On the leapfrogging phenomenon in fluid mechanics
|
|
- Maurice Goodwin
- 5 years ago
- Views:
Transcription
1 On the leapfrogging phenomenon in fluid mechanics Didier Smets Université Pierre et Marie Curie - Paris Based on works with Robert L. Jerrard U. of Toronto) CIRM, Luminy, June 27th / 22
2 Single vortex ring : a movie example c D. Kleckner & W. Irvine, Nature, / 22
3 Helmholtz master paper on vorticity Vortex rings : Leapfrogging : 3 / 22
4 Leapfrogging vortices : a movie example c T.T. Lim 4 / 22
5 Leapfrogging vortices : a movie example B. Balog, Q. Kriaa, S. Yehudi 5 / 22
6 Existence of vortex rings : a practical point of view 6 / 22
7 Euler equation for incompressible fluids The 3 dimensional Euler equations read { tv + v )v = p, div v = 0, where v : R 3 R R 3 is the velocity field, p : R 3 R R is the pressure field. For a number of physically meaningful flows, an important quantity is given by the vorticity which satisfies ω = curl v tω + v ω = ω v. The velocity v can always be recovered from the vorticity ω through the Biot-Savart law vt, x) = 1 y x ωt, y) dy. 4π R 3 y x 3 7 / 22
8 Gross-Pitaevskii equation for quantum fluids The 3 dimensional Gross-Pitaevskii equation reads where i tu u = 1 ε 2 u1 u 2 ) u : R 3 R C is the complex field function ε > 0 is a given length-scale. In the framework of thin vortex tubes, it bears some resemblance with the Euler equation. An important quantity here is given by the vorticity Ju = 1 curl ju) = curl u u). 2 In contrast with the Euler equation, the current ju) is not fully determined by the Jacobian Ju since an additional degree of freedom is present through the phase: u = ρ expiϕ) = ju) = ρ 2 ϕ, ρ 1 = Ju = 1 curl ϕ / 22
9 On the analogy between GP and Euler For axisymmetric solutions of GP) we have the equation for the vorticity d Ju ϕ drdz = F u, ϕ), dt H where k r F u, ϕ) := ε ij j u, k u) i ϕ + ε ij j u, k u) ik ϕ. H r H For the axisymmetric Euler equation we have d ω ϕ drdz = Fv, ϕ). dt H In contrast to Euler, for GP) we also need an equation for the compressible part of ju) : ε t u 2 1 ε = 2 r divrju)). 9 / 22
10 Existence of vortex rings : a mathematical point of view One looks for cylindrically symmetric traveling wave solutions of the Euler equation. These can be obtained by the variational problem maximize E := ωx)gx, x )ωx ) dνdν H H P := r 2 ωdrdz = given Cst, under constrains H ω ω0 is a transport measure of r r, where ω 0 is given. Here, H := {r, z), r 0, z R}, dν = r dr dz and G refers to the Green function of the Laplacian in cylindrical coordinates. Important contributions by Arnold 1964, Fraenkel-Berger 1974, Benjamin 1976, Friedman-Turkington 1981, Burton For the Gross-Pitaevskii, a related strategy is given by minimize E := [ 1 H 2 u ε 2 u 2 1) 2 ] rdrdz under constrain P := r 2 Ju drdz = given Cst. H Analysis by Bethuel-Orlandi-S / 22
11 Leapfrogging for GP : notion of reference vortex tubes Let C be a smooth oriented closed curve in R 3 and let J be the vector distribution corresponding to 2π times the circulation along C, namely J, X = 2π X τ X DR 3, R 3 ), C where τ is the tangent vector to C. To the current density J is associated the induction B, which satisfies the equations div B) = 0, curl B) = J in R 3, and is obtained from J by the Biot-Savart law. To B is then associated a vector potential A, which satisfies div A) = 0, curl A) = B in R 3, so that A = curl curl A) = J in R 3. For a H, let C a be the circle of radius ra) parallel to the xy-plane in R 3, centered at the point 0, 0, za)), and oriented so that its binormal vector points towards the positive z-axis. By cylindrical symmetry, we may write the corresponding vector potential as A a A ar, z) e θ. The expression of the vector laplacian in cylindrical coordinates yields the equation for the scalar function A a : r r r 1 ) r z A a = 2πδ a in H A a = 0 on H, 11 / 22
