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We consider almost Riemann solitons $$(V,\lambda )$$ in a Riemannian manifold and underline their relation to almost Ricci solitons. When V is of gradient type, using Bochner formula, we explicitly express the function $$\lambda $$ by means of the gradient vector field V and illustrate the result with suitable examples. Moreover, we deduce some properties for the particular cases when the potential vector field of the soliton is solenoidal or torse-forming, with a special view towards curvature.

In this note, we obtain a simpler expression for the constant number given by Costin–Maz’ya on the sharp Hardy–Leray inequality for a class of solenoidal (namely divergence-free) vector fields, with respect to any radial power-weighted measure. The dependence of the constant on the weight exponent will be clear.

We give an example of a bounded divergence free autonomous vector field in $${\mathbb {R}}^3$$ (and of a nonautonomous bounded divergence free vector field in $${\mathbb {R}}^2$$ ) and of a smooth initial data for which the Cauchy problem for the corresponding transport equation has 2 distinct solutions. We then show that both solutions are limits of classical solutions of transport equations for appropriate smoothings of the vector fields and of the initial data.

New formulas are derived for once-differentiable 3-dimensional fields, using the operator . This new operator has a property similar to that of the Laplacian operator; however, unlike the Laplacian operator, the new operator requires only once-differentiability. A simpler formula is derived for the classical Helmholtz decomposition. Orthogonality of the solenoidal and irrotational parts of a vector field, the uniqueness of the familiar inverse-square laws, and the existence of solution of a system of first-order PDEs in 3 dimensions are proved. New proofs are given for the Helmholtz Decomposition Theorem and the Divergence theorem. The proofs use the relations between the rectangular-Cartesian and spherical-polar coordinate systems. Finally, an application is made to the study of Maxwell’s equations.

In this article, discrete variants of several results from vector calculus are studied for classical finite difference summation by parts operators in two and three space dimensions. It is shown that existence theorems for scalar/vector potentials of irrotational/solenoidal vector fields cannot hold discretely because of grid oscillations, which are characterised explicitly. This results in a non-vanishing remainder associated with grid oscillations in the discrete Helmholtz Hodge decomposition. Nevertheless, iterative numerical methods based on an interpretation of the Helmholtz Hodge decomposition via orthogonal projections are proposed and applied successfully. In numerical experiments, the discrete remainder vanishes and the potentials converge with the same order of accuracy as usual in other first-order partial differential equations. Motivated by the successful application of the Helmholtz Hodge decomposition in theoretical plasma physics, applications to the discrete analysis of magnetohydrodynamic (MHD) wave modes are presented and discussed.

We present a methodology to impose micromechanical constraints, i.e. stress equilibrium at grain and sub-grain scale, to an arbitrary (non-equilibrated) voxelized stress field obtained, for example, by means of synchrotron X-ray diffraction techniques. The method consists in finding the equilibrated stress field closest (in $L^2$-norm sense) to the measured non-equilibrated stress field, via the solution of an optimization problem. The extraction of the divergence-free (equilibrated) part of a general (non-equilibrated) field is performed using the Hodge decomposition of a symmetric matrix field, which is the generalization of the Helmholtz decomposition of a vector field into the sum of an irrotational field and a solenoidal field. The combination of: a) the Euler-Lagrange equations that solve the optimization problem, and b) the Hodge decomposition, gives a differential expression that contains the bi-harmonic operator and two times the curl operator acting on the measured stress field. These high-order derivatives can be efficiently performed in Fourier space. The method is applied to filter the non-equilibrated parts of a synthetic piecewise constant stress fields with a known ground truth, and stress fields in Gum Metal, a beta-Ti-based alloy measured in-situ using Diffraction Contrast Tomography (DCT). In both cases, the largest corrections were obtained near grain boundaries., Comment: 19 pages, 5 figures, submitted to mechanics of materials

We set up the analytical framework for studying the threshold for the appearance of a lift force exerted by a viscous steady fluid (the wind) on the deck of a bridge. We model this interaction as in a wind tunnel experiment, where at the inlet and outlet sections the velocity field of the fluid has a Poiseuille flow profile. Since in a symmetric configuration the appearance of lift forces is a consequence of non-uniqueness of solutions, we compute an explicit threshold on the incoming flow ensuring uniqueness. This requires building an explicit solenoidal extension of the prescribed Poiseuille flow and bounding some embedding and cutoff constants.

