Divergence-free Vector Fields Research Articles

Rectifiability of divergence-free fields along invariant 2-tori

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

Limit sets and global dynamic for 2-D divergence-free vector fields

Limit sets and global dynamic for 2-D divergence-free vector fields

Efficient WENO-Based Prolongation Strategies for Divergence-Preserving Vector Fields

Adaptive mesh refinement (AMR) is the art of solving PDEs on a mesh hierarchy with increasing mesh refinement at each level of the hierarchy. Accurate treatment on AMR hierarchies requires accurate prolongation of the solution from a coarse mesh to a newly defined finer mesh. For scalar variables, suitably high-order finite volume WENO methods can carry out such a prolongation. However, classes of PDEs, such as computational electrodynamics (CED) and magnetohydrodynamics (MHD), require that vector fields preserve a divergence constraint. The primal variables in such schemes consist of normal components of the vector field that are collocated at the faces of the mesh. As a result, the reconstruction and prolongation strategies for divergence constraint-preserving vector fields are necessarily more intricate. In this paper we present a fourth-order divergence constraint-preserving prolongation strategy that is analytically exact. Extension to higher orders using analytically exact methods is very challenging. To overcome that challenge, a novel WENO-like reconstruction strategy is invented that matches the moments of the vector field in the faces, where the vector field components are collocated. This approach is almost divergence constraint-preserving, therefore, we call it WENO-ADP. To make it exactly divergence constraint-preserving, a touch-up procedure is developed that is based on a constrained least squares (CLSQ) method for restoring the divergence constraint up to machine accuracy. With the touch-up, it is called WENO-ADPT. It is shown that refinement ratios of two and higher can be accommodated. An item of broader interest in this work is that we have also been able to invent very efficient finite volume WENO methods, where the coefficients are very easily obtained and the multidimensional smoothness indicators can be expressed as perfect squares. We demonstrate that the divergence constraint-preserving strategy works at several high orders for divergence-free vector fields as well as vector fields, where the divergence of the vector field has to match a charge density and its higher moments. We also show that our methods overcome the late time instability that has been known to plague adaptive computations in CED.

Growth of Sobolev norms and loss of regularity in transport equations.

We consider transport of a passive scalar advected by an irregular divergence-free vector field. Given any non-constant initial data [Formula: see text], [Formula: see text], we construct a divergence-free advecting velocity field [Formula: see text] (depending on [Formula: see text]) for which the unique weak solution to the transport equation does not belong to [Formula: see text] for any positive time. The velocity field [Formula: see text] is smooth, except at one point, controlled uniformly in time, and belongs to almost every Sobolev space [Formula: see text] that does not embed into the Lipschitz class. The velocity field [Formula: see text] is constructed by pulling back and rescaling a sequence of sine/cosine shear flows on the torus that depends on the initial data. This loss of regularity result complements that in Ann. PDE, 5(1):Paper No. 9, 19, 2019. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 1)'.

Nonuniqueness and Existence of Continuous, Globally Dissipative Euler Flows

We show that Hölder continuous incompressible Euler flows that satisfy the local energy inequality (“globally dissipative” solutions) exhibit nonuniqueness and contain examples that strictly dissipate kinetic energy. The collection of such solutions emanating from a fixed initial data may have positive Hausdorff dimension in the energy space even if the local energy equality is imposed, and the set of initial data giving rise to such an infinite family of solutions is $$C^0$$ dense in the space of continuous, divergence free vector fields on the torus $${{\mathbb {T}}}^3$$ . The construction of these solutions involves a new and explicit convex integration approach mirroring Kraichnan’s LDIA theory of turbulent energy cascades that overcomes the limitations of previous schemes, which had been restricted to bounded measurable solutions or to continuous solutions that dissipate total kinetic energy.

Kinematic N-expansive continuous dynamical systems

Expansiveness has been used to study dynamic systems and has been developed for various forms of expansiveness. In this paper, we introduce the concept of kinematic N-expansiveness for flows on a C∞ compact connected manifold M, which is an extension of N-expansive homeomorphisms. We prove that if a vector field X on M is C1 robustly kinematic N-expansive, then it is quasi-Anosov. Furthermore, we consider the divergence-free vector fields and Hamiltonian systems with the kinematic N-expansive property; then, we study their robustness.

On Woltjer's force‐free minimizers and Moffatt's magnetic relaxation

In this note, we exhibit a situation where a stationary state of Moffatt's ideal magnetic relaxation problem is different than the corresponding force-free L 2 $L^2$ energy minimizer of Woltjer's variational principle. Such examples have been envisioned in Moffatt's seminal work on the subject and involve divergence free vector fields supported on collections of essentially linked magnetic tubes. Justification of Moffatt's examples requires the strong convergence of a minimizing sequence. What is proven in the current note is that there is a gap between the global minimum (Woltjer's minimizer) and the minimum over the weak L 2 $L^2$ closure of the class of vector fields obtained from a topologically nontrivial field by energy-decreasing diffeomorphisms. In the context of Woltjer-Taylor relaxation, our result shows that the Taylor state cannot be reached during the viscous relaxation in the perfectly conducting magneto-fluid, if the initial field has a nontrivial topology. The result also applies to any other relaxation process which evolves a divergence free field by means of energy-decreasing diffeomorphisms, such processes were proposed by Vallis et al. and more recently by Nishiyama.