12 Leapfrogging for GP : notion of reference vortex tubes or equivalently ) 1 div r raa) = 2πδ a in H A a = 0 on H, which can be integrated explicitly in terms of complete elliptic integrals. Up to a constant phase factor, there exists a unique unimodular map ua C H \ {a}, S 1 ) W 1,1 loc H, S1 ) such that riua, u a ) = rju a ) = ra a). In the sense of distributions in H, we have { divrju a )) = 0 curljua )) = 2πδa, and the function ua corresponds therefore to a singular vortex ring. In order to describe a reference vortex ring for the Gross-Pitaevskii equation, we shall make the notion of core more precise. In R 2, the Gross-Pitaevskii equation possesses a distinguished stationary solution called vortex : in polar coordinates, it has the special form u εr, θ) = f εr) expiθ) where the profile f ε : R + [0, 1] satisfies f ε0) = 0, f ε+ ) = 1, and rr f ε r fε r r 2 fε + 1 ε 2 fε1 f ε 2 ) = 0. The reference vortex ring associated to the point a H is defined to be u ε,ar, z) = f ε r, z) a ) u a r, z). 12 / 22
13 Leapfrogging for GP : notion of reference vortex tubes More generally, when a = {a 1,, a n} is a family of n distinct points in H, we set u a r, z) := n ua k r, z), and uε,ar, z) := k=1 n uε,a k r, z), where the products are meant in C. The field u ε,a hence corresponds to a collection of n reference vortex rings sharing the same axis and oriented in the same direction), and is the typical kind of object which we shall study the evolution of. It can be shown that Ju ε,a π Finally, classical computations lead to k=1 n Ẇ δ ai = Oε) as ε 0. i=1 1,1 H) E u ε,a) = Hεa 1,, a n) + o1), where H εa 1,, a n) := n i=1 [ ra i ) π log ra i ) ε ) + γ + π 3 log2) 2 ) + π j i ] A aj a i ), and γ is a numerical constant. 13 / 22
14 The ODE of leapfrogging We consider the associated hamiltonian system LF ) ȧ i t) = 1 π J a i Hε a1 t),, a nt) ), i = 1,, n, where, with a slight abuse of notation, J := ) 0 1 ra i ) 1. 0 ra i ) In addition to the hamiltonian H ε, the system LF) also conserves the momentum Pa 1,, a n) := π n r 2 a k ), which may be interpreted as the total area of the disks determined by the vortex rings. As a matter of fact, also note that k=1 Puε,a) := Juε,a r 2 drdz = π H n r 2 a k ) + o1), as ε 0, and that, at least formally, the momentum P is a conserved quantity for the Gross- Pitaevskii equation. k=1 14 / 22
15 Mathematical convergence results We consider axisymmetric initial data u 0 ε for GP), we write a 0 i,ε = a0 + b0 i,ε log ε and denote by ai,ε s the corresponding solution of LF )ε. We let S > 0 and fix a constant K > 0 such that bi,ε s K for all s [0, S]. We define Ju0 ε π n i=1 δ a i,ε 0 Ẇ =: ra 0 concentration scale) 1,1 [ + Euε) 0 H εa1,ε n,ε)] 0,, a0 =: Σ 0 a energy excess). Theorem Jerrard-S.) Under the above assumptions, there exist constants ε 0, σ 0, C 0 depending only on a 0, n, K) such that if ε ε 0 and if Σ 0 a + r 0 a log ε σ 0, then n ra s := Juε s π δ a s C i,ε 0 ra i=1 Ẇ 0 + 1,1 for s [0, S], where δ > 0 is arbitrary. Σ0 a log ε + ) C δ log ε 1 δ expc 0 s) 15 / 22
