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Arxiv:2009.07259V1 [Math.AP] 15 Sep 2020
A GEOMETRIC TRAPPING APPROACH TO GLOBAL REGULARITY FOR 2D NAVIER-STOKES ON MANIFOLDS AYNUR BULUT AND KHANG MANH HUYNH Abstract. In this paper, we use frequency decomposition techniques to give a direct proof of global existence and regularity for the Navier-Stokes equations on two-dimensional Riemannian manifolds without boundary. Our techniques are inspired by an approach of Mattingly and Sinai [15] which was developed in the context of periodic boundary conditions on a flat background, and which is based on a maximum principle for Fourier coefficients. The extension to general manifolds requires several new ideas, connected to the less favorable spectral localization properties in our setting. Our argu- ments make use of frequency projection operators, multilinear estimates that originated in the study of the non-linear Schr¨odingerequation, and ideas from microlocal analysis. 1. Introduction Let (M; g) be a closed, oriented, connected, compact smooth two-dimensional Riemannian manifold, and let X(M) denote the space of smooth vector fields on M. We consider the incompressible Navier-Stokes equations on M, with viscosity coefficient ν > 0, @ U + div (U ⊗ U) − ν∆ U = − grad p in M t M ; (1) div U = 0 in M with initial data U0 2 X(M); where I ⊂ R is an open interval, and where U : I ! X(M) and p : I × M ! R represent the velocity and pressure of the fluid, respectively. Here, the operator ∆M is any choice of Laplacian defined on vector fields on M, discussed below. arXiv:2009.07259v1 [math.AP] 15 Sep 2020 The theory of two-dimensional fluid flows on flat spaces is well-developed, and a variety of global regularity results are well-known. -
Lecture Notes in Mathematics
Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann Subseries: Mathematisches Institut der Universit~it und Max-Planck-lnstitut for Mathematik, Bonn - voL 5 Adviser: E Hirzebruch 1111 Arbeitstagung Bonn 1984 Proceedings of the meeting held by the Max-Planck-lnstitut fur Mathematik, Bonn June 15-22, 1984 Edited by E Hirzebruch, J. Schwermer and S. Suter I IIII Springer-Verlag Berlin Heidelberg New York Tokyo Herausgeber Friedrich Hirzebruch Joachim Schwermer Silke Suter Max-Planck-lnstitut fLir Mathematik Gottfried-Claren-Str. 26 5300 Bonn 3, Federal Republic of Germany AMS-Subject Classification (1980): 10D15, 10D21, 10F99, 12D30, 14H10, 14H40, 14K22, 17B65, 20G35, 22E47, 22E65, 32G15, 53C20, 57 N13, 58F19 ISBN 3-54045195-8 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-15195-8 Springer-Verlag New York Heidelberg Berlin Tokyo CIP-Kurztitelaufnahme der Deutschen Bibliothek. Mathematische Arbeitstagung <25. 1984. Bonn>: Arbeitstagung Bonn: 1984; proceedings of the meeting, held in Bonn, June 15-22, 1984 / [25. Math. Arbeitstagung]. Ed. by E Hirzebruch ... - Berlin; Heidelberg; NewYork; Tokyo: Springer, 1985. (Lecture notes in mathematics; Vol. 1t11: Subseries: Mathematisches I nstitut der U niversit~it und Max-Planck-lnstitut for Mathematik Bonn; VoL 5) ISBN 3-540-t5195-8 (Berlin...) ISBN 0-387q5195-8 (NewYork ...) NE: Hirzebruch, Friedrich [Hrsg.]; Lecture notes in mathematics / Subseries: Mathematischee Institut der UniversitAt und Max-Planck-lnstitut fur Mathematik Bonn; HST This work ts subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re~use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. -
Laplacians in Geometric Analysis
