Regular Vector Fields Research Articles

Fine asymptotic expansion of the ODE's flow

We study the dynamics of the flow X solution to the ODE: X′(t,x)=b(X(t,x)) with X(0,x)=x∈Rd, where b is a regular Zd-periodic vector field in Rd. We provide conditions on b to get the fine asymptotic expansion: |X(t,x)−x−tζ(x)|≤M<∞. To this end, we try to express X(t,x)−x−tζ(x) as Φ(X(t,x))−Φ(x), which yields the desired expansion when Φ is bounded. Then, assuming that the 2D Kolmogorov theorem and some extension for d>2 hold, we establish several regimes depending on the commensurability of the rotation vectors ζ(x) for which the expansion of X is valid. Moreover, we prove that for any 2D flow with a non vanishing smooth b inducing a unique incommensurable rotation vector ξ, the fine expansion holds in R2 if, and only if, ξ1/ξ2 is a Diophantine number. The case where ξ is commensurable is also investigated. Finally, several examples illustrate the different results, including the case of a vanishing b which blows up the asymptotic expansion in some direction. In particular, the case of some Euler flows is investigated.

О единственности потока, порождаемого нерегулярным векторным полем

The flow in time of an initial state ensemble in a multidimensional phase space, as a rule, models some dynamic process. Under what conditions is such a flow generated by a vector field in such a way that the given flow corresponds to the vector field in a unique way? A positive answer to this question is given by the classical uniqueness theorems for the solution of the initial value problem in the case of a regular vector field with the required properties of the modulus of continuity in space variables. In mathematical models of stochastic differential equations, in models of irregular hydrodynamic flows, and in a number of other cases when the flow is generated by a “bad” vector field that has a modulus of continuity in space variables that does not meet the conditions of the uniqueness theorem for solving the initial problem for a vector field, generating this flow, we cannot speak about the correctness of the initial problem for the vector field and, thus, about the correctness of finding the trajectories connecting the initial and actual states of the ensemble of particles in the phase space. In this case, the uniqueness of the flow generated by the vector field remains to be judged only by the properties of the flow itself. The only known result of this type is van Kampen's theorem, which states that the uniqueness of a flow generated by a vector field continuous in space variables is guaranteed by the properties of homeomorphism and the Lipschitz property of the flow in space variables. If the vector velocity field loses the property of continuity in space variables, then van Kampen's theorem does not work and some other properties of the flow are required to guarantee its uniqueness. In this paper, we establish such properties of a flow that guarantee its uniqueness even in the case of a violation of the continuity of the vector field that generates this flow. The conditions of van Kampen's theorem in a certain sense are a special case of the properties of the flow established in this paper, which guarantee its uniqueness as a solution to the initial problem for an irregular vector field. The general construction constructed here makes it possible to establish such properties of flows in various mathematical models that guarantee its uniqueness for a generating vector field.

Modified general relativity and dark matter

Modified General Relativity (MGR) is the natural extension of General Relativity (GR). MGR explicitly uses the smooth regular line element vector field [Formula: see text], which exists in all Lorentzian spacetimes, to construct a connection-independent symmetric tensor that represents the energy–momentum of the gravitational field. It solves the problem of the nonlocalization of gravitational energy–momentum in GR, preserves the ontology of the Einstein equation, and maintains the equivalence principle. The line element field provides MGR with the extra freedom required to describe dark energy and dark matter. An extended Schwarzschild solution for the matter-free Einstein equation of MGR is developed, from which the Tully–Fisher relation is derived, and the gravitational energy density is calculated. The mass of the invisible matter halo of galaxy NGC 3198 calculated with MGR is identical to the result obtained from GR using a dark matter profile. Although dark matter in MGR is described geometrically, it has an equivalent representation as a particle with the property of a vector boson or a pair of fermions; the geometry of spacetime and the quantum nature of matter are linked together by the unit line element covectors that belong to both the Lorentzian metric and the spin-1 Klein–Gordon wave equation. The three classic tests of GR provide a comparison of the theories in the solar system and several parts of the cosmos. MGR provides the flexibility to describe inflation after the Big Bang and galactic anisotropies.