16 Analysis of the leapfrogging system for two vortex rings When n = 2, the system LF ) may be analyzed in great details. Since P is conserved and since H ε is invariant by a joint translation of both rings in the z direction, it is classical to introduce the variables η, ξ) by { r 2 a 1 ) = P 2 η r 2 a 2 ) = P 2 + η, ξ = za 1) za 2 ), and to draw the level curves of the function H ε in those two real variables, the momentum P being considered as a parameter. Size of order O1) P = 2 log ε = Pass through η 0 Attract then repel Leapfrogging log log ε O ) log ε 0.5 Equilibria corresponding to a pair of traveling vortices ξ 16 / 22
17 Collision of vortex rings c T.T. Lim 17 / 22
18 Strategy of the proof We rely mostly on the already mentioned evolution equation for the Jacobian d F u, ϕ) Ju ϕ drdz =, ds H log ε where k r F u, ϕ) := ε ij j u, k u) i ϕ + ε ij j u, k u) ik ϕ. H r H We wish to prove that Ju remains well concentrated around a sum of Dirac masses F u, ϕ) is a good approximation of the ode LF) ε. It is clear that both are not independent and we shall use a common Gronwall argument on r s a. We will decompose j u, k u ) = j u k u + ju) j ju) k u u ) ju) = j u k u + j u j ) ju) + j u j + j ) j ), j k where j is a suitable approximation of ju). ju) u j ) j ) ) ju) + j k u k k j ) j 18 / 22
19 Some elements in the proof: excess energy and concentration Proposition There exist constants ε 1, σ 1, C 1 > 0, depending only on n, r 0 and K, with the following properties. If ε ε 1 and Σ r a := Σ a + r a log ε σ 1 log ε, then there exist ξ 1,, ξ n in H such that and H\ i Bξ i,ε 2 3 ) where we have written Moreover, Note the pointwise identity r ξ := Ju π n δ ξi Ẇ 1,1 C 1 ε log ε C 1 e C 1Σ r a, i=1 [ r e ε u ) + ju) u ju ξ ) 2 ] C 1 Σξ + ε 1 3 log ε C 1 e C 1Σ r a ), Σ ξ := [Eu) H εξ 1,, ξ n)] +. Σ ξ Σ a + C 1 r a log ε + C 1 ε log ε C 1 e C 1Σ r a. e εu) = 1 2 ju ξ ) 2 + juξ ju) ) u juξ )) + e ε u ) + 1 ju) juξ 2 u ) / 22
20 Some elements in the proof: approximation including the core Idea : smoothen u ξ not at scale ε but at the larger scale r ξ i.e. at the best known localization scale). It actually suffices to regularize ju ξ ), we call it j u ξ ). Proposition In addition to the statement of the previous Proposition, [ r e ε u ) + ju) ] j uξ H u ) 2 C 1 Σ r a + log log ε ). e εu) = 1 2 j uξ ) 2 + j uξ ju) ) u j uξ )) + e ε u ) + 1 ju) j uξ 2 u ) / 22
21 Some elements in the proof: get rid of the main translation Recall that and write Then where H εa 1,, a n) := n i=1 [ ra i ) π log ra i ) ε ) + γ + π 3 log2) 2 ) + π a i := r 0 + rb i ), z 0 + zb i ) ), i = 1,, n. log ε log ε j i ] A aj a i ), H εa 1,, a n) = Γ εr 0, n) + W ε,r0 b 1,, b n) + o1) as ε 0, 1) W ε,r0 b 1,, b n) = π i=1 i j Also, expansion of the squares leads directly to n Pa 1,, a n) = πnr πr rb i ) 0 + π i=1 log ε and therefore H ε where log ε P 2r 0 n rb i ) log ε πr 0 log b i b j. 2) n i=1 rb i ) 2 log ε, ) a 1,, a n) = π 2 nr 0 log ε + Γ εr 0, n) + πr 0 W b 1,, b n) + o1), as ε 0, W b 1,, b n) := i j log b i b j 1 2r 2 0 n rb i ) 2. i=1 21 / 22
22 Some elements in the proof: get rid of the main translation Proposition For σ sufficiently small, if Σ a + H εa 1,, a n) H εξ,, ξ n) σ log ε, then [ ] 1 Σ ξ 2Σ a + C r a log ε + log ε + log ε Pu) Pa 1,, a n) For a quantity f write δf := f a 1,, a n) f ξ 1,, ξ n). By the triangle inequality we have ) δh ε δ H ε + and also log ε P 2r 0 log ε δp, 2r 0 δp Pu) Pξ 1,, ξ n) + Pu) Pa 1,, a n). In view of the discussion in the previous slide δ H ε log ε P 2r 0 ) C ξ 1 a 1,, ξ n a n) log ε. We also have a good control on P since it involves only the Jacobian : log ε 2r 0 Pu) Pξ 1,, ξ n) C 1 + Σ ξ ) 2 C 1 + Σ a + δh ε) 2 2r 0 log ε 2r 0 log ε and we may then absorb the last term involving δh ε in the left-hand side above. 22 / 22
Mean Field Limits for Ginzburg-Landau Vortices