LAPLACIANS IN GEOMETRIC ANALYSIS Syafiq Johar syafi[email protected] Contents 1 Trace Laplacian 1 1.1 Connections on Vector Bundles . .1 1.2 Local and Explicit Expressions . .2 1.3 Second Covariant Derivative . .3 1.4 Curvatures on Vector Bundles . .4 1.5 Trace Laplacian . .5 2 Harmonic Functions 6 2.1 Gradient and Divergence Operators . .7 2.2 Laplace-Beltrami Operator . .7 2.3 Harmonic Functions . .8 2.4 Harmonic Maps . .8 3 Hodge Laplacian 9 3.1 Exterior Derivatives . .9 3.2 Hodge Duals . 10 3.3 Hodge Laplacian . 12 4 Hodge Decomposition 13 4.1 De Rham Cohomology . 13 4.2 Hodge Decomposition Theorem . 14 5 Weitzenb¨ock and B¨ochner Formulas 15 5.1 Weitzenb¨ock Formula . 15 5.1.1 0-forms . 15 5.1.2 k-forms . 15 5.2 B¨ochner Formula . 17 1 Trace Laplacian In this section, we are going to present a notion of Laplacian that is regularly used in differential geometry, namely the trace Laplacian (also called the rough Laplacian or connection Laplacian). We recall the definition of connection on vector bundles which allows us to take the directional derivative of vector bundles. 1.1 Connections on Vector Bundles Definition 1.1 (Connection). Let M be a differentiable manifold and E a vector bundle over M. A connection or covariant derivative at a point p 2 M is a map D : Γ(E) ! Γ(T ∗M ⊗ E) 1 with the properties for any V; W 2 TpM; σ; τ 2 Γ(E) and f 2 C (M), we have that DV σ 2 Ep with the following properties: 1. -
1.2 Topological Tensor Calculus
PH211 Physical Mathematics Fall 2019 1.2 Topological tensor calculus 1.2.1 Tensor fields Finite displacements in Euclidean space can be represented by arrows and have a natural vector space structure, but finite displacements in more general curved spaces, such as on the surface of a sphere, do not. However, an infinitesimal neighborhood of a point in a smooth curved space1 looks like an infinitesimal neighborhood of Euclidean space, and infinitesimal displacements dx~ retain the vector space structure of displacements in Euclidean space. An infinitesimal neighborhood of a point can be infinitely rescaled to generate a finite vector space, called the tangent space, at the point. A vector lives in the tangent space of a point. Note that vectors do not stretch from one point to vector tangent space at p p space Figure 1.2.1: A vector in the tangent space of a point. another, and vectors at different points live in different tangent spaces and so cannot be added. For example, rescaling the infinitesimal displacement dx~ by dividing it by the in- finitesimal scalar dt gives the velocity dx~ ~v = (1.2.1) dt which is a vector. Similarly, we can picture the covector rφ as the infinitesimal contours of φ in a neighborhood of a point, infinitely rescaled to generate a finite covector in the point's cotangent space. More generally, infinitely rescaling the neighborhood of a point generates the tensor space and its algebra at the point. The tensor space contains the tangent and cotangent spaces as a vector subspaces. A tensor field is something that takes tensor values at every point in a space. -
Vector Calculus and Differential Forms with Applications To
Vector Calculus and Differential Forms with Applications to Electromagnetism Sean Roberson May 7, 2015 PREFACE This paper is written as a final project for a course in vector analysis, taught at Texas A&M University - San Antonio in the spring of 2015 as an independent study course. Students in mathematics, physics, engineering, and the sciences usually go through a sequence of three calculus courses before go- ing on to differential equations, real analysis, and linear algebra. In the third course, traditionally reserved for multivariable calculus, stu- dents usually learn how to differentiate functions of several variable and integrate over general domains in space. Very rarely, as was my case, will professors have time to cover the important integral theo- rems using vector functions: Green’s Theorem, Stokes’ Theorem, etc. In some universities, such as UCSD and Cornell, honors students are able to take an accelerated calculus sequence using the text Vector Cal- culus, Linear Algebra, and Differential Forms by John Hamal Hubbard and Barbara Burke Hubbard. Here, students learn multivariable cal- culus using linear algebra and real analysis, and then they generalize familiar integral theorems using the language of differential forms. This paper was written over the course of one semester, where the majority of the book was covered. Some details, such as orientation of manifolds, topology, and the foundation of the integral were skipped to save length. The paper should still be readable by a student with at least three semesters of calculus, one course in linear algebra, and one course in real analysis - all at the undergraduate level. -