Тензор кривизны n-поверхности и ее сферического образа в En+k

The main task of classical multidimensional differential geometry is to study the properties of various n-surfaces. Often these studies use torsion coefficients that are defined for any n-surface with codimension k &amp;amp;gt; 1 in (n + k)-dimensional Euclidean space. For hypersurfaces, the torsion coefficients are not defined.Another important concept used to study the properties of n-surfaces is the spherical Gaussian mapping. The Gaussian mapping defined on submanifolds of Euclidean and pseudo-Euclidean spaces allows one to study the external properties of a submanifold immersed in a Euclidean or pseudo-Euclidean space. In a number of papers, the properties of the Gaussian mapping are studied, as well as the geometric characteristics of the images of submanifolds under a spherical mapping, which are submanifolds of a hypersphere or a Grassmannian.In this article, we study the local properties of the spherical image of a regular n-surface of arbitrary codimension. The spherical mapping is defined for n-surfaces with codimension greater than one in Euclidean space by means of a regular vector field. Each vector of this field at a point of the submanifold is orthogonal to the tangent space of the submanifold at the chosen point.The article uses the methods of differential and Riemannian geometry, as well as tensor analysis to study n-surfaces with a codimension greater than one. Under some additional conditions, a connection is established between the curvature tensor of a given surface and the curvature tensor of its spherical image. Under the same additional conditions, some geometric characteristics of the points of the spherical image of the original n-surface are studied.

On expansions for nonlinear systems Error estimates and convergence issues

Explicit formulas expressing the solution to non-autonomous differential equations are of great importance in many application domains such as control theory or numerical operator splitting. In particular, intrinsic formulas allowing to decouple time-dependent features from geometry-dependent features of the solution have been extensively studied. First, we give a didactic review of classical expansions for formal linear differential equations, including the celebrated Magnus expansion (associated with coordinates of the first kind) and Sussmann's infinite product expansion (associated with coordinates of the second kind). Inspired by quantum mechanics, we introduce a new mixed expansion, designed to isolate the role of a time-invariant drift from the role of a time-varying perturbation. Second, in the context of nonlinear ordinary differential equations driven by regular vector fields, we give rigorous proofs of error estimates between the exact solution and finite approximations of the formal expansions. In particular, we derive new estimates focusing on the role of time-varying perturbations. For scalar-input systems, we derive new estimates involving only a weak Sobolev norm of the input. Third, we investigate the local convergence of these expansions. We recall known positive results for nilpotent dynamics and for linear dynamics. Nevertheless, we also exhibit arbitrarily small analytic vector fields for which the convergence of the Magnus expansion fails, even in very weak senses. We state an open problem concerning the convergence of Sussmann's infinite product expansion. Eventually, we derive approximate direct intrinsic representations for the state and discuss their link with the choice of an appropriate change of coordinates.

Specific Properties of the ODE’s Flow in Dimension Two Versus Dimension Three

This paper deals with the asymptotics of the ODE’s flow induced by a regular vector field b on the d-dimensional torus \({\mathbb {R}}^d/{\mathbb {Z}}^d\). First, we start by revisiting the Franks-Misiurewicz theorem which claims that the Herman rotation set of any two-dimensional continuous flow is a closed line segment of \({\mathbb {R}}^2\). Various general examples illustrate this result, among which a complete study of the Stepanoff flow associated with a vector field \(b=a\,\zeta \), where \(\zeta \) is a constant vector in \({\mathbb {R}}^2\). Furthermore, several extensions of the Franks-Misiurewicz theorem are obtained in the two-dimensional ODE’s context. On the one hand, we provide some interesting stability properties in the case where the Herman rotation set has a commensurable direction. On the other hand, we present new results highlighting the exceptional character of the opposite case, i.e. when the Herman rotation set is a closed line segment with \(0_{{\mathbb {R}}^2}\) at one end and with an irrational slope, if it is not reduced to a single point. Besides this, given a pair \((\mu ,\nu )\) of invariant probability measures for the flow, we establish new Fourier relations between the determinant \(\det \,(\widehat{\mu b}(j),\widehat{\nu b}(k))\) and the determinant \(\det \,(j,k)\) for any pair (j, k) of non null integer vectors, which can be regarded as an extension of the Franks-Misiurewicz theorem. Next, in contrast with dimension two, any three-dimensional closed convex polyhedron with rational vertices is shown to be the rotation set associated with a suitable vector field b. Finally, in the case of an invariant measure \(\mu \) with a regular density and a non null mass \(\mu (b)\) with respect to b, we show that the homogenization of the two-dimensional transport equation with the oscillating velocity \(b(x/\varepsilon )\) as \(\varepsilon \) tends to 0, leads us to a nonlocal limit transport equation, but with the effective constant velocity \(\mu (b)\).