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).

The magnetic excitation of eddy currents in thin metal sheets is a mathematically challenging problem with many practical applications. Since the eddy current density is a solenoidal field on a 2-D plane, we may express it as the curl of a separate vector field with only one scalar component. This new field is governed by a unique form of the vector-Poisson equation that is readily solved numerically through standard finite-difference techniques. For the special case of a weakly induced eddy current, the self-inductance terms can be neglected to produce a sparse system matrix that is easily inverted. For a strongly induced eddy current, self-inductance must be included at the cost of a more complex and dense system matrix. The method is validated against the CST EM Studio software suite by producing nearly identical results in only a tiny fraction of the computational time.

Velocity statistics is a direct probe of the dynamics of interstellar turbulence. Its observational measurements are very challenging due to the convolution between density and velocity and projection effects. We introduce the projected velocity structure function, which can be generally applied to statistical studies of both sub- and super-sonic turbulence in different interstellar phases. It recovers the turbulent velocity spectrum from the projected velocity field in different regimes, and when the thickness of a cloud is less than the driving scale of turbulence, it can also be used to determine the cloud thickness and the turbulence driving scale. By applying it to the existing core velocity dispersion measurements of the Taurus cloud, we find a transition from the Kolmogorov to the Burgers scaling of turbulent velocities with decreasing length scales, corresponding to the large-scale solenoidal motions and small-scale compressive motions, respectively. The latter occupy a small fraction of the volume and can be selectively sampled by clusters of cores with the typical cluster size indicated by the transition scale., 5 pages, 2 figures, accepted for publication in MNRAS

In the present paper, a Vortex Particle Method is combined with a Boundary Element Method for the study of viscous incompressible planar flow around solid bodies. The method is based on Chorins operator splitting approach for the Navier–Stokes equations written in vorticity–velocity formulation, and consists of an advection step followed by a diffusion step. The evaluation of the advection velocity exploits the Helmholtz–Hodge Decomposition, while the no-slip condition is enforced by an indirect boundary integral equation. The above decomposition and splitting are discussed in comparison to the analogous decomposition for the pressure-velocity formulation of the governing equations. The Vortex Particle Method is implemented with a completely meshless algorithm, as neither advection nor diffusion requires topological connection of the point lattice. The results of the meshless approach are compared with those obtained by a mesh-based Finite Volume Method, where the pseudo-compressible iteration is exploited to enforce the solenoidal constraint on the velocity field. Several benchmark tests were performed for verification and validation purposes. In particular, we analyzed the two-dimensional flow past a circle, past an ellipse with incidence and past a triangle for different Reynolds numbers.

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.

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 aim of this paper is to give an existence and uniqueness of solutions for the Cauchy problem of the Navier–Stokes type equations associated to the sublaplacian provided by the left invariant vector fields on the Heisenberg group. To avoid the difficulty of the non-commutative which is intrinsic in 2-step stratified Lie groups, we construct the solenoidal space by using the right invariant vector fields on the Heisenberg group.

We give a survey on local and semi-local bifurcations of divergence-free vector fields. These differ for low dimensions from generic' bifurcations of structure-less dissipative' vector fields, up to a dimension-threshold that grows with the co-dimension of the bifurcation.

Cascades of temperature and entropy fluctuations are studied by numerical simulations of stationary three-dimensional compressible turbulence with a heat source. The fluctuation spectra of velocity, compressible velocity component, density and pressure exhibit the $-5/3$ scaling in an inertial range. The strong acoustic equilibrium relation between spectra of the compressible velocity component and pressure is observed. The $-5/3$ scaling behaviour is also identified for the fluctuation spectra of temperature and entropy, with the Obukhov–Corrsin constants close to that of a passive scalar spectrum. It is shown by Kovasznay decomposition that the dynamics of the temperature field is dominated by the entropic mode. The average subgrid-scale (SGS) fluxes of temperature and entropy normalized by the total dissipation rates are close to 1 in the inertial range. The cascade of temperature is dominated by the compressible mode of the velocity field, indicating that the theory of a passive scalar in incompressible turbulence is not suitable to describe the inter-scale transfer of temperature in compressible turbulence. In contrast, the cascade of entropy is dominated by the solenoidal mode of the velocity field. The different behaviours of cascades of temperature and entropy are partly explained by the geometrical properties of SGS fluxes. Moreover, the different effects of local compressibility on the SGS fluxes of temperature and entropy are investigated by conditional averaging with respect to the filtered dilatation, demonstrating that the effect of compressibility on the cascade of temperature is much stronger than on the cascade of entropy.

