the terrestrial magnetosphere

AbstractThis study presents a theoretical approach to calculate the inductive electric field, and it is further applied to global MHD simulations of the magnetosphere. The contribution of the inductive component to the total electric field is found by decomposing the motional electric field into a superposition of an irrotational and a solenoidal vector and assuming that the time‐varying magnetic field vanishes on the boundary. We find that a localized change in the magnetic field generates an inductive electric field whose effect extends over all space, meaning that the effect of the inductive electric field is global even if the changes in the magnetic field are localized. Application of this formalism to disturbed times provides strong evidence that during periods of increased activity the electric field induced by the localized change in magnetic field can be comparable to (or larger than) the potential electric fields in certain regions. This induced field exhibits significant spatial and temporal variations, which means that particles that drift into different regions of space are being exposed to different means of acceleration. These results suggest that the inductive electric field could have a substantial contribution to particle energization in the near‐Earth region even though the changes in the magnetic fields occur at distances of several tens of Earth radii. This finding is particularly important for ring current modeling which in many cases excludes inductive contributions to the total particle drift.