Incompressible Vector Fields Research Articles

Classification of Vector Fields and Soliton Structures on a Tangent Bundle with a Ricci Quarter-Symmetric Metric Connection

Consider $TM$ as the tangent bundle of a (pseudo-)Riemannian manifold $M$, equipped with a Ricci quarter-symmetric metric connection $\overline{\nabla }$. This research article aims to accomplish two primary objectives. Firstly, the paper undertakes the classification of specific types of vector fields, including incompressible vector fields, harmonic vector fields, concurrent vector fields, conformal vector fields, projective vector fields, and $% \widetilde{\varphi }(Ric)$ vector fields, within the framework of $\overline{% \nabla }$ on $T\dot{M}$. Secondly, the paper establishes the necessary and sufficient conditions for the tangent bundle $TM$ to become as a Riemannian soliton and a generalized Ricci-Yamabe soliton with regard to the connection $\overline{\nabla }$.

Dissipation in Onsager’s Critical Classes and Energy Conservation in BV∩L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$BV\\cap L^\\infty $$\\end{document} with and Without Boundary

This paper is concerned with the incompressible Euler equations. In Onsager’s critical classes we provide explicit formulas for the Duchon–Robert measure in terms of the regularization kernel and a family of vector-valued measures {μz}z⊂Mx,t\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{\\mu _z\\}_z\\subset {\\mathcal {M}}_{x,t}$$\\end{document}, having some Hölder regularity with respect to the direction z∈B1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$z\\in B_1$$\\end{document}. Then, we prove energy conservation for Lx,t∞∩Lt1BVx\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^\\infty _{x,t}\\cap L^1_t BV_x$$\\end{document} solutions, in both the absence or presence of a physical boundary. This result generalises the previously known case of Vortex Sheets, showing that energy conservation follows from the structure of L∞∩BV\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^\\infty \\cap BV$$\\end{document} incompressible vector fields rather than the flow having “organized singularities”. The interior energy conservation features the use of Ambrosio’s anisotropic optimization of the convolution kernel and it differs from the usual energy conservation arguments by heavily relying on the incompressibility of the vector field. This is the first energy conservation proof, for a given class of solutions, which fails to simultaneously apply to both compressible and incompressible models, coherently with compressible shocks having non-trivial entropy production. To run the boundary analysis we introduce a notion of “normal Lebesgue trace” for general vector fields, very reminiscent of the one for BV functions. We show that having such a null normal trace is basically equivalent to have vanishing boundary energy flux. This goes beyond the previous approaches, laying down a setup which applies to every Lipschitz bounded domain. Allowing any Lipschitz boundary introduces several technicalities to the proof, with a quite geometrical/measure-theoretical flavour.

A Note on Incompressible Vector Fields

In this paper, we use incompressible vector fields for characterizing Killing vector fields. We show that on a compact Riemannian manifold, a nontrivial incompressible vector field has a certain lower bound on the integral of the Ricci curvature in the direction of the incompressible vector field if, and only if, the vector field ξ is Killing. We also show that a nontrivial incompressible vector field ξ on a compact Riemannian manifold is a Jacobi-type vector field if, and only if, ξ is Killing. Finally, we show that a nontrivial incompressible vector field ξ on a connected Riemannian manifold has a certain lower bound on the Ricci curvature in the direction of ξ, and if ξ is also a geodesic vector field, it necessarily implies that ξ is Killing.

On the failure of the chain rule for the divergence of Sobolev vector fields

We construct a large class of incompressible vector fields with Sobolev regularity, in dimension [Formula: see text], for which the chain rule problem has a negative answer. In particular, for any renormalization map [Formula: see text] (satisfying suitable assumptions) and any (distributional) renormalization defect [Formula: see text] of the form [Formula: see text], where [Formula: see text] is an [Formula: see text] vector field, we can construct an incompressible Sobolev vector field [Formula: see text] and a density [Formula: see text] for which [Formula: see text] but [Formula: see text], provided [Formula: see text].