There are given algebraic and integral identities for a pair or a triple of plane solenoidal fields. As applications, we obtain sufficient potentiality conditions for a plane vector field. The integral identities are also important for exact a priori estimates.

In the context of intra-cluster medium turbulence, it is essential to be able to split the turbulent velocity field in a compressive and a solenoidal component. We describe and implement a new method for this aim, i.e., performing a Helmholtz-Hodge decomposition, in multi-grid, multi-resolution descriptions, focusing on (but not being restricted to) the outputs of AMR cosmological simulations. The method is based on solving elliptic equations for a scalar and a vector potential, from which the compressive and the solenoidal velocity fields, respectively, are derived through differentiation. These equations are addressed using a combination of Fourier (for the base grid) and iterative (for the refinement grids) methods. We present several idealised tests for our implementation, reporting typical median errors in the order of $1\unicode{x2030}$-$1\%$, and with 95-percentile errors below a few percents. Additionally, we also apply the code to the outcomes of a cosmological simulation, achieving similar accuracy at all resolutions, even in the case of highly non-linear velocity fields. We finally take a closer look to the decomposition of the velocity field around a massive galaxy cluster., Comment: 12 pages, 8 figures; accepted for publication in Computer Physics Communications

The cross-spectral density (CSD) of the non-linear forcing in resolvent analyses is here quantified for the first time for turbulent channel flows. Direct numerical simulations (DNS) at $Re_{\tau} =179$ and $Re_{\tau} =543$ are performed. The CSDs are computed for highly energetic structures typical of buffer-layer and large-scale motions, for different temporal frequencies. The CSD of the non-linear forcing is shown not to be uncorrelated (white) in space, which implies the forcing is structured. Since the non-linear forcing is non-solenoidal by construction and the velocity of an incompressible flow is affected only by the solenoidal part of the forcing, this solenoidal part is evaluated. It is shown that the solenoidal part of the non-linear forcing is the combination of oblique streamwise vortices and a streamwise component which counteract each other, as in a destructive interference. It is shown that a rank-2 approximation of the forcing, with only the most energetic SPOD (spectral proper orthogonal decomposition) modes, leads to the bulk of the response. The projections of the non-linear forcing onto the right-singular vectors of the resolvent are evaluated. The left-singular vectors of the resolvent associated with very low-magnitude singular values are non-negligible since the non-linear forcing term has a non-negligible projection onto the linear sub-optimals of resolvent analysis. The same projections are computed when the forcing is modelled with an eddy-viscosity approach. It is clarified that this modelling improves the accuracy of the prediction since the projections are closer to those associated with the non-linear forcing from DNS data.

Divergence-free (div-free) and curl-free vector fields are pervasive in many areas of science and engineering, from fluid dynamics to electromagnetism. A common problem that arises in applications is that of constructing smooth approximants to these vector fields and/or their potentials based only on discrete samples. Additionally, it is often necessary that the vector approximants preserve the div-free or curl-free properties of the field to maintain certain physical constraints. Div/curl-free radial basis functions (RBFs) are a particularly good choice for this application as they are meshfree and analytically satisfy the div-free or curl-free property. However, this method can be computationally expensive due to its global nature. In this paper, we develop a technique for bypassing this issue that combines div/curl-free RBFs in a partition of unity framework, where one solves for local approximants over subsets of the global samples and then blends them together to form a div-free or curl-free global approximant. The method is applicable to div/curl-free vector fields in $\R^2$ and tangential fields on two-dimensional surfaces, such as the sphere, and the curl-free method can be generalized to vector fields in $\R^d$. The method also produces an approximant for the scalar potential of the underlying sampled field. We present error estimates and demonstrate the effectiveness of the method on several test problems.