Chaos in the incompressible Euler equation on manifolds of high dimension

We construct finite dimensional families of non-steady solutions to the Euler equations, existing for all time, and exhibiting all kinds of qualitative dynamics in the phase space, for example: strange attractors and chaos, invariant manifolds of arbitrary topology, and quasiperiodic invariant tori of any dimension. The main theorem of the paper, from which these families of solutions are obtained, states that for any given vector field X on a closed manifold N, there is a Riemannian manifold M on which the following holds: N is diffeomorphic to a finite dimensional manifold in the phase space of fluid velocities (the space of divergence-free vector fields on M) that is invariant under the Euler evolution, and on which the Euler equation reduces to a finite dimensional ODE that is given by an arbitrarily small smooth perturbation of the vector field X on N.

Flows with ergodic pseudo orbit tracing property

<abstract><p>In the manuscript, we deal with a type of pseudo orbit tracing property and hyperbolicity about a vector field (or a divergence free vector field). We prove that a vector field (or a divergence free vector field) of a smooth closed manifold $ M $ has the robustly ergodic pseudo orbit tracing property then it does not contain any singularities and it is Anosov. Additionally, there is a dense and open set $ \mathcal{R} $ in the set of $ C^1 $ a vector field (or a divergence free vector field) of a smooth closed manifold $ M $ such that given a vector field (or a divergence free vector field) has the ergodic pseudo orbit tracing property then it does not contain singularities and it is Anosov.</p></abstract>

Mixing rates for Hamiltonian Monte Carlo algorithms in finite and infinite dimensions

We establish the geometric ergodicity of the preconditioned Hamiltonian Monte Carlo (HMC) algorithm defined on an infinite-dimensional Hilbert space, as developed in Beskos et al. (Stoch Process Appl 121(10):2201–2230, 2011). This algorithm can be used as a basis to sample from certain classes of target measures which are absolutely continuous with respect to a Gaussian measure. Our work addresses an open question posed in Beskos et al. (2011), and provides an alternative to a recent proof based on exact coupling techniques given in Bou-Rabee and Eberle (Two-scale coupling for preconditioned Hamiltonian Monte Carlo in infinite dimensions , 2019). The approach here establishes convergence in a suitable Wasserstein distance by using the weak Harris theorem together with a generalized coupling argument. We also show that a law of large numbers and central limit theorem can be derived as a consequence of our main convergence result. Moreover, our approach yields a novel proof of mixing rates for the classical finite-dimensional HMC algorithm. As such, the methodology we develop provides a flexible framework to tackle the rigorous convergence of other Markov Chain Monte Carlo algorithms. Additionally, we show that the scope of our result includes certain measures that arise in the Bayesian approach to inverse PDE problems, cf. Stuart (Acta Numer 19:451–559, 2010). Particularly, we verify all of the required assumptions for a certain class of inverse problems involving the recovery of a divergence free vector field from a passive scalar, Borggaard et al. (SIAM/ASA J Uncertain Quant 8(3):1036–1060, 2020).

Dirichlet eigenvalue problems of irreversible Langevin diffusion

The basic objective of this research work is to investigate the asymptotic behavior of the first eigenvalue of an irreversible Langevin diffusion with zero boundary values. In particular, a reversible diffusion is perturbed by adding an antisymmetric drift which preserves the invariant measure. Then, a necessary and sufficient condition is provided for the boundness and the limiting behavior of the first eigenvalue, under Dirichlet boundary conditions and with respect to the invariant measure. In other words, we prove that the first eigenvalue is bounded if and only if the associated stochastic dynamical system has a first integral. Furthermore, we demonstrate that the limiting eigenvalue is the minimum of the Dirichlet functional over all first integrals of the divergence-free vector field. An extension of this model with a time parameter in the boundary conditions is studied, where we give another characterization to achieve the same main result.

Differentiability properties of the flow of 2d autonomous vector fields

We investigate under which assumptions the flow associated to autonomous planar vector fields inherits the Sobolev or BV regularity of the vector field. We consider nearly incompressible and divergence-free vector fields, taking advantage in both cases of the underlying Hamiltonian structure. Finally, we provide an example of an autonomous planar Sobolev divergence-free vector field, such that the corresponding regular Lagrangian flow has no bounded variation.