Mean Field Limits for Ginzburg-Landau Vortices Sylvia Serfaty Université P. et M. Curie Paris 6, Laboratoire Jacques-Louis Lions & Institut Universitaire de France Jean-Michel Coron s 60th birthday, June
More informationOn the stability of filament flows and Schrödinger maps
On the stability of filament flows and Schrödinger maps Robert L. Jerrard 1 Didier Smets 2 1 Department of Mathematics University of Toronto 2 Laboratoire Jacques-Louis Lions Université Pierre et Marie
More informationMean Field Limits for Ginzburg-Landau Vortices
Mean Field Limits for Ginzburg-Landau Vortices Sylvia Serfaty Courant Institute, NYU & LJLL, Université Pierre et Marie Curie Conference in honor of Yann Brenier, January 13, 2017 Fields Institute, 2003
More informationComplex Analysis MATH 6300 Fall 2013 Homework 4
Complex Analysis MATH 6300 Fall 2013 Homework 4 Due Wednesday, December 11 at 5 PM Note that to get full credit on any problem in this class, you must solve the problems in an efficient and elegant manner,
More informationWeek 2 Notes, Math 865, Tanveer
Week 2 Notes, Math 865, Tanveer 1. Incompressible constant density equations in different forms Recall we derived the Navier-Stokes equation for incompressible constant density, i.e. homogeneous flows:
More informationAsymptotic stability for solitons of the Gross-Pitaevskii and Landau-Lifshitz equations
Asymptotic stability for solitons of the Gross-Pitaevskii and Landau-Lifshitz equations Philippe Gravejat Cergy-Pontoise University Joint work with F. Béthuel (Paris 6), A. de Laire (Lille) and D. Smets
More informationRemarks on the blow-up criterion of the 3D Euler equations
Remarks on the blow-up criterion of the 3D Euler equations Dongho Chae Department of Mathematics Sungkyunkwan University Suwon 44-746, Korea e-mail : chae@skku.edu Abstract In this note we prove that the
More informationContinuum Mechanics Lecture 5 Ideal fluids
Continuum Mechanics Lecture 5 Ideal fluids Prof. http://www.itp.uzh.ch/~teyssier Outline - Helmholtz decomposition - Divergence and curl theorem - Kelvin s circulation theorem - The vorticity equation
More informationWEAK VORTICITY FORMULATION FOR THE INCOMPRESSIBLE 2D EULER EQUATIONS IN DOMAINS WITH BOUNDARY
WEAK VORTICITY FORMULATION FOR THE INCOMPRESSIBLE 2D EULER EQUATIONS IN DOMAINS WITH BOUNDARY D. IFTIMIE, M. C. LOPES FILHO, H. J. NUSSENZVEIG LOPES AND F. SUEUR Abstract. In this article we examine the
More informationPRACTICE PROBLEMS. Please let me know if you find any mistakes in the text so that i can fix them. 1. Mixed partial derivatives.
PRACTICE PROBLEMS Please let me know if you find any mistakes in the text so that i can fix them. 1.1. Let Show that f is C 1 and yet How is that possible? 1. Mixed partial derivatives f(x, y) = {xy x
More informationZdzislaw Brzeźniak. Department of Mathematics University of York. joint works with Mauro Mariani (Roma 1) and Gaurav Dhariwal (York)
Navier-Stokes equations with constrained L 2 energy of the solution Zdzislaw Brzeźniak Department of Mathematics University of York joint works with Mauro Mariani (Roma 1) and Gaurav Dhariwal (York) LMS
More informationOn the p-laplacian and p-fluids
LMU Munich, Germany Lars Diening On the p-laplacian and p-fluids Lars Diening On the p-laplacian and p-fluids 1/50 p-laplacian Part I p-laplace and basic properties Lars Diening On the p-laplacian and
More informationTWO EXISTENCE RESULTS FOR THE VORTEX-WAVE SYSTEM
TWO EXISTENCE RESULTS FOR THE VORTEX-WAVE SYSTEM EVELYNE MIOT Abstract. The vortex-wave system is a coupling of the two-dimensional Euler equations for the vorticity together with the point vortex system.