The structural bifurcation of a 2D divergence free vector field \begin{document}$ \mathbf{u}(\cdot, t) $\end{document} when \begin{document}$ \mathbf{u}(\cdot, t_0) $\end{document} has an interior isolated singular point \begin{document}$ \mathbf{x}_0 $\end{document} of zero index has been studied by Ma and Wang [ 23 ]. Although in the class of divergence free fields which undergo a local bifurcation around a singular point, the ones with index zero singular points are generic, this class excludes some important families of symmetric flows. In particular, when \begin{document}$ \mathbf{u}(\cdot, t_0) $\end{document} is anti-symmetric with respect to \begin{document}$ \mathbf{x}_0 $\end{document} , or symmetric with respect to the axis located on \begin{document}$ \mathbf{x}_0 $\end{document} and normal to the unique eigendirection of the Jacobian \begin{document}$ D\mathbf{u}(\cdot, t_0) $\end{document} , the vector field must have index 1 or -1 at the singular point. Thus we study the structural bifurcation when \begin{document}$ \mathbf{u}(\cdot, t_0) $\end{document} has an interior isolated singular point \begin{document}$ \mathbf{x}_0 $\end{document} with index -1, 1. In particular, we show that if such a vector field with its acceleration at \begin{document}$ t_0 $\end{document} both satisfy the aforementioned symmetries then generically the flow will undergo a local bifurcation. Under these generic conditions, we rigorously prove the existence of flow patterns such as pairs of co-rotating vortices and double saddle connections. We also present numerical evidence of the Stokes flow in a rectangular cavity showing that the bifurcation scenarios we present are indeed realizable.

We show well-posedness of a diffuse interface model for a two-phase flow of two viscous incompressible fluids with different densities locally in time. The model leads to an inhomogeneous Navier-Stokes/Cahn-Hilliard system with a solenoidal velocity field for the mixture, but a variable density of the fluid mixture in the Navier-Stokes type equation. We prove existence of strong solutions locally in time with the aid of a suitable linearization and a contraction mapping argument. To this end we show maximal $L^2$-regularity for the Stokes part of the linearized system and use maximal $L^p$-regularity for the linearized Cahn-Hilliard system., 25 pages

In this paper, we deal with the solenoidal conservative Lie algebra associated to the classical normal form of Hopf-zero singular system. We concentrate on the study of some representations and Z 2 -equivariant normal form for such singular differential equations. First, we list some of the representations that this Lie algebra admits. The vector fields from this Lie algebra could be expressed by the set of ordinary differential equations where the first two of them are in the canonical form of a one-degree of freedom Hamiltonian system and the third one depends upon the first two variables. This representation is governed by the associated Poisson algebra to one sub-family of this Lie algebra. Euler’s form, vector potential, and Clebsch representation are other representations of this Lie algebra that we list here. We also study the non-potential property of vector fields with Hopf-zero singularity from this Lie algebra. Finally, we examine the unique normal form with non-zero cubic terms of this family in the presence of the symmetry group Z 2 . The theoretical results of normal form theory are illustrated with the modified Chua’s oscillator.

A mathematically correct description is presented on the interrelations between the dynamics of divergence free vector fields on an oriented 3-dimensional manifold $M$ and the dynamics of Hamiltonian systems. It is shown that for a given divergence free vector field $X$ with a global cross-section there exist some 4-dimensional symplectic manifold $\tilde{M}\supset M$ and a smooth Hamilton function $H: \tilde{M}\to \mathbb R$ such that for some $c\in \mathbb R$ one gets $M = \{H=c\}$ and the Hamiltonian vector field $X_H$ restricted on this level coincides with $X$. For divergence free vector fields with singular points such the extension is impossible but the existence of local cross-section allows one to reduce the dynamics to the study of symplectic diffeomorphisms in some sub-domains of $M$. We also consider the case of a divergence free vector field $X$ with a smooth integral having only finite number of critical levels. It is shown that such a noncritical level is always a 2-torus and restriction of $X$ on it possesses a smooth invariant 2-form. The linearization of the flow on such a torus (i.e. the reduction to the constant vector field) is not always possible in contrast to the case of an integrable Hamiltonian system but in the analytic case ($M$ and $X$ are real analytic), due to the Kolmogorov theorem, such the linearization is possible for tori with Diophantine rotation numbers., 17 pages, no figures