Turing Universality of the Incompressible Euler Equations and a Conjecture of Moore

Abstract In this article, we construct a compact Riemannian manifold of high dimension on which the time-dependent Euler equations are Turing complete. More precisely, the halting of any Turing machine with a given input is equivalent to a certain global solution of the Euler equations entering a certain open set in the space of divergence-free vector fields. In particular, this implies the undecidability of whether a solution to the Euler equations with an initial datum will reach a certain open set or not in the space of divergence-free fields. This result goes one step further in Tao’s programme to study the blow-up problem for the Euler and Navier–Stokes equations using fluid computers. As a remarkable spin-off, our method of proof allows us to give a counterexample to a conjecture of Moore dating back to 1998 on the non-existence of analytic maps on compact manifolds that are Turing complete.

Anisotropic Helmholtz decomposition for controlled fluid simulation

The combination of fluids and tensors has been explored in recent works to constrain the fluid flow along specific directions for simulation applications. Such an approach points towards the necessity to model flows using anisotropic fluid dynamics. The formalism behind anisotropic Helmholtz decomposition plays a central role given the foundations to model directional constraints. In this paper, we apply the anisotropic Helmholtz theory to obtain a divergent free velocity field that respects the constraints of a symmetric positive tensor field. A major contribution is a well-defined anisotropic projection method suitable for anisotropic transport. An anisotropic projection is paramount for stable incompressible advection in anisotropic medium. Aiming to show how to benefit from the anisotropic projection, we customize the Navier-Stokes equations to use tensor information for locally modifying fluid momentum and material advection. Besides, we develop a stable numerical method to integrate the obtained system of partial differential equations and a methodology to optimize the tensor field to reduce numerical errors. Experiments show that tensor fields with different anisotropic features can provide distinct projected divergence-free vector fields. Our results also show that the proposed method forms a basis for fluid flow simulation following meaningful paths induced by the tensor field geometry and topology.

Jacobi multipliers and Hojman symmetry

The geometric intrinsic approach to Hojman symmetry is developed and use is made of the theory of the Jacobi last multipliers to find the corresponding conserved quantity for non divergence-free vector fields. The particular cases of autonomous Lagrangian and Hamiltonian systems are studied as well as the generalization of these results to normalizer vector fields of the dynamics. The nonautonomous cases, where normalizer vector fields play a relevant role, are also developed.

The helicity uniqueness conjecture in 3d hydrodynamics

We prove that the helicity is the only regular Casimir function for the coadjoint action of the volume-preserving diffeomorphism group $\text{SDiff}(M)$ on smooth exact divergence-free vector fields on a closed three-dimensional manifold $M$. More precisely, any regular $C^1$ functional defined on the space of $C^\infty$ (more generally, $C^k$, $k\ge 4$) exact divergence-free vector fields and invariant under arbitrary volume-preserving diffeomorphisms can be expressed as a $C^1$ function of the helicity. This gives a complete description of Casimirs for adjoint and coadjoint actions of $\text{SDiff}(M)$ in 3D and completes the proof of Arnold-Khesin's 1998 conjecture for a manifold $M$ with trivial first homology group. Our proofs make use of different tools from the theory of dynamical systems, including normal forms for divergence-free vector fields, the Poincar\'e-Birkhoff theorem, and a division lemma for vector fields with hyperbolic zeros.

Regularity estimates for the flow of BV autonomous divergence-free vector fields in

We consider the regular Lagrangian flow X associated to a bounded divergence-free vector field b with bounded variation. We prove a Lusin-Lipschitz regularity result for X and we show that the Lipschitz constant grows at most linearly in time. As a consequence we deduce that both geometric and analytical mixing have a lower bound of order as

Self-gravitating vector dark matter

We derive the non-relativistic limit of a massive vector field. We show that the Cartesian spatial components of the vector behave as three identical, non-interacting scalar fields. We find classes of spherical, cylindrical, and planar self-gravitating vector solitons in the Newtonian limit. The gravitational properties of the lowest-energy vector solitons$\mathrm{-}$the gravitational potential and density field$\mathrm{-}$depend only on the net mass of the soliton and the vector particle mass. In particular, these self-gravitating, ground-state vector solitons are independent of the distribution of energy across the vector field components, and are indistinguishable from their scalar-field counterparts. Fuzzy Vector Dark Matter models can therefore give rise to halo cores with identical observational properties to the ones in scalar Fuzzy Dark Matter models. We also provide novel hedgehog vector soliton solutions, which cannot be observed in scalar-field theories. The gravitational binding of the lowest-energy hedgehog halo is about three times weaker than the ground-state vector soliton. Finally, we show that no spherically symmetric solitons exist with a divergence-free vector field.

A simpler expression for Costin–Maz’ya’s constant in the Hardy–Leray inequality with weight

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.

Smoothing does not give a selection principle for transport equations with bounded autonomous fields

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.