Gel′fand-Fuchs cohomology in algebraic geometry and factorization algebras

Let X X be a smooth affine variety over a field k \mathbf k of characteristic 0 0 and T ( X ) T(X) be the Lie algebra of regular vector fields on X X . We compute the Lie algebra cohomology of T ( X ) T(X) with coefficients in k \mathbf k . The answer is given in topological terms relative to any embedding k ⊂ C \mathbf k\subset \mathbb {C} and is analogous to the classical Gel′fand-Fuchs computation for smooth vector fields on a C ∞ C^\infty -manifold. Unlike the C ∞ C^\infty -case, our setup is purely algebraic: no topology on T ( X ) T(X) is present. The proof is based on the techniques of factorization algebras, both in algebro-geometric and topological contexts.

Relaxation oscillation in planar discontinuous piecewise smooth fast–slow systems

This paper provides a geometric analysis of relaxation oscillations in the context of planar fast–slow systems with a discontinuous right-hand side. We give conditions that guarantee the existence of a stable crossing limit cycle Γε when the singular perturbation parameter ε is positive and small enough. Moreover, in the singular limit ε→0, the cycle Γε converges to a crossing closed singular trajectory. We also study the regularization of the crossing relaxation oscillator Γε and show that a (smooth) relaxation oscillation exists for the regularized vector field, which is a smooth fast–slow vector field with singular perturbation parameter ε. Our approach uses tools in geometric singular perturbation theory. We demonstrate the results to a number of examples including a model of an arch bridge with nonlinear viscous damping.

Quasisymmetric magnetic fields in asymmetric toroidal domains

We explore the existence of quasisymmetric magnetic fields in asymmetric toroidal domains. These vector fields can be identified with a class of magnetohydrodynamic equilibria in the presence of pressure anisotropy. First, using Clebsch potentials, we derive a system of two coupled nonlinear first order partial differential equations expressing a family of quasisymmetric magnetic fields in bounded domains. In regions where flux surfaces and surfaces of constant field strength are not tangential, this system can be further reduced to a single degenerate nonlinear second order partial differential equation with externally assigned initial data. Subclasses of solutions are then constructed by specifying as input the form the flux function, which enforces boundary shape and nested flux surfaces. In particular, we exhibit smooth quasisymmetric vector fields, which correspond to local solutions of anisotropic magnetohydrodynamics in asymmetric toroidal domains such that tangential boundary conditions are fulfilled on a portion of the bounding surface. These solutions are local because they lack periodicity in the toroidal angle. The problems of boundary shape and locality are also discussed. We find that magnetic fields with Euclidean isometries can be fitted into asymmetric domains and that the mathematical difficulty encountered in the derivation of global quasisymmetric magnetic fields lies in the topological obstruction toward global extension affecting local solutions of the governing nonlinear first order partial differential equations.

Convergence of an iterative process generated by a regular vector field

Convergence of an iterative process generated by a regular vector field

Two iterative processes generated by regular vector fields in Banach spaces

Given a convex objective function on a Banach space, which is Lipschitz on bounded sets, we consider the class of regular vector fields introduced in our previous work on descent methods. We analyze the behaviour of the values of the objective function for two iterative processes generated by a regular vector field in the presence of computational errors and show that if the computational errors tend to zero, then the values of the objective function converge to its infimum.

A Note on the Lagrangian Flow Associated to a Partially Regular Vector Field

In this paper we derive quantitative estimates for the Lagrangian flow associated to a partially regular vector field of the form $$\begin{aligned} b(t,x_1,x_2) = (b_1(t,x_1),b_2(t,x_1,x_2)) \in {\mathbb {R}}^{n_1}\times {\mathbb {R}}^{n_2} \,, \qquad (x_1,x_2)\in {\mathbb {R}}^{n_1}\times {\mathbb {R}}^{n_2}\,. \end{aligned}$$We assume that the first component $$b_1$$ does not depend on the second variable $$x_2$$, and has Sobolev $$W^{1,p}$$ regularity in the variable $$x_1$$, for some $$p>1$$. On the other hand, the second component $$b_2$$ has Sobolev $$W^{1,p}$$ regularity in the variable $$x_2$$, but only fractional Sobolev $$W^{\alpha ,1}$$ regularity in the variable $$x_1$$, for some $$\alpha >1/2$$. These estimates imply well-posedness, compactness, and quantitative stability for the Lagrangian flow associated to such a vector field.