In this study, topological features of an incompressible two-dimensional flow far from any boundaries is considered. A rigorous theory has been developed for degenerate streamline patterns and their bifurcation. The homotopy invariance of the index is used to simplify the differential equations of fluid flows which are parameter families of divergence-free vector fields. When the degenerate flow pattern is perturbed slightly, a structural bifurcation for flows with symmetry is obtained. We give possible flow structures near a bifurcation point. A flow pattern is found where a degenerate cusp point appears on the x-axis. Moreover, we also show that bifurcation of the flow structure near a non-simple degenerate critical point with double symmetry is generic away from boundaries. Finally, we give an application of the degenerate flow patterns emerging when index 0 and -2 in a double lid driven cavity and in two dimensional peristaltic flow.

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.

Several perturbation tools are established in the volume preserving setting allowing for the pasting, extension, localized smoothing and local linearization of vector fields. The pasting and local linearization hold in all classes of regularity ranging from $C^{1}$ to $C^{\infty}$ (H\"older included). For diffeomorphisms, a conservative linearized version of Franks lemma is proved in the $C^{r,\alpha}$ ($r\in\mathbb{Z}^+$, $0, Comment: final version, to appear in ETDS; important footnote on page 9

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 ( v 0 h , 0 ) ( x h ) + ( w 0 h , w 0 3 ) ( x h , x 3 ) , we shall prove the global wellposedness of (1.1) . The main difficulty here lies in the fact that we will have to obtain the L 1 ( R + ; Lip ( R 3 ) ) 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.

The history of Bergman spaces goes back to the book [4] in the early fifties by S. Bergman, where the first systematic treatment of the subject was given, and since then there have been a lot of papers devoted to this area. Some standard works here are [5] , [8] , [15] , [28] and the references therein, which contain a broad summary and historical notes of the subject, that frees us from referring to missing details. In their recent works Gonzalez-Cervantes, Luna-Elizarraras and Shapiro [11] , [12] , laid the foundations for the generalization of the theory of Bergman spaces induced by Laplacian (sometimes called solenoidal and irrotational, or harmonic) vector fields by taking advantage on the intimate connections between harmonic vector fields theory and quaternionic analysis for the Moisil-Theodorescu operator (MT-operator for short). A deeper discussion of the last mentioned relation can be found in [1] . On the setting of general bounded domains in R 3 , we extend the aforementioned study in a very natural way to the case of an introduced v-MT-operator for v ∈ R 3 , proving several properties of induced Bergman spaces and some relative results about Stokes and Borel-Pompieu formulas for v-MT-hyperholomorphic functions, i.e., functions which belong to kernel of the v-MT-operator. In addition, we show that this v-MT operator satisfies a conformal co-variant property. Application of all the above allows to study of Bergman type spaces induced by v-Laplacian vector fields theory, which represents the main goal of this paper.

In this note, we prove the flowbox theorem for divergence-free Lipschitz vector fields., Dans cette note, nous prouvons le théorème du flot tubulaire pour les champs vectoriels Lipschitz à divergence nulle.

A numerical solution of the problem of recovering the solenoidal part of three-dimensional vector field using the incomplete tomographic data is proposed. Namely, values of the ray transform for all straight lines, which are parallel to one of the coordinate planes, are known. The recovery algorithms are based on the method of approximate inverse.

In this paper, we extend our study of mass transport in multicomponent isothermal fluids to the incompressible case. For a mixture, incompressibility is defined as the independence of average volume on pressure, and a weighted sum of the partial mass densities stays constant. In this type of models, the velocity field in the Navier–Stokes equations is not solenoidal and, due to different specific volumes of the species, the pressure remains connected to the densities by algebraic formula. By means of a change of variables in the transport problem, we equivalently reformulate the PDE system as to eliminate positivity and incompressibility constraints affecting the density, and prove two type of results: the local-in-time well-posedness in classes of strong solutions, and the global-in-time existence of solutions for initial data sufficiently close to a smooth equilibrium solution.

Finite dilatational effects in wall-bounded compressible turbulence are investigated. We find that finite correlation between the solenoidal and the dilatational parts of the velocity field account for a nonnegligible fraction of the turbulent shear stress near walls.