Applications of the Hodge Decomposition to Biological Structure and Function Modeling
Applications of the Hodge Decomposition to Biological Structure and Function Modeling Andrew Gillette joint work with Chandrajit Bajaj and John Luecke Department of Mathematics, Institute of Computational Engineering and Sciences University of Texas at Austin, Austin, Texas 78712, USA http://www.math.utexas.edu/users/agillette university-logo Computational Visualization Center , I C E S ( DepartmentThe University of Mathematics, of Texas atInstitute Austin of Computational EngineeringDec2008 and Sciences 1/32 University Introduction Molecular dynamics are governed by electrostatic forces of attraction and repulsion. These forces are described as the solutions of a PDE over the molecular surfaces. Molecular surfaces may have complicated topological features affecting the solution. The Hodge Decomposition relates topological properties of the surface to solution spaces of PDEs over the surface. ∼ (space of forms) = (solutions to ∆u = f 6≡ 0) ⊕ (non-trivial deRham classes)university-logo Computational Visualization Center , I C E S ( DepartmentThe University of Mathematics, of Texas atInstitute Austin of Computational EngineeringDec2008 and Sciences 2/32 University Outline 1 The Hodge Decomposition for smooth differential forms 2 The Hodge Decomposition for discrete differential forms 3 Applications of the Hodge Decomposition to biological modeling university-logo Computational Visualization Center , I C E S ( DepartmentThe University of Mathematics, of Texas atInstitute Austin of Computational EngineeringDec2008 and Sciences 3/32 University Outline 1 The Hodge Decomposition for smooth differential forms 2 The Hodge Decomposition for discrete differential forms 3 Applications of the Hodge Decomposition to biological modeling university-logo Computational Visualization Center , I C E S ( DepartmentThe University of Mathematics, of Texas atInstitute Austin of Computational EngineeringDec2008 and Sciences 4/32 University Differential Forms Let Ω denote a smooth n-manifold and Tx (Ω) the tangent space of Ω at x. -
[Math.GT] 7 Mar 2013 Ae Stems Ftetinuaintnst Zero
GEOMETRIC STRUCTURES ON THE COCHAINS OF A MANIFOLD SCOTT O. WILSON Abstract. In this paper we develop several algebraic structures on the sim- plicial cochains of a triangulated manifold that are analogues of objects in differential geometry. We study a cochain product and prove several state- ments about its convergence to the wedge product on differential forms. Also, for cochains with an inner product, we define a combinatorial Hodge star oper- ator, and describe some applications, including a combinatorial period matrix for surfaces. We show that for a particularly nice cochain inner product, these combinatorial structures converge to their continuum analogues as the mesh of a triangulation tends to zero. 1. Introduction In this paper we develop combinatorial analogues of several objects in differential and complex geometry, including the Hodge star operator and the period matrix of a Riemann surface. We define these structures on the appropriate combinatorial analogue of differential forms, namely simplicial cochains. As we recall in section 3, the two essential ingredients to the smooth Hodge star operator are Poincar´eDuality and a metric, or inner product. We’ll define the combinatorial star operator in much the same way, using both an inner product and Poincar´eDuality, expressed on cochains in the form of a (graded) commutative product. Using the inner product introduced in [6], we prove the following: Theorem 1.1. The combinatorial star operator, defined on the simplicial cochains of a triangulated Riemannian manifold, converges to the smooth Hodge star operator as the mesh of the triangulation tends to zero. We show in section 7 that, on a closed surface, this combinatorial star operator gives rise to a combinatorial period matrix and prove: arXiv:math/0505227v4 [math.GT] 7 Mar 2013 Theorem 1.2. -