Pointwise descriptions of nearly incompressible vector fields with bounded curl

Among those nearly incompressible vector fields v:Rn→Rn with |x|log⁡|x| growth at infinity, we give a pointwise characterization of the ones for which curlv=Dv−Dtv belongs to L∞. When n=2 we can go further and describe, still in pointwise terms, the vector fields v:R2→R2 for which |divv|+|curlv|∈L∞.

Suppression of epitaxial thin film growth by mixing

We consider following fourth-order parabolic equation with gradient nonlinearity on the two-dimensional torus with and without advection of an incompressible vector field in the case 2<p<3:∂tu+(−Δ)2u=−∇⋅(|∇u|p−2∇u). The study of this form of equations arises from mathematical models that simulate the epitaxial growth of the thin film. We prove the local existence of mild solutions for any initial data lies in L2 in both cases. Our main result is: in the advective case, if the imposed advection is sufficiently mixing, then the global existence of solution can be proved, and the solution will converge exponentially to a homogeneous mixed state. While in the absence of advection, there exist initial data in H2∩W1,∞ such that the solution will blow up in finite time.

Initial–boundary value problems for merely bounded nearly incompressible vector fields in one space dimension

We establish existence and uniqueness results for initial-boundary value problems for transport equations in one space dimension with nearly incompressible velocity fields, under the sole assumption that the fields are bounded. In the case where the velocity field is either nonnegative or nonpositive, one can rely on similar techniques as in the case of the Cauchy problem. Conversely, in the general case we introduce a new and more technically demanding construction, which heuristically speaking relies on a “lagrangian formulation” of the problem, albeit in a highly irregular setting. We also establish stability of the solution in weak and strong topologies, and propagation of the BV regularity. In the case of either nonnegative or nonpositive velocity fields we also establish a BV-in-time regularity result, and we exhibit a counterexample showing that the result is false in the case of sign-changing vector fields. To conclude, we establish a trace renormalization property.

SOME VECTORS FIELDS ON THE TANGENT BUNDLE WITH A SEMI-SYMMETRIC METRIC CONNECTION

Let $M$ is a (pseudo-)Riemannian manifold and $TM$ be its tangent bundlewith the semi-symmetric metric connection $\overline{\nabla }$. In thispaper, we examine some special vector fields, such as incompressible vectorfields, harmonic vector fields, concurrent vector fields, conformal vectorfields and projective vector fields on $TM$ with respect to thesemi-symmetric metric connection $\overline{\nabla }$ and obtain someproperties related to them.

Global existence for the two-dimensional Kuramoto-Sivashinsky equation with advection

We study the Kuramoto-Sivashinsky equation (KSE) in scalar form on the two-dimensional torus with and without advection by an incompressible vector field. We prove local existence of mild solutions for arbitrary data in L 2. We then study the issue of global existence. We prove global existence for the KSE in the presence of advection for arbitrary data, provided the advecting velocity field v satisfies certain conditions that ensure the dissipation time of the associated hyperdiffusion-advection equation is sufficiently small. In the absence of advection, global existence can be shown only if the linearized operator does not admit any growing mode and for sufficiently small initial data.

Lagrangian Representations for Linear and Nonlinear Transport

In this note we present a unifying approach for two classes of first-order partial differential equations: we introduce the notion of Lagrangian representation in the settings of continuity equation and scalar conservation laws. This yields, on the one hand, the uniqueness of weak solutions to transport equation driven by a two-dimensional BV nearly incompressible vector field. On the other hand, it is proved that the entropy dissipation measure for scalar conservation laws in one space dimension is concentrated on countably many Lipschitz curves.

Infinitesimal quantum group from helicity in fluid mechanics

Arnold showed that the Euler equations of an ideal fluid describe geodesics in the Lie algebra of incompressible vector fields. We will show that helicity induces a splitting of the Lie algebra into two isotropic subspaces, forming a Manin triple. Viewed another way, this shows that there is an infinitesimal quantum group (a.k.a. Lie bi-algebra) underlying classical fluid mechanics.