Smooth vector fields on $\mathbb{R}^n$ 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 $\textbf{F}(\textbf{x}) = A \textbf{x}$ 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., 22 pages, 5 figures. To appear in J. Phys. A: Math. Theor

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.

The aim of this work is twofold: proving the existence of solution ( u , π ) ∈ H 1 ( Ω ) × L 2 ( Ω ) in bounded domains of R 2 and the whole plane for the Oseen problem ( O ) for solenoidal vector fields v in L 2 ( Ω ) , and analyzing the same problem in bounded domains of R n for n = 2 , 3 when h = 0 , g = 0 and the solenoidal field v belongs to L s ( Ω ) for s n .

We provide precise estimates of the Poincare constants firstly for scalar functions and secondly for solenoidal (i.e. divergence free) vector fields (in both cases with vanishing Dirichlet traces on the boundary) on 2d-annuli by the use of the first eigenvalues of the scalar Laplacian and the Stokes operator, respectively. In our non-dimensional setting each annulus ΩA is defined via two concentrical circles with radii A/2 and A/2+1. Additionally, corresponding problems on domains Ωσ*, the 2d-annuli from [7], are investigated - for comparison but also to provide limits for A→0. In particular, the Green's function of the Laplacian on Ωσ* with vanishing Dirichlet traces on ∂Ωσ* is used to show that for σ→0 the first eigenvalue here tends to the first eigenvalue of the corresponding problem on the open unit circle. On the other hand, we take advantage of the so-called small-gap limit for A→∞.

This paper studies the analyticity of the Stokes semigroup in an infinite cylinder or more generally a cylindrical domain with several exits to infinity in the space \({C_{0,\sigma}}\), the \({L^\infty}\)-closure of all smooth compactly supported solenoidal vector fields. These domains are not strictly admissible in the sense of the first two authors (Abe and Giga in J Evol Equ 14:1–28, 2014). However, it is shown that these domains are still admissible which yields the analyticity in \({C_{0,\sigma}}\). A new proof based on a blow-up argument is given to derive an \({L^\infty}\)-type resolvent estimate which enables us to conclude that the analyticity angle of the Stokes semigroup in \({C_{0,\sigma}}\) is \({\pi/2}\).

Summary We have developed a new divergence preserving method for the reconstruction of the Cartesian components of a vector field from the orthogonal projection of a vector field to the normals to edges in two dimensional. In this method, discrete divergences computed from the nodal components and from the normal ones are exactly the same. Our new method consists of two stages. At the first stage, we use an extended version of the local procedure described in [J. Comput. Phys., 139:406–409, 1998] to obtain a ‘reference’ nodal vector. This local procedure is exact for linear vector fields; however, the discrete divergence is not preserved. Then, we formulate a constrained optimization problem, in which this reference vector plays the role of a target, and the divergence constraints are enforced by using Lagrange multipliers. It leads to the solution of ‘elliptic’ like discrete equations for the cell-centered Lagrange multipliers. The new global divergence preserving method is exact for linear vector fields. We describe all details of our new method and present numerical results, which confirm our theory. Copyright © 2016 John Wiley & Sons, Ltd.

The paper presents the methods describing the behavior of lines of a three-dimensional vector field near null points of order greater than one assuming that the potential of that field satisfies the Laplace equation. Some general analytic solutions are presented for the second order case.

In this note, we establish the estimate on the Lorentz space $L(3/2,1)$ for vector fields in bounded domains under the assumption that the normal or the tangential component of the vector fields on the boundary vanishing. We prove that the $L(3/2,1)$ norm of the vector field can be controlled by the norms of its divergence and curl in the atomic Hardy spaces and the $L^1$ norm of the vector field itself., Comment: 11pages

We propose a framework for the spectral processing of tangential vector fields on surfaces. The basis is a Fourier-type representation of tangential vector fields that associates frequencies with tangential vector fields. To implement the representation for piecewise constant tangential vector fields on triangle meshes, we introduce a discrete Hodge–Laplace operator that fits conceptually to the prominent cotan discretization of the Laplace–Beltrami operator. Based on the Fourier representation, we introduce schemes for spectral analysis, filtering and compression of tangential vector fields. Moreover, we introduce a spline-type editor for modelling of tangential vector fields with interpolation constraints for the field itself and its divergence and curl. Using the spectral representation, we propose a numerical scheme that allows for real-time modelling of tangential vector fields.

