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We consider positive classical solutions of $$ \left\{ \begin{array}{rcl} -{\Delta} u + \lambda a(x) \cdot \nabla u &=& u^{p} \qquad\text{in} {\Omega}, \\ u&=& 0 \qquad \quad \text{on} {\partial {\Omega}}, \end{array}\right. $$ where p > 1, a is a smooth divergence free vector field and λ > 0 is a large parameter. Under certain assumptions on a(x) and (or) assumptions on the existence of first integrals of a(x) we show there is a subsequence of smooth positive solutions which converge to a nonzero first integral of a(x) as $ \lambda \rightarrow \infty $ .

We show that graph map with zero topological entropy is Li-Yorke chaotic if and only if it has an NS-pair (a pair of non-separable points containing in a same solenoidal $\omega$-limit set), and a non-diagonal pair is an NS-pair if and only if it is an IN-pair if and only if it is an IT-pair. This completes characterization of zero topological sequence entropy for graph maps., Comment: 14 pages. Published online in Proc. Amer. Math. Soc

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.

We show that for any given solenoidal initial data in $L^2$ and any solenoidal external force in $L_{\text{loc}}^q \bigcap L^{3/2}$ with $q>3$, there exist partially regular weak solutions of the Navier-Stokes equations in $\R^4 \times [0,\infty[$ which satisfy certain local energy inequalities and whose singular sets have locally finite $2$-dimensional parabolic Hausdorff measure. With the help of a parabolic concentration-compactness theorem we are able to overcome the possible lack of compactness arising in the spatially $4$-dimensional setting by using defect measures, which we then incorporate into the partial regularity theory., Various improvements

We prove a new decomposition result for rank $m$ symmetric tensor fields which generalizes the well-known solenoidal and potential decomposition of tensor fields. This decomposition is then used to...

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.

Fluid flows around an obstacle generate vortices which, in turn, generate forces on the obstacle. This phenomenon is studied for planar viscous flows governed by the stationary Navier–Stokes equations with inhomogeneous Dirichlet boundary data in a (virtual) square containing an obstacle. In a symmetric framework the appearance of forces is strictly related to the multiplicity of solutions. Precise bounds on the data ensuring uniqueness are then sought and several functional inequalities (concerning relative capacity, Sobolev embedding, solenoidal extensions) are analyzed in detail: explicit bounds are obtained for constant boundary data. The case of “almost symmetric” frameworks is also considered. A universal threshold on the Reynolds number ensuring that the flow generates no lift is obtained regardless of the shape and the nature of the obstacle. Based on the asymmetry/multiplicity principle, the performance of different obstacle shapes is then compared numerically. Finally, connections of the results with elasticity and mechanics are emphasized.

The asymptotics from the two-fluid incompressible Navier-Stokes-Maxwell system with Ohm's law to solenoidal Ohm's law was pointed out in Arsenio-Saint-Raymond's work [3] . We justify rigorously this limit in the context of global-in-time classical solutions. The key is to derive the global-in-time uniform in e energy estimate of the rescaled system with Ohm's law by employing the decay properties of both the electric field E e and the wave equation with linear damping of the divergence free magnetic field B e , then take the limit as e → 0 to obtain the solutions of the system with solenoidal Ohm's law.

An arbitrary diffeomorphism of an annular set of the form is considered, where is a ball in a Banach space and is a (finite- or infinite-dimensional) torus. A system of effective sufficient conditions is proposed which ensure that has a global attractor that can be represented as a generalized solenoid, that is, the inverse limit , where is an expanding linear endomorphism of the torus . Furthermore, the restriction is topologically conjugate to a shift map of the solenoid. Bibliography: 25 titles.

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.

This paper aims to study a degenerate parabolic problem for a solenoidal vector field in which the time derivative acts on a moving body. We propose a fully-discrete finite element scheme combined with backward Euler’s method for the saddle-point variational formulation. The convergence of this numerical scheme is proved and error estimates for some stable finite element pairs are also established.

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.

In this paper we present a generalization of Poincar\'e's Rotation Theory of homeomorphisms of the circle to the case of one-dimensional compact abelian groups which are solenoidal groups, {\it i.e.}, groups which fiber over the circle with fiber a Cantor abelian group. We define rotation elements, \emph{\`a la} Poincar\'e and discuss the dynamical properties of translations on these solenoidal groups. We also study the semiconjugation problem when the rotation element generates a dense subgroup of the solenoidal group. Finally, we comment on the relation between Rotation Theory and entropy for these homeomorphisms, since unlike the case of the circle, for the solenoids considered here there are homeomorphisms (not homotopic to the identity) with positive entropy.

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 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 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.

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.

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.