More informationREMARKS ON THE VANISHING OBSTACLE LIMIT FOR A 3D VISCOUS INCOMPRESSIBLE FLUID
REMARKS ON THE VANISHING OBSTACLE LIMIT FOR A 3D VISCOUS INCOMPRESSIBLE FLUID DRAGOŞ IFTIMIE AND JAMES P. KELLIHER Abstract. In [Math. Ann. 336 (2006), 449-489] the authors consider the two dimensional
More informationVortex stretching in incompressible and compressible fluids
Vortex stretching in incompressible and compressible fluids Esteban G. Tabak, Fluid Dynamics II, Spring 00 1 Introduction The primitive form of the incompressible Euler equations is given by ( ) du P =
More informationZdzislaw Brzeźniak. Department of Mathematics University of York. joint works with Mauro Mariani (Roma 1) and Gaurav Dhariwal (York)
Navier-Stokes equations with constrained L 2 energy of the solution Zdzislaw Brzeźniak Department of Mathematics University of York joint works with Mauro Mariani (Roma 1) and Gaurav Dhariwal (York) Stochastic
More informationMATH 332: Vector Analysis Summer 2005 Homework
MATH 332, (Vector Analysis), Summer 2005: Homework 1 Instructor: Ivan Avramidi MATH 332: Vector Analysis Summer 2005 Homework Set 1. (Scalar Product, Equation of a Plane, Vector Product) Sections: 1.9,
More information1-D cubic NLS with several Diracs as initial data and consequences
1-D cubic NLS with several Diracs as initial data and consequences Valeria Banica (Univ. Pierre et Marie Curie) joint work with Luis Vega (BCAM) Roma, September 2017 1/20 Plan of the talk The 1-D cubic
More informationExistence, stability and instability for Einstein-scalar field Lichnerowicz equations by Emmanuel Hebey
Existence, stability and instability for Einstein-scalar field Lichnerowicz equations by Emmanuel Hebey Joint works with Olivier Druet and with Frank Pacard and Dan Pollack Two hours lectures IAS, October
More informationExercise Set 4. D s n ds + + V. s dv = V. After using Stokes theorem, the surface integral becomes
Exercise Set Exercise - (a) Let s consider a test volum in the pellet. The substract enters the pellet by diffusion and some is created and disappears due to the chemical reaction. The two contribute to
More informationAE301 Aerodynamics I UNIT B: Theory of Aerodynamics
AE301 Aerodynamics I UNIT B: Theory of Aerodynamics ROAD MAP... B-1: Mathematics for Aerodynamics B-: Flow Field Representations B-3: Potential Flow Analysis B-4: Applications of Potential Flow Analysis
More informationENGI Gradient, Divergence, Curl Page 5.01
ENGI 94 5. - Gradient, Divergence, Curl Page 5. 5. The Gradient Operator A brief review is provided here for the gradient operator in both Cartesian and orthogonal non-cartesian coordinate systems. Sections
More information18.325: Vortex Dynamics
8.35: Vortex Dynamics Problem Sheet. Fluid is barotropic which means p = p(. The Euler equation, in presence of a conservative body force, is Du Dt = p χ. This can be written, on use of a vector identity,
More informationV (r,t) = i ˆ u( x, y,z,t) + ˆ j v( x, y,z,t) + k ˆ w( x, y, z,t)
IV. DIFFERENTIAL RELATIONS FOR A FLUID PARTICLE This chapter presents the development and application of the basic differential equations of fluid motion. Simplifications in the general equations and common
More informationIsoperimetric inequalities and cavity interactions
Laboratoire Jacques-Louis Lions Université Pierre et Marie Curie - Paris 6, CNRS May 17, 011 Motivation [Gent & Lindley 59] [Lazzeri & Bucknall 95 Dijkstra & Gaymans 93] [Petrinic et al. 06] Internal rupture
More informationChapter 5. The Differential Forms of the Fundamental Laws
Chapter 5 The Differential Forms of the Fundamental Laws 1 5.1 Introduction Two primary methods in deriving the differential forms of fundamental laws: Gauss s Theorem: Allows area integrals of the equations
More informationVortex motion. Wasilij Barsukow, July 1, 2016
The concept of vorticity We call Vortex motion Wasilij Barsukow, mail@sturzhang.de July, 206 ω = v vorticity. It is a measure of the swirlyness of the flow, but is also present in shear flows where the
More informationEuler Equations: local existence
Euler Equations: local existence Mat 529, Lesson 2. 1 Active scalars formulation We start with a lemma. Lemma 1. Assume that w is a magnetization variable, i.e. t w + u w + ( u) w = 0. If u = Pw then u
More informationTopics in Relativistic Astrophysics
Topics in Relativistic Astrophysics John Friedman ICTP/SAIFR Advanced School in General Relativity Parker Center for Gravitation, Cosmology, and Astrophysics Part I: General relativistic perfect fluids
More informationLecture Notes Introduction to Vector Analysis MATH 332
Lecture Notes Introduction to Vector Analysis MATH 332 Instructor: Ivan Avramidi Textbook: H. F. Davis and A. D. Snider, (WCB Publishers, 1995) New Mexico Institute of Mining and Technology Socorro, NM
More informationImproved estimates for the Ginzburg-Landau equation: the elliptic case
Improved estimates for the Ginzburg-Landau equation: the elliptic case F. Bethuel, G. Orlandi and D. Smets May 9, 2007 Abstract We derive estimates for various quantities which are of interest in the analysis
More informationInteraction energy between vortices of vector fields on Riemannian surfaces
Interaction energy between vortices of vector fields on Riemannian surfaces Radu Ignat 1 Robert L. Jerrard 2 1 Université Paul Sabatier, Toulouse 2 University of Toronto May 1 2017. Ignat and Jerrard (To(ulouse,ronto)
More informationVortex knots dynamics and momenta of a tangle:
Lecture 2 Vortex knots dynamics and momenta of a tangle: - Localized Induction Approximation (LIA) and Non-Linear Schrödinger (NLS) equation - Integrable vortex dynamics and LIA hierarchy - Torus knot
More informationBIHARMONIC WAVE MAPS INTO SPHERES
BIHARMONIC WAVE MAPS INTO SPHERES SEBASTIAN HERR, TOBIAS LAMM, AND ROLAND SCHNAUBELT Abstract. A global weak solution of the biharmonic wave map equation in the energy space for spherical targets is constructed.