The set of the dual vectors which are introduced by $\vec{A}=\vec{a}+\varepsilon\vec{a^{*}}$, $\varepsilon^{2}=0$ called as "dual vector field". In our paper, we introduce the directional derivative of the dual vector fields and investigate some properties of them. Then we give a numeric example of the dual vector field aided by E.Study theorem.

We present a divergence free vector field in the Sobolev space H 1 such that the flow associated to the field does not belong to any Sobolev space. The vector field is deterministic but constructed as the realization of a random field combining simple elements. This construction illustrates the optimality of recent quantitative regularity estimates as it gives a straightforward example of a well-posed flow which has nevertheless only very weak regularity.

Benefiting from the progress of manufacturing technology of fiber-reinforced composite structures, the angle of a fiber is allowed to change continuously, and stiffness can be different at different positions of a structure. Therefore, the design optimization has large freedom to improve the performance of a structure. A fundamental issue in the design optimization is how to describe the spatially varying fiber angles. In the present study, the fiber angle arrangement is described by using a parametric divergence-free vector field (pDVF) that is constructed through an expansion by using a set of basis vector fields. A basis vector field is obtained by multiplying the gradient vectors of a compact supported radial basis function with an anti-symmetric matrix. Then the expansion coefficients are regarded as design variables in the optimization. The proposed method is able to describe continuous curvilinear fibers, and it ensures that the fibers in one ply do not cross each other.

Mixing of passive scalars in compressible turbulence does not obey the same classical Reynolds number scaling as its incompressible counterpart. We first show from a large database of direct numerical simulations that even the solenoidal part of the velocity field fails to follow the classical incompressible scaling when the forcing includes a substantial dilatational component. Though the dilatational effects on the flow remain significant, our main results are that both the solenoidal energy spectrum and the passive scalar spectrum scale assume incompressible forms, and that the scalar gradient aligns with the most compressive eigenvalue of the solenoidal part, provided that only the solenoidal components are used for scaling in a consistent manner. Minor modifications to this statement are also pointed out., Comment: 6 pages and 7 figures

In this work we propose and analyze a new stabilized finite element method for the coupled Navier–Stokes/temperature (or Boussinesq) equations. The method is built using low order conforming elements for velocity and temperature, and piecewise constant elements for pressure. With the help of the lowest order Raviart–Thomas space, a lifting of the jumps of the discrete pressure is built in such a way that when this lifting is added to the conforming velocity field, the resulting velocity is solenoidal (at the price of being non-conforming). This field is then fed to the momentum and temperature equations, guaranteeing that the convective terms in these equations are antisymmetric, without the need of altering them, thus simplifying the analysis of the resulting method. Existence of solutions, discrete stability, and optimal convergence are proved for both the conforming velocity field, and its corresponding divergence-free non-conforming counterpart. Numerical results confirm the theoretical findings, as well as the gain provided by the solenoidal discrete velocity field over the conforming one.

A vector field on a pseudo-Riemannian manifold N is called concircular if it satisfies ΔXv = μX for any vector X tangent to N, where ∆ is the Levi-Civita connection of N. A concircular vector field satisfying ∆Xv = µX is called a nontrivial concircular vector field if the function µ is non-constant. A concircular vector field ν is called a concurrent vector field if the function μ is a non-zero constant. In this article we prove that every pseudo-Kaehler manifold of complex dimension > 1 does not admit a non-trivial concircular vector field. We also prove that this result is false whenever the pseudo-Kaehler manifold is of complex dimension one. In the last section we provide some remarks on pseudo-Kaehler manifolds which admit a concurrent vector field.

In this paper, we introduce the notion of measure expansivity for volume preserving diffeomorphisms and divergence free vector fields. We prove that the following three theorems. (1) The C 1 -interior of measure expansive volume preserving diffeomorphism is Anosov. (2) A C 1 -generic volume preserving diffeomorphism is Anosov. (3) The C 1 -interior of measure expansive divergence free vector field is Anosov.