In this paper, we propose and analyze the stability and thedissipative structureof a new dynamic term-by-term stabilizedfinite element formulationfor the Navier–Stokes problem that can be viewed as a variationalmultiscale(VMS) method under some assumptions. The essential point of the formulation is the time dependent nature of the subscales and, contrary to residual-based formulations, the introduction of two velocity subscale components. They represent the components of the convective and the pressure gradientterms, respectively, of themomentum equationthat cannot be captured by thefinite element mesh. A key idea of the proposed method is that the convective subscale is close to a solenoidal field and the pressure gradient subscale is close to a potential field. The method ensures stability inanisotropicspace–timediscretizations, which is proved usingnumerical analysisfor alinearized problemand demonstrated in classical numerical tests. The work includes a detailed description of the proposed formulation and severalnumerical examplesthat serve to justify our claims.

In this paper, we study the lowest-order mixed finite element method for the axisymmetric Darcy problem using Raviart–Thomas elements. In contrast to the Cartesian setting, the method is non-conforming in the sense that the discretely divergence-free functions are not solenoidal. We derive several estimates that measure the inconsistency of the method and derive error estimates of the discrete pressure and velocity solutions. We show that if the domain is convex, then the errors converge with optimal order modulo logarithmic terms. Numerical experiments are presented, and they indicate that the estimates are sharp.

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.

A Beltrami field is an eigenvector of the curl operator. Beltrami fields describe steady flows in fluid dynamics and force free magnetic fields in plasma turbulence. By application of the Lie-Darboux theorem of differential geoemtry, we prove a local representation theorem for Beltrami fields. We find that, locally, a Beltrami field has a standard form amenable to an Arnold-Beltrami-Childress flow with two of the parameters set to zero. Furthermore, a Beltrami flow admits two local invariants, a coordinate representing the physical plane of the flow, and an angular momentum-like quantity in the direction across the plane. As a consequence of the theorem, we derive a method to construct Beltrami fields with given proportionality factor. This method, based on the solution of the eikonal equation, guarantees the existence of Beltrami fields for any orthogonal coordinate system such that at least two scale factors are equal. We construct several solenoidal and non-solenoidal Beltrami fields with both homogeneous and inhomogeneous proportionality factors., 5 figures

For smooth compact connected manifolds with strictly convex boundary, no conjugate points and a hyperbolic trapped set, we prove an equivalence principle concerning the injectivity of the X-ray transform $I_m$ on symmetric solenoidal tensors and the surjectivity of an operator ${\pi_m}_*$ on the set of solenoidal tensors. This allows us to establish the injectivity of the X-ray transform on solenoidal tensors of any order in the case of a surface satisfying these assumptions., Comment: 24 pages, 2 figures. Theorem 1.2 improved thanks to a referee's suggestion: the condition "I^e_m is s-injective" is relaxed to "I_m is s-injective"

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 previous work Hamamoto (Calc Var Partial Differ Equ 58(4):23, 2019), following from an idea of Costin–Maz’ya (Costin and Maz’ya in Calc Var Partial Differ Equ 32(4):523–532, 2008), we computed the best constant in Rellich–Leray inequality for axisymmetric solenoidal fields, including any radial power weight. In the present paper, we recompute it without such a symmetry assumption. As a result, it turns out that the best constant in the same inequality for solenoidal fields is distinct from the one for unconstrained fields, only when the weight exponent stays within a bounded range.

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.

Flow development and eddy structure in an L‐shaped cavity with lids moving in the same directions have been investigated using both tools from topological and numerical methods. In particular, structural bifurcation near a nonsimple degenerate point is investigated by making a local analysis of the velocity field based on a Taylor series expansion. The streamlines of a Hamiltonian vector field system are simplified by using the homotopy invariance of the index theory. A series of bifurcation curves are constructed to determine the sequence of flow structures by which eddies are generated in the L‐shaped cavity.

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

Two approaches are proposed for recovering a symmetric m-tensor field in a unit disk from the given values of ray transforms. The approaches are based on the method of approximate inverse. The first approach allows us to reconstruct all components of the tensor field, while the second recovers the potentials of the solenoidal part and m potential parts of the tensor field.

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

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

The LS-STAG immersed boundary cut-cell method modification for viscoelastic flow computations is presented. Rate type viscoelastic flow models (linear and quasilinear) are considered. Formulae for differential types of convected time derivatives the LS-STAG discretization was obtained. Normal non-newtonian stresses are computed at the centers of base LS-STAG mesh cells and shear non-newtonian stresses are computed at the cell corners. The LS-STAG-discretization of extra-stress equations for viscoelastic Maxwell, Jeffreys, upper-convected Maxwell, Maxwell-A, Oldroyd-B, Oldroyd-A, Johnson - Segalman fluids was developed. Time-stepping algorithm is defined by the following three steps. Firstly, a prediction of the velocity and pressure correction are computed by means of semi-implicit Euler scheme. Secondly, the provisional velocity is corrected to get a solenoidal velocity and the corresponding pressure field. After this the extra-stress equations are solved. Applications to popular benchmarks for viscoelastic flows with stationary boundaries and comparisons with experimental and numerical studies are presented. The results show that the developed LS-STAG method modification demonstrates an accuracy comparable to body-fitted methods. The obtained modification is implemented in the «LS-STAG» software package developed by the author. This software allows to simulate viscous incompressible flows around a moving airfoil of arbitrary shape or airfoils system with one or two degrees of freedom. For example, it allows to simulate rotors autorotation and airfoils system wind resonance. Intel ® Cilk™ Plus, Intel ® TBB and OpenMP parallel programming technologies are used in the «LS-STAG».