Non-renormalized solutions to the continuity equation

We show that there are continuous, W^{1,p} (p<d-1), incompressible vector fields for which uniqueness of solutions to the continuity equation fails.

The saga of a fish: from a survival guide to closing lemmas

In the paper by D. Burago, S. Ivanov and A. Novikov, “A survival guide for feeble fish”, it has been shown that a fish with limited velocity can reach any point in the (possibly unbounded) ocean provided that the fluid velocity field is incompressible, bounded and has vanishing mean drift. This result extends some known global controllability theorems though being substantially nonconstructive. We give a fish a different recipe of how to survive in a turbulent ocean, and show its relationship to structural stability of dynamical systems by providing a constructive way to change slightly the velocity field to produce conservative (in the sense of not having wandering sets of positive measure) dynamics. In particular, this leads to the extension of C. Pugh's closing lemma to incompressible vector fields over unbounded domains. The results are based on an extension of the Poincaré recurrence theorem to some σ-finite measures and on specially constructed Newtonian potentials.

A link between harmonicity of 2-distance functions and incompressibility of canonical vector fields

Let $M$ be a Riemannian submanifold of a Riemannian manifold $\tilde M$ equipped with a concurrent vector field $\tilde Z$. Let $Z$ denote the restriction of $\tilde Z$ along $M$ and let $Z^T$ be the tangential component of $Z$ on $M$, called the canonical vector field of $M$. The 2-distance function $\delta^2_Z$ of $M$ (associated with $Z$) is defined by $\delta^2_Z=\$. In this article, we initiate the study of submanifolds $M$ of $\tilde M$ with incompressible canonical vector field $Z^T$ arisen from a concurrent vector field $\tilde Z$ on the ambient space $\tilde M$. First, we derive some necessary and sufficient conditions for such canonical vector fields to be incompressible. In particular, we prove that the 2-distance function $\delta^2_Z$ is harmonic if and only if the canonical vector field $Z^T$ on $M$ is an incompressible vector field. Then we provide some applications of our main results.

On the one-dimensional continuity equation with a nearly incompressible vector field

We consider the Cauchy problem for the continuity equation with a bounded nearly incompressible vector field \begin{document} $b\colon (0,T) × \mathbb{R}^d \to \mathbb{R}^d$ \end{document} , \begin{document} $T>0$ \end{document} . This class of vector fields arises in the context of hyperbolic conservation laws (in particular, the Keyfitz-Kranzer system, which has applications in nonlinear elasticity theory). It is well known that in the generic multi-dimensional case ( \begin{document}$d≥ 1$\end{document} ) near incompressibility is sufficient for existence of bounded weak solutions, but uniqueness may fail (even when the vector field is divergence-free), and hence further assumptions on the regularity of \begin{document} $b$ \end{document} (e.g. Sobolev regularity) are needed in order to obtain uniqueness. We prove that in the one-dimensional case ( \begin{document}$d = 1$\end{document} ) near incompressibility is sufficient for existence and uniqueness of locally integrable weak solutions. We also study compactness properties of the associated Lagrangian flows.

Lagrangian Representations for Linear and Nonlinear Transport

In this note we present a unifying approach for two classes of first order partial differential equations: we introduce the notion of Lagrangian representation in the settings of continuity equation and scalar conservation laws. This yields, on the one hand, the uniqueness of weak solutions to transport equation driven by a two dimensional BV nearly incompressible vector field. On the other hand, it is proved that the entropy dissipation measure for scalar conservation laws in one space dimension is concentrated on countably many Lipschitz curves.

Initial-boundary value problems for nearly incompressible vector fields, and applications to the Keyfitz and Kranzer system

We establish existence and uniqueness results for initial boundary value problems with nearly incompressible vector fields. We then apply our results to establish well-posedness of the initial-boundary value problem for the Keyfitz and Kranzer system of conservation laws in several space dimensions.