More informationRotational & Rigid-Body Mechanics. Lectures 3+4
Rotational & Rigid-Body Mechanics Lectures 3+4 Rotational Motion So far: point objects moving through a trajectory. Next: moving actual dimensional objects and rotating them. 2 Circular Motion - Definitions
More informationAn unfortunate misprint
An unfortunate misprint Robert L. Jerrard Department of Mathematics University of Toronto March 22, 21, BIRS Robert L. Jerrard (Toronto ) An unfortunate misprint March 22, 21, BIRS 1 / 12 Consider the
More informationRegularity for Poisson Equation
Regularity for Poisson Equation OcMountain Daylight Time. 4, 20 Intuitively, the solution u to the Poisson equation u= f () should have better regularity than the right hand side f. In particular one expects
More informationThe Divergence Theorem Stokes Theorem Applications of Vector Calculus. Calculus. Vector Calculus (III)
Calculus Vector Calculus (III) Outline 1 The Divergence Theorem 2 Stokes Theorem 3 Applications of Vector Calculus The Divergence Theorem (I) Recall that at the end of section 12.5, we had rewritten Green
More informationOn the limiting behaviour of regularizations of the Euler Equations with vortex sheet initial data
On the limiting behaviour of regularizations of the Euler Equations with vortex sheet initial data Monika Nitsche Department of Mathematics and Statistics University of New Mexico Collaborators: Darryl
More informationMean Field Limits for Ginzburg-Landau Vortices
Mean Field Limits for Ginzburg-Landau Vortices Sylvia Serfaty Courant Institute, NYU Rivière-Fabes symposium, April 28-29, 2017 The Ginzburg-Landau equations u : Ω R 2 C u = u ε 2 (1 u 2 ) Ginzburg-Landau
More informationCreated by T. Madas VECTOR OPERATORS. Created by T. Madas
VECTOR OPERATORS GRADIENT gradϕ ϕ Question 1 A surface S is given by the Cartesian equation x 2 2 + y = 25. a) Draw a sketch of S, and describe it geometrically. b) Determine an equation of the tangent
More informationDIRECTION OF VORTICITY AND A REFINED BLOW-UP CRITERION FOR THE NAVIER-STOKES EQUATIONS WITH FRACTIONAL LAPLACIAN
DIRECTION OF VORTICITY AND A REFINED BLOW-UP CRITERION FOR THE NAVIER-STOKES EQUATIONS WITH FRACTIONAL LAPLACIAN KENGO NAKAI Abstract. We give a refined blow-up criterion for solutions of the D Navier-
More informationContents. MATH 32B-2 (18W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables. 1 Multiple Integrals 3. 2 Vector Fields 9
MATH 32B-2 (8W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables Contents Multiple Integrals 3 2 Vector Fields 9 3 Line and Surface Integrals 5 4 The Classical Integral Theorems 9 MATH 32B-2 (8W)
More informationLifshitz Hydrodynamics
Lifshitz Hydrodynamics Yaron Oz (Tel-Aviv University) With Carlos Hoyos and Bom Soo Kim, arxiv:1304.7481 Outline Introduction and Summary Lifshitz Hydrodynamics Strange Metals Open Problems Strange Metals
More informationPractice Final Solutions
Practice Final Solutions Math 1, Fall 17 Problem 1. Find a parameterization for the given curve, including bounds on the parameter t. Part a) The ellipse in R whose major axis has endpoints, ) and 6, )
More informationVorticity and Dynamics
Vorticity and Dynamics In Navier-Stokes equation Nonlinear term ω u the Lamb vector is related to the nonlinear term u 2 (u ) u = + ω u 2 Sort of Coriolis force in a rotation frame Viscous term ν u = ν
More informationAPPENDIX 2.1 LINE AND SURFACE INTEGRALS
2 APPENDIX 2. LINE AND URFACE INTEGRAL Consider a path connecting points (a) and (b) as shown in Fig. A.2.. Assume that a vector field A(r) exists in the space in which the path is situated. Then the line
More informationSOLUTIONS TO THE FINAL EXAM. December 14, 2010, 9:00am-12:00 (3 hours)