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.

Generalized Stokes operators $$A_S$$ arise as linearizations of various models for non-Newtonian fluid flows. Here, it is proved that such operators in fairly general settings of domains and boundary conditions admit a bounded $${\mathcal H}^\infty $$ -calculus in the framework of solenoidal $$L_q$$ -spaces, $$1

We construct a solenoidal vector field u belonging to $$ {W}^{2,q}\left(\Omega \right)\cap {W}_0^{1,s}\left(\Omega \right),q\in \left(1,n\right),s\in \left(1,\infty \right) $$ , such that (1 + |Du|)p − 2, p ∈ (1, ∞), p ≠ 2, does not belong to the Muckenhoupt class A∞(Ω). Thus, one cannot use the Korn inequality in weighted Lebesgue spaces to prove the natural regularity of the p-Stokes problem.

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.

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.

[EN] In this paper, we describe the periodic points of automorphisms of a one dimensional solenoid, considering it as the inverse limit, lim←k (S 1 , γk) of a sequence (γk) of maps on the circle S 1 . The periodic points are discussed for a class of automorphisms on some higher dimensional solenoids also., Department of Science and Technology, Govt. of India, project ECR/2017/000741.

We study existence of minimizers of the least gradient problem inf v ∈ B V g ⁡ ∫ Ω φ ( x , D v ) , where B V g = { v ∈ B V ( Ω ) : ∫ ∂ Ω g v = 1 } , φ ( x , p ) : Ω × R n → R is a convex, continuous, and homogeneous function of degree 1 with respect to the p variable, and g satisfies the compatibility condition ∫ ∂ Ω g d S = 0 . We prove that for every 0 ≢ g ∈ L ∞ ( ∂ Ω ) there are infinitely many minimizers in B V ( Ω ) . Moreover there exists a divergence free vector field T ∈ ( L ∞ ( Ω ) ) n that determines the structure of level sets of all minimizers, i.e. T determines D u | D u | , | D u | -a.e. in Ω, for every minimizer u . We also prove some existence results for general 1-Laplacian type equations with Neumann boundary condition. A numerical algorithm is presented that simultaneously finds T and a minimizer of the above least gradient problem. Applications of the results in conductivity imaging are discussed.

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.

We describe, analyse and reduce micro-current effects in one\ud class of lattice Boltzmann equation simulation method describing im-miscible fluids within the continuum approximation, due to Lishchuk et al. (Phys. Rev. E 67 036701 (2003)). This model's micro-current flow �field and associated density adjustment, when considered in the\ud linear, low-Reynolds number regime, may be decomposed into independent, superposable contributions arising from various error terms in its immersed boundary force. Error force contributions which are rotational (solenoidal) are mainly responsible for the micro-current (corresponding density adjustment). Rotationally anisotropic error\ud terms arise from numerical derivatives and from the sampling of the interface-supporting force. They may be removed, either by eliminating the causal error force or by negating it. It is found to be straightforward to design more effective stencils with significantly improved performance.\ud Practically, the micro-current activity arising in Lishchuk's method is reduced by approximately three quarters by using an appropriate stencil and approximately by an order of magnitude when the effects of sampling are removed.

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.

We establish an asymptotic profile that sharply describes the behavior as $t\to\infty$ for solutions to a non-solenoidal approximation of the incompressible Navier–Stokes equations introduced by Temam. The solutions of Temam's model are known to converge to the corresponding solutions of the classical Navier–Stokes, e.g., in $L^3_{\rm loc} (R^+ \times R^3)$, provided $\epsilon\to0$, where $\epsilon>0$ is the physical parameter related to the artificial compressibility term. However, we show that such model is no longer a good approximation of Navier–Stokes for large times: indeed, its solutions can decay much slower as $t\to+\infty$ than the corresponding solutions of Navier–Stokes.

An example of a diffeomorphism of the 3-sphere with positive topological entropy which has a one-dimensional solenoidal basis set with a two-dimensional unstable and a one-dimensional stable invariant manifold at each point (in particular, the basis set is neither an attractor nor a repeller) is given. On the basis of this diffeomorphism, a nondissipative fast kinematic dynamo with a one-dimensional invariant solenoidal set is constructed.