6.10 the Generalized Stokes's Theorem
6.10 The generalized Stokes’s theorem 645 6.9.2 In the text we proved Proposition 6.9.7 in the special case where the mapping f is linear. Prove the general statement, where f is only assumed to be of class C1. n m 1 6.9.3 Let U R be open, f : U R of class C , and ξ a vector field on m ⊂ → R . a. Show that (f ∗Wξ)(x)=W . Exercise 6.9.3: The matrix [Df(x)]!ξ(x) adj(A) of part c is the adjoint ma- b. Let m = n. Show that if [Df(x)] is invertible, then trix of A. The equation in part b (f ∗Φ )(x) = det[Df(x)]Φ 1 . is unsatisfactory: it does not say ξ [Df(x)]− ξ(x) how to represent (f ∗Φ )(x) as the ξ c. Let m = n, let A be a square matrix, and let A be the matrix obtained flux of a vector field when the n n [j,i] × from A by erasing the jth row and the ith column. Let adj(A) be the matrix matrix [Df(x)] is not invertible. i+j whose (i, j)th entry is (adj(A))i,j =( 1) det A . Show that Part c deals with this situation. − [j,i] A(adj(A)) = (det A)I and f ∗Φξ(x)=Φadj([Df(x)])ξ(x) 6.10 The generalized Stokes’s theorem We worked hard to define the exterior derivative and to define orientation of manifolds and of boundaries. Now we are going to reap some rewards for our labor: a higher-dimensional analogue of the fundamental theorem of calculus, Stokes’s theorem. -
Operator-Valued Tensors on Manifolds: a Framework for Field
Operator-Valued Tensors on Manifolds: A Framework for Field Quantization 1 2 H. Feizabadi , N. Boroojerdian ∗ 1,2Department of pure Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, No. 424, Hafez Ave., Tehran, Iran. Abstract In this paper we try to prepare a framework for field quantization. To this end, we aim to replace the field of scalars R by self-adjoint elements of a commu- tative C⋆-algebra, and reach an appropriate generalization of geometrical concepts on manifolds. First, we put forward the concept of operator-valued tensors and extend semi-Riemannian metrics to operator valued metrics. Then, in this new geometry, some essential concepts of Riemannian geometry such as curvature ten- sor, Levi-Civita connection, Hodge star operator, exterior derivative, divergence,... will be considered. Keywords: Operator-valued tensors, Operator-Valued Semi-Riemannian Met- rics, Levi-Civita Connection, Curvature, Hodge star operator MSC(2010): Primary: 65F05; Secondary: 46L05, 11Y50. 0 Introduction arXiv:1501.05065v2 [math-ph] 27 Jan 2015 The aim of the present paper is to extend the theory of semi-Riemannian metrics to operator-valued semi-Riemannian metrics, which may be used as a framework for field quantization. Spaces of linear operators are directly related to C∗-algebras, so C∗- algebras are considered as a basic concept in this article. This paper has its roots in Hilbert C⋆-modules, which are frequently used in the theory of operator algebras, allowing one to obtain information about C⋆-algebras by studying Hilbert C⋆-modules over them. Hilbert C⋆-modules provide a natural generalization of Hilbert spaces arising when the field of scalars C is replaced by an arbitrary C⋆-algebra. -
Hodge Theory
HODGE THEORY PETER S. PARK Abstract. This exposition of Hodge theory is a slightly retooled version of the author's Harvard minor thesis, advised by Professor Joe Harris. Contents 1. Introduction 1 2. Hodge Theory of Compact Oriented Riemannian Manifolds 2 2.1. Hodge star operator 2 2.2. The main theorem 3 2.3. Sobolev spaces 5 2.4. Elliptic theory 11 2.5. Proof of the main theorem 14 3. Hodge Theory of Compact K¨ahlerManifolds 17 3.1. Differential operators on complex manifolds 17 3.2. Differential operators on K¨ahlermanifolds 20 3.3. Bott{Chern cohomology and the @@-Lemma 25 3.4. Lefschetz decomposition and the Hodge index theorem 26 Acknowledgments 30 References 30 1. Introduction Our objective in this exposition is to state and prove the main theorems of Hodge theory. In Section 2, we first describe a key motivation behind