Fast escape in incompressible vector fields

Swimmers caught in a rip current flowing away from the shore are advised to swim orthogonally to the current to escape it. We describe a mathematical principle in a similar spirit. More precisely, we consider flows $$\gamma $$ in the plane induced by incompressible vector fields $$\mathbf v :{\mathbb {R}}^2 \rightarrow {\mathbb {R}}^2$$ satisfying $$ c_1< \Vert v\Vert < c_2.$$ The length $$\ell $$ a flow curve $$\dot{\gamma }(t) = \mathbf v (\gamma (t))$$ until $$\gamma $$ leaves a disk of radius 1 centered at the initial position can be as long as $$\ell \sim c_2/c_1$$ . The same is true for the orthogonal flow $$\mathbf v ^{\perp } = (-\mathbf v _2, \mathbf v _1)$$ . We show that a combination does strictly better: there always exists a curve flowing first along $$\mathbf v ^{\perp }$$ and then along $$\mathbf v $$ which escapes the unit disk before reaching the length $$ \sqrt{4\pi c_2 / c_1}$$ . Moreover, if the escape length of $$\mathbf v $$ is uniformly $$\sim c_2/c_1$$ , then the escape length of $$\mathbf v ^{\perp }$$ is uniformly $$\sim 1$$ (allowing for a fast escape from the current). We also prove an elementary quantitative Poincare–Bendixson theorem that seems to be new.

Steady Nearly Incompressible Vector Fields in Two-Dimension: Chain Rule and Renormalization

Given bounded vector field \({b : {\mathbb{R}^{d}} \to {\mathbb{R}^{d}}}\), scalar field \({u : {\mathbb{R}^{d}} \to {\mathbb{R}}}\), and a smooth function \({\beta : {\mathbb{R}} \to {\mathbb{R}}}\), we study the characterization of the distribution \({{\rm div}(\beta(u)b)}\) in terms of div b and div(ub). In the case of BV vector fields b (and under some further assumptions), such characterization was obtained by L. Ambrosio, C. De Lellis and J. Malý, up to an error term which is a measure concentrated on the so-called tangential set of b. We answer some questions posed in their paper concerning the properties of this term. In particular, we construct a nearly incompressible BV vector field b and a bounded function u for which this term is nonzero. For steady nearly incompressible vector fields b (and under some further assumptions), in the case when d = 2, we provide complete characterization of div(\({\beta(u)b}\)) in terms of div b and div(ub). Our approach relies on the structure of level sets of Lipschitz functions on \({{\mathbb{R}^{2}}}\) obtained by G. Alberti, S. Bianchini and G. Crippa. Extending our technique, we obtain new sufficient conditions when any bounded weak solution u of \({\partial_t u + b \cdot \nabla u=0}\) is renormalized, that is when it also solves \({\partial_t \beta(u) + b \cdot \nabla \beta(u)=0}\) for any smooth function \({\beta \colon{\mathbb{R}} \to {\mathbb{R}}}\). As a consequence, we obtain new a uniqueness result for this equation.

Co-invariants of Lie algebras of vector fields on algebraic varieties

We prove that the space of coinvariants of functions on an affine variety by a Lie algebra of vector fields whose flow generates finitely many leaves is finite-dimensional. Cases of the theorem include Poisson (or more generally Jacobi) varieties with finitely many symplectic leaves under Hamiltonian flow, complete intersections in Calabi-Yau varieties with isolated singularities under the flow of incompressible vector fields, quotients of Calabi-Yau varieties by finite volume-preserving groups under the incompressible vector fields, and arbitrary varieties with isolated singularities under the flow of all vector fields. We compute this quotient explicitly in many of these cases. The proofs involve constructing a natural D-module representing the invariants under the flow of the vector fields, which we prove is holonomic if it has finitely many leaves (and whose holonomicity we study in more detail). We give many counterexamples to naive generalizations of our results. These examples have been a source of motivation for us.