On the centralizer of vector fields: criteria of triviality and genericity results

In this paper, we investigate the question of whether a typical vector field on a compact connected Riemannian manifold $M^d$ has a `small' centralizer. In the $C^1$ case, we give two criteria, one of which is $C^1$-generic, which guarantees that the centralizer of a $C^1$-generic vector field is indeed small, namely \textit{collinear}. The other criterion states that a $C^1$ \textit{separating} flow has a collinear $C^1$-centralizer. When all the singularities are hyperbolic, we prove that the collinearity property can actually be promoted to a stronger one, refered as \textit{quasi-triviality}. In particular, the $C^1$-centralizer of a $C^1$-generic vector field is quasi-trivial. In certain cases, we obtain the triviality of the centralizer of a $C^1$-generic vector field, which includes $C^1$-generic Axiom A (or sectional Axiom A) vector fields and $C^1$-generic vector fields with countably many chain recurrent classes. For sufficiently regular vector fields, we also obtain various criteria which ensure that the centralizer is \textit{trivial} (as small as it can be), and we show that in higher regularity, collinearity and triviality of the $C^d$-centralizer are equivalent properties for a generic vector field in the $C^d$ topology. We also obtain that in the non-uniformly hyperbolic scenario, with regularity $C^2$, the $C^1$-centralizer is trivial.

Inexact descent methods for convex minimization problems in Banach spaces

"Given a Lipschitz and convex objective function of an unconstrained optimization problem, defined on a Banach space, we revisit the class of regular vector fields which was introduced in our previous work on descent methods. We study, in particular, the asymptotic behavior of the sequence of values of the objective function for a certain inexact process generated by a regular vector field when the sequence of computational errors converges to zero and show that this sequence of values converges to the infimum of the given objective function of the unconstrained optimization problem."

Smooth approximation is not a selection principle for the transport equation with rough vector field

In this paper we analyse the selection problem for weak solutions of the transport equation with rough vector field. We answer in the negative the question whether solutions of the equation with a regularized vector field converge to a unique limit, which would be the selected solution of the limit problem. To this aim, we give a new example of a vector field which admits infinitely many flows. Then we construct a smooth approximating sequence of the vector field for which the corresponding solutions have subsequences converging to different solutions of the limit equation.

Descent methods with computational errors in Banach spaces

ABSTRACT Given a Lipschitz convex and coercive objective function on a Banach space, we revisit the class of regular vector fields introduced in our previous work on descent methods. Taking into account computational errors, we study the behaviour of the values of the objective function for the process generated by a regular vector field and show that if the computational errors are small enough, then the values of the objective function become close to its infimum.

Isotropic Realizability of Fields and Reconstruction of Invariant Measures under Positivity Properties. Asymptotics of the Flow by a Non-Ergodic Approach

The paper is devoted to the isotropic realizability of a regular gradient field u or a more general vector field b, namely the existence of a continuous positive function σ such that σb is divergence free in R d or in an open set of R d. First, we prove that under some suitable positivity condition satisfied by u, the isotropic realizability of u holds either in R d if u does not vanish, or in the open sets {c j < u < c j+1 } if the c j are the critical values of u (including inf R d u and sup R d u) which are assumed to be in finite number. It turns out that this positivity condition is not sufficient to ensure the existence of a continuous positive invariant measure σ on the torus when u is periodic. Then, we establish a new criterium of the existence of an invariant measure for the flow associated with a regular periodic vector field b, which is based on the equality b · v = 1 in R d. We show that this gradient invertibility is not related to the classical ergodic assumption, but it actually appears as an alternative to get the asymptotics of the flow.

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

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

Metrically Regular Vector Field and Iterative Processes for Generalized Equations in Hadamard Manifolds

This paper is focused on the problem of finding a singularity of the sum of two vector fields defined on a Hadamard manifold, or more precisely, the study of a generalized equation in a Riemannian setting. We extend the concept of metric regularity to the Riemannian setting and investigate its relationship with the generalized equation in this new context. In particular, a version of Graves’s theorem is presented and we also define some concepts related to metric regularity, including the Aubin property and the strong metric regularity of set-valued vector fields. A conceptual method for finding a singularity of the sum of two vector fields is also considered. This method has as particular instances: the proximal point method, Newton’s method, and Zincenko’s method on Hadamard manifolds. Under the assumption of metric regularity at the singularity, we establish that the methods are well defined in a suitable neighborhood of the singularity. Moreover, we also show that each sequence generated by these methods converges to this singularity at a superlinear rate.

Dynamics and Regularization of the Kepler Problem on Surfaces of Constant Curvature

AbstractWe classify and analyze the orbits of the Kepler problemon surfaces of constant curvature (both positive and negative, 𝕊2and ℍ2, respectively) as functions of the angular momentum and the energy. Hill's regions are characterized, and the problem of time-collision is studied. We also regularize the problem in Cartesian and intrinsic coordinates, depending on the constant angular momentum, and we describe the orbits of the regularized vector field. The phase portraits both for 𝕊2and ℍ2are pointed out.