Divergence-free Vector Fields Research Articles

On the structure of divergence-free measures on ℝ2

Abstract We consider the structure of divergence-free vector measures on the plane. We show that such measures can be decomposed into measures induced by closed simple curves. More generally, we show that if the divergence of a planar vector-valued measure is a signed measure, then the vector-valued measure can be decomposed into measures induced by simple curves (not necessarily closed). As an application we generalize certain rigidity properties of divergence-free vector fields to vector-valued measures. Namely, we show that if a locally finite vector-valued measure has zero divergence, vanishes in the lower half-space and the normal component of the unit tangent vector of the measure is bounded from below (in the upper half-plane), then the measure is identically zero.

Nonuniqueness of Weak Solutions for the Transport Equation at Critical Space Regularity

We consider the linear transport equations driven by an incompressible flow in dimensions $$d\ge 3$$ . For divergence-free vector fields $$u \in L^1_t W^{1,q}$$ , the celebrated DiPerna-Lions theory of the renormalized solutions established the uniqueness of the weak solution in the class $$L^\infty _t L^p$$ when $$\frac{1}{p} + \frac{1}{q} \le 1$$ . For such vector fields, we show that in the regime $$\frac{1}{p} + \frac{1}{q} > 1$$ , weak solutions are not unique in the class $$ L^1_t L^p$$ . One crucial ingredient in the proof is the use of both temporal intermittency and oscillation in the convex integration scheme.

Cartoon and Texture Image Decomposition Driven by Weighted Curvature

In this paper, we propose a curvature-guided model with divergence-free constraints to facilitate image decomposition. Since basic TV regularization has difficulty in processing edge geometry information of cartoon image, in order to preserve the edge features, we introduce a level set curvature term to smooth the uniform area, and use the edge indicator function as the weighted regularization to preserve the edge. In addition, to control the smoothness, a gradient function is proposed to balance the edge indicator function and the curvature term. On the other hand, we find that the existing partial decomposition models measure the oscillation function with the $H^{-1}$ functional, but this functional ignores the divergence-free vector field during the Hodge decomposition process, then the texture will lose some vector direction information, and then affects the decomposition results. By analyzing the theory of divergence-free vector field, a new decomposition model with the constraint of divergence-free vector field is proposed in this paper. Numerical experiments show that the proposed model can well preserves the edges of cartoon and protects the texture of repetitive patterns.

Structure preservation for the Deep Neural Network Multigrid Solver

The simulation of partial differential equations is a central subject of numerical analysis and an indispensable tool in science, engineering and related fields. Existing approaches, such as finite elements, provide (highly) efficient tools but deep neural network-based techniques emerged in the last few years as an alternative with very promising results. We investigate the combination of both approaches for the approximation of the Navier-Stokes equations and to what extent structural properties such as divergence freedom can and should be respected. Our work is based on DNN-MG, a deep neural network multigrid technique, that we introduced recently and which uses a neural network to represent fine grid fluctuations not resolved by a geometric multigrid finite element solver. Although DNN-MG provides solutions with very good accuracy and is computationally highly efficient, we noticed that the neural network-based corrections substantially violate the divergence freedom of the velocity vector field. In this contribution, we discuss these findings and analyze three approaches to address the problem: a penalty term to encourage divergence freedom of the network output; a penalty term for the corrected velocity field; and a network that learns the stream function, i.e. the scalar potential of the divergence free velocity vector field and which hence yields by construction divergence free corrections. Our experimental results show that the third approach based on the stream function outperforms the other two and not only improves the divergence freedom but in particular also the overall fidelity of the simulation.

A Partition of Unity Method for Divergence-Free or Curl-Free Radial Basis Function Approximation

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 ...

On the convexity condition for the Semi-Geostrophic system

We show that conservative distributional solutions to the Semi-Geostrophic systems in a rigid domain are in some well-defined sense critical points of a time-shifted energy functional involving measure-preserving rearrangements of the absolute density and momentum, which arise as one-parameter flow maps of continuously differentiable, compactly supported divergence free vector fields. We also show directly, with no recourse to Monge-Kantorovich theory, that the convexity requirement on the modified pressure potentials arises naturally if these critical points are local minimizers of said energy functional for any admissible vector field. The obligatory connection with the Monge-Kantorovich theory is addressed briefly.