SOLUTIONS TO THE 18.02 FINAL EXAM BJORN POONEN December 14, 2010, 9:00am-12:00 (3 hours) 1) For each of (a)-(e) below: If the statement is true, write TRUE. If the statement is false, write FALSE. (Please
More informationPAPER 84 QUANTUM FLUIDS
MATHEMATICAL TRIPOS Part III Wednesday 6 June 2007 9.00 to 11.00 PAPER 84 QUANTUM FLUIDS Attempt TWO questions. There are FOUR questions in total. The questions carry equal weight. STATIONERY REQUIREMENTS
More informationStokes Theorem. MATH 311, Calculus III. J. Robert Buchanan. Summer Department of Mathematics. J. Robert Buchanan Stokes Theorem
tokes Theorem MATH 311, alculus III J. Robert Buchanan Department of Mathematics ummer 2011 Background (1 of 2) Recall: Green s Theorem, M(x, y) dx + N(x, y) dy = R ( N x M ) da y where is a piecewise
More information. (70.1) r r. / r. Substituting, we have the following equation for f:
7 Spherical waves Let us consider a sound wave in which the distribution of densit velocit etc, depends only on the distance from some point, ie, is spherically symmetrical Such a wave is called a spherical
More informationHamiltonian Dynamics from Lie Poisson Brackets
1 Hamiltonian Dynamics from Lie Poisson Brackets Jean-Luc Thiffeault Department of Applied Physics and Applied Mathematics Columbia University http://plasma.ap.columbia.edu/~jeanluc 12 February 2002 2
More informationOn Some Variational Optimization Problems in Classical Fluids and Superfluids
On Some Variational Optimization Problems in Classical Fluids and Superfluids Bartosz Protas Department of Mathematics & Statistics McMaster University, Hamilton, Ontario, Canada URL: http://www.math.mcmaster.ca/bprotas
More informationS chauder Theory. x 2. = log( x 1 + x 2 ) + 1 ( x 1 + x 2 ) 2. ( 5) x 1 + x 2 x 1 + x 2. 2 = 2 x 1. x 1 x 2. 1 x 1.
Sep. 1 9 Intuitively, the solution u to the Poisson equation S chauder Theory u = f 1 should have better regularity than the right hand side f. In particular one expects u to be twice more differentiable
More informationHamiltonian aspects of fluid dynamics
Hamiltonian aspects of fluid dynamics CDS 140b Joris Vankerschaver jv@caltech.edu CDS 01/29/08, 01/31/08 Joris Vankerschaver (CDS) Hamiltonian aspects of fluid dynamics 01/29/08, 01/31/08 1 / 34 Outline
More informationTopics in Fluid Dynamics: Classical physics and recent mathematics
Topics in Fluid Dynamics: Classical physics and recent mathematics Toan T. Nguyen 1,2 Penn State University Graduate Student Seminar @ PSU Jan 18th, 2018 1 Homepage: http://math.psu.edu/nguyen 2 Math blog:
More information3 Generation and diffusion of vorticity
Version date: March 22, 21 1 3 Generation and diffusion of vorticity 3.1 The vorticity equation We start from Navier Stokes: u t + u u = 1 ρ p + ν 2 u 1) where we have not included a term describing a
More informationξ,i = x nx i x 3 + δ ni + x n x = 0. x Dξ = x i ξ,i = x nx i x i x 3 Du = λ x λ 2 xh + x λ h Dξ,
1 PDE, HW 3 solutions Problem 1. No. If a sequence of harmonic polynomials on [ 1,1] n converges uniformly to a limit f then f is harmonic. Problem 2. By definition U r U for every r >. Suppose w is a
More informationUniformly accurate averaging numerical schemes for oscillatory evolution equations
Uniformly accurate averaging numerical schemes for oscillatory evolution equations Philippe Chartier University of Rennes, INRIA Joint work with M. Lemou (University of Rennes-CNRS), F. Méhats (University
More informationPotential Descending Principle, Dynamic Law of Physical Motion and Statistical Theory of Heat
Potential Descending Principle, Dynamic Law of Physical Motion and Statistical Theory of Heat Tian Ma and Shouhong Wang Supported in part by NSF, ONR and Chinese NSF http://www.indiana.edu/ fluid Outline