the Hodge theory for compact, closed, oriented Riemannian manifolds: the observation that the differential forms that satisfy certain par- tial differential equations depending on the choice of Riemannian metric (forms in the kernel of the associated Laplacian operator, or harmonic forms) turn out to be precisely the norm-minimizing representatives of the de Rham cohomology classes. This naturally leads to the statement our first main theorem, the Hodge decomposition|for a given compact, closed, oriented Riemannian manifold|of the space of smooth k-forms into the image of the Laplacian and its kernel, the sub- space of harmonic forms. We then develop the analytic machinery|specifically, Sobolev spaces and the theory of elliptic differential operators|that we use to prove the aforementioned decom- position, which immediately yields as a corollary the phenomenon of Poincar´eduality. -
Differential Geometry: Discrete Exterior Calculus
Differential Geometry: Discrete Exterior Calculus [Discrete Exterior Calculus (Thesis). Hirani, 2003] [Discrete Differential Forms for Computational Modeling. Desbrun et al., 2005] [Build Your Own DEC at Home. Elcott et al., 2006] Chains Recall: A k-chain of a simplicial complex Κ is linear combination of the k-simplices in Κ: c = ∑c(σ )⋅σ σ∈Κ k where c is a real-valued function. The dual of a k-chain is a k-cochain which is a linear map taking a k-chain to a real value. Chains and cochains related through evaluation: –Cochains: What we integrate –Chains: What we integrate over Boundary Operator Recall: The boundary ∂σ of a k-simplex σ is the signed union of all (k-1)-faces. ∂ ∂ ∂ ∂ +1 -1 The boundary operator extends linearly to act on chains: ∂c = ∂ ∑c(σ )⋅σ = ∑c(σ )⋅∂σ σ∈Κ k σ∈Κ k Discrete Exterior Derivative Recall: The exterior derivative d:Ωk→ Ωk+1 is the operator on cochains that is the complement of the boundary operator, defined to satisfy Stokes’ Theorem. -1 -2 -1 -3.5 -2 0.5 0.5 2 0 0.5 2 0 0.5 2 1.5 1 1 Since the boundary of the boundary is empty: ∂∂ = 0 dd = 0 The Laplacian Definition: The Laplacian of a function f is a function defined as the divergence of the gradient of f: ∆f = div(∇f ) = ∇ ⋅∇f The Laplacian Definition: The Laplacian of a function f is a function defined as the divergence of the gradient of f: ∆f = div(∇f ) = ∇ ⋅∇f Q. -
Homogeneous and Inhomogeneous Maxwell's
GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 37 (2017) 15-27 HOMOGENEOUS AND INHOMOGENEOUS MAXWELL’S EQUATIONS IN TERMS OF HODGE STAR OPERATOR Zakir Hossine1,* and Md. Showkat Ali2 1Department of Mathematics, Bangladesh University of Engineering and Technology, Dhaka-1000, Bangladesh 2Department of Applied Mathematics, University of Dhaka, Dhaka 1000, Bangladesh *Corresponding author: [email protected] Received 04.05.2016 Accepted 25.05.2017 ABSTRACT The main purpose of this work is to provide application of differential forms in physics. For this purpose, we describe differential forms, exterior algebra in details and then we express Maxwell’s equations by using differential forms. In the theory of pseudo-Riemannian manifolds there will be an important operator, called Hodge Star Operator. Hodge Star Operator arises in the coordinate free formulation of Maxwell’s equation in flat space-time. This operator is an important ingredient in the formulation of Stoke’stheorem. Keywords: Homogeneous, Inhomogeneous, Hodge Star operator, differential forms. 1. Introduction In mathematical field of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of co-ordinates. Differential forms[5] provide a unified approach in defining integrands over curves, surfaces, volumes and higher dimensional manifolds. The modern notion of differential forms as well as the idea of the differential form being the wedge product of the exterior derivative, forming an exterior algebra, was pioneered by ElieCartan[3]. 2. Differential Forms ( ) A differential -form is a sum of terms of the forms ,,…, ⋀⋀…⋀. Addition of forms and multiplication of forms by functions, is defined in the usual way.