Modelling white matter in gyral blades as a continuous vector field

Many brain imaging studies aim to measure structural connectivity with diffusion tractography. However, biases in tractography data, particularly near the boundary between white matter and cortical grey matter can limit the accuracy of such studies. When seeding from the white matter, streamlines tend to travel parallel to the convoluted cortical surface, largely avoiding sulcal fundi and terminating preferentially on gyral crowns. When seeding from the cortical grey matter, streamlines generally run near the cortical surface until reaching deep white matter. These so-called “gyral biases” limit the accuracy and effective resolution of cortical structural connectivity profiles estimated by tractography algorithms, and they do not reflect the expected distributions of axonal densities seen in invasive tracer studies or stains of myelinated fibres. We propose an algorithm that concurrently models fibre density and orientation using a divergence-free vector field within gyral blades to encourage an anatomically-justified streamline density distribution along the cortical white/grey-matter boundary while maintaining alignment with the diffusion MRI estimated fibre orientations. Using in vivo data from the Human Connectome Project, we show that this algorithm reduces tractography biases. We compare the structural connectomes to functional connectomes from resting-state fMRI, showing that our model improves cross-modal agreement. Finally, we find that after parcellation the changes in the structural connectome are very minor with slightly improved interhemispheric connections (i.e, more homotopic connectivity) and slightly worse intrahemispheric connections when compared to tracers.

A parametric divergence-free vector field method for the optimization of composite structures with curvilinear fibers

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.

Scalar elliptic equations with a singular drift

ABSTRACT We investigate the weak solvability and properties of weak solutions to the Dirichlet problem for a scalar elliptic equation in a bounded domain containing the origin, where with q>2 and , b is a divergence-free vector field and is a parameter.

Simple Vectorial Lie Algebras in Characteristic 2 and their Superizations

We overview the classifications of simple finite-dimensional modular Lie algebras. In characteristic 2, their list is wider than that in other characteristics; e.g., it contains desuperizations of modular analogs of complex simple vectorial Lie superalgebras. We consider odd parameters of deformations. For all 15 Weisfeiler gradings of the 5 exceptional families, and one Weisfeiler grading for each of 2 serial simple complex Lie superalgebras (with 2 exceptional subseries), we describe their characteristic-2 analogs - new simple Lie algebras. Descriptions of several of these analogs, and of their desuperizations, are far from obvious. One of the exceptional simple vectorial Lie algebras is a previously unknown deform (the result of a deformation) of the characteristic-2 version of the Lie algebra of divergence-free vector fields; this is a new simple Lie algebra with no analogs in characteristics distinct from 2. In characteristic 2, every simple Lie superalgebra can be obtained from a simple Lie algebra by one of the two methods described in arXiv:1407.1695. Most of the simple Lie superalgebras thus obtained from simple Lie algebras we describe here are new.

Averaging of Hamilton-Jacobi equations along divergence-free vector fields

<p style='text-indent:20px;'>We study the asymptotic behavior of solutions to the Dirichlet problem for Hamilton-Jacobi equations with large drift terms, where the drift terms are given by divergence-free vector fields. This is an attempt to understand the averaging effect for fully nonlinear degenerate elliptic equations. In this work, we restrict ourselves to the case of Hamilton-Jacobi equations. The second author has already established averaging results for Hamilton-Jacobi equations with convex Hamiltonians (<inline-formula><tex-math id="M1">\begin{document}$ G $\end{document}</tex-math></inline-formula> below) under the classical formulation of the Dirichlet condition. Here we treat the Dirichlet condition in the viscosity sense and establish an averaging result for Hamilton-Jacobi equations with relatively general Hamiltonian <inline-formula><tex-math id="M2">\begin{document}$ G $\end{document}</tex-math></inline-formula>.

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.

Ambit Field Modelling of Isotropic, Homogeneous, Divergence-Free and Skewed Vector Fields in Two Dimensions

We discuss the application of ambit fields to the construction of stochastic vector fields in two dimensions that are divergence-free and statistically homogeneous and isotropic but are not invariant under the parity operation. These vector fields are derived from a stochastic stream function defined as a weighted integral with respect to a Lévy basis. By construction, the stream function is homogeneous and isotropic and the corresponding vector field is, in addition, divergence-free. From a decomposition of the kernel in the Lévy-based integral, necessary conditions for the violation of parity symmetry are inferred. In particular, we focus on such fields that allow for skewness of projected increments, which is one of the cornerstones of the Kraichnan–Leith–Bachelor theory of two-dimensional turbulence.

Global, Local and Dense Non-mixing of the 3D Euler Equation

We prove a non-mixing property of the flow of the 3D Euler equation which has a local nature: in any neighbourhood of a “typical” steady solution there is a generic set of initial conditions, such that the corresponding Euler flows will never enter a vicinity of the original steady one. More precisely, we establish that there exist stationary solutions $$u_0$$ of the Euler equation on $${\mathbb {S}}^3$$ and divergence-free vector fields $$v_0$$ arbitrarily close to $$u_0$$ , whose (non-steady) evolution by the Euler flow cannot converge in the $$C^k$$ Hölder norm ( $$k>10$$ non-integer) to any stationary state in a small (but fixed a priori) $$C^k$$ -neighbourhood of $$u_0$$ . The set of such initial conditions $$v_0$$ is open and dense in the vicinity of $$u_0$$ . A similar (but weaker) statement also holds for the Euler flow on $${\mathbb {T}}^3$$ . Two essential ingredients in the proof of this result are a geometric description of all steady states near certain nondegenerate stationary solutions, and a KAM-type argument to generate knotted invariant tori from elliptic orbits.

Flow of Spatiotemporal Turbulentlike Random Fields

We study the Lagrangian trajectories of statistically isotropic, homogeneous, and stationary divergence-free spatiotemporal random vector fields. We design this advecting Eulerian velocity field such that it gets asymptotically rough and multifractal, both in space and time, as it is demanded by the phenomenology of turbulence at infinite Reynolds numbers. We then solve numerically the flow equations for a differentiable version of this field. We observe that trajectories get also rough, characterized by nearly the same Hurst exponent as the one of our prescribed advecting field. Moreover, even when considering the simplest situation of the advection by a fractional Gaussian field, we evidence in the Lagrangian framework additional intermittent corrections. The present approach involves properly defined random fields, and asks for a rigorous treatment that would explain our numerical findings and deepen our understanding of this long lasting problem.

A stable algorithm for divergence-free radial basis functions in the flat limit

The direct method used for calculating smooth radial basis function (RBF) interpolants in the flat limit becomes numerically unstable. The RBF-QR algorithm bypasses this ill-conditioning using a clever change of basis technique. We extend this method for computing interpolants involving matrix-valued kernels, specifically surface divergence-free RBFs on the sphere, in the flat limit. Results illustrating the effectiveness of this algorithm are presented for a divergence-free vector field on the sphere from samples at scattered points.

Vector Fields with the Asymptotic Orbital Pseudo-orbit Tracing Property

In this paper, we introduce the notion of the asymptotic orbital pseudo-orbit tracing property for a vector field X of a compact smooth manifold M. We show that if a vector field X has the C1 robust asymptotic orbital pseudo-orbit tracing property, then it is Anosov, and if a C1 generic vector field X has the asymptotic orbital pseudo-orbit tracing property, then it is Anosov. Moreover, we also show our results for divergence-free vector fields and Hamiltonian systems.

Liouville type result and long time behavior for Fisher-KPP equation with sign-changing and decaying potentials

This paper concerns the Liouville type result for the general semilinear elliptic equation(S)aij(x)∂iju(x)+Kqi(x)∂iu(x)+f(x,u(x))=0a.e. in RN, where f is of the KPP-monostable nonlinearity, as a continuation of the previous works of the second author [31,32]. The novelty of this work is that we allow fs(x,0) to be sign-changing and to decay fast up to a Hardy potential near infinity. First, we introduce a weighted generalized principal eigenvalue and use it to characterize the Liouville type result for Eq. (S) that was proposed by H. Berestycki. Secondly, if (aij) is the identity matrix and q is a compactly supported divergence-free vector field, we find a condition that Eq. (S) admits no positive solution for K>K⋆, where K⋆ is a certain positive threshold. To achieve this, we derive some new techniques, thanks to the recent results on the principal spectral theory for elliptic operators [14], to overcome some fundamental difficulties arising from the lack of compactness in the domain. This extends a nice result of Berestycki-Hamel-Nadirashvili [5] on the limit of eigenvalues with large drift to the case without periodic condition. Lastly, the well-posedness and long time behavior of the evolution equation corresponding to (S) are further investigated. The main tool of our work is based on the maximum principle for elliptic and parabolic equations however it is far from being obvious to see if the comparison principle for (S) holds or not.

Rigidity and trace properties of divergence-measure vector fields

Abstract We consider a φ-rigidity property for divergence-free vector fields in the Euclidean n-space, where φ ⁢ ( t ) {\varphi(t)} is a non-negative convex function vanishing only at t = 0 {t=0} . We show that this property is always satisfied in dimension n = 2 {n=2} , while in higher dimension it requires some further restriction on φ. In particular, we exhibit counterexamples to quadratic rigidity (i.e. when φ ⁢ ( t ) = c ⁢ t 2 {\varphi(t)=ct^{2}} ) in dimension n ≥ 4 {n\geq 4} . The validity of the quadratic rigidity, which we prove in dimension n = 2 {n=2} , implies the existence of the trace of a divergence-measure vector field ξ on an ℋ 1 {\mathcal{H}^{1}} -rectifiable set S, as soon as its weak normal trace [ ξ ⋅ ν S ] {[\xi\cdot\nu_{S}]} is maximal on S. As an application, we deduce that the graph of an extremal solution to the prescribed mean curvature equation in a weakly-regular domain becomes vertical near the boundary in a pointwise sense.

Magnetic Trajectories on Lightlike Surfaces

Divergence free vector fields are called magnetic vector fields in three-dimensional semi-Riemannian manifolds. When a charged particle enters the magnetic vector field, it traces a new trajectory called magnetic curve by the influenced of magnetic field. In the present paper, we investigate the magnetic curves on the lightlike surfaces corresponding to the Killing magnetic fields in 3D semi-Riemannian manifolds. Moreover, we give some characterizations of these curves. As an application, we determine all magnetic curves on the lightlike cone. Finally, we give various examples to confirm the main results.