More informationQuasi-neutral limit for Euler-Poisson system in the presence of plasma sheaths
in the presence of plasma sheaths Department of Mathematical Sciences Ulsan National Institute of Science and Technology (UNIST) joint work with Masahiro Suzuki (Nagoya) and Chang-Yeol Jung (Ulsan) The
More informationMath 265H: Calculus III Practice Midterm II: Fall 2014
Name: Section #: Math 65H: alculus III Practice Midterm II: Fall 14 Instructions: This exam has 7 problems. The number of points awarded for each question is indicated in the problem. Answer each question
More informationAero III/IV Conformal Mapping
Aero III/IV Conformal Mapping View complex function as a mapping Unlike a real function, a complex function w = f(z) cannot be represented by a curve. Instead it is useful to view it as a mapping. Write
More informationChapter 2 Dynamics of Perfect Fluids
hapter 2 Dynamics of Perfect Fluids As discussed in the previous chapter, the viscosity of fluids induces tangential stresses in relatively moving fluids. A familiar example is water being poured into
More informationTHE L 2 -HODGE THEORY AND REPRESENTATION ON R n
THE L 2 -HODGE THEORY AND REPRESENTATION ON R n BAISHENG YAN Abstract. We present an elementary L 2 -Hodge theory on whole R n based on the minimization principle of the calculus of variations and some
More information14 Higher order forms; divergence theorem
Tel Aviv University, 2013/14 Analysis-III,IV 221 14 Higher order forms; divergence theorem 14a Forms of order three................ 221 14b Divergence theorem in three dimensions.... 225 14c Order four,
More information5.1 Fluid momentum equation Hydrostatics Archimedes theorem The vorticity equation... 42
Chapter 5 Euler s equation Contents 5.1 Fluid momentum equation........................ 39 5. Hydrostatics................................ 40 5.3 Archimedes theorem........................... 41 5.4 The
More informationSpotlight on Laplace s Equation
16 Spotlight on Laplace s Equation Reference: Sections 1.1,1.2, and 1.5. Laplace s equation is the undriven, linear, second-order PDE 2 u = (1) We defined diffusivity on page 587. where 2 is the Laplacian
More informationFinite-time singularity formation for Euler vortex sheet
Finite-time singularity formation for Euler vortex sheet Daniel Coutand Maxwell Institute Heriot-Watt University Oxbridge PDE conference, 20-21 March 2017 Free surface Euler equations Picture n N x Ω Γ=
More informationYAN GUO, JUHI JANG, AND NING JIANG
LOCAL HILBERT EXPANSION FOR THE BOLTZMANN EQUATION YAN GUO, JUHI JANG, AND NING JIANG Abstract. We revisit the classical ork of Caflisch [C] for compressible Euler limit of the Boltzmann equation. By using
More informationPhysics 451/551 Theoretical Mechanics. G. A. Krafft Old Dominion University Jefferson Lab Lecture 18
Physics 451/551 Theoretical Mechanics G. A. Krafft Old Dominion University Jefferson Lab Lecture 18 Theoretical Mechanics Fall 018 Properties of Sound Sound Waves Requires medium for propagation Mainly
More informationF11AE1 1. C = ρν r r. r u z r
F11AE1 1 Question 1 20 Marks) Consider an infinite horizontal pipe with circular cross-section of radius a, whose centre line is aligned along the z-axis; see Figure 1. Assume no-slip boundary conditions
More informationThe central force problem
1 The central force problem Moment. Das Moment des Eindrucks, den ein Mann auf das gemeine Volk macht, ist ein Produkt aus dem Wert des Rocks in den Titel. Georg Christoph Lichtenberg We start by dealing
More informationThe Convergence of Mimetic Discretization
The Convergence of Mimetic Discretization for Rough Grids James M. Hyman Los Alamos National Laboratory T-7, MS-B84 Los Alamos NM 87545 and Stanly Steinberg Department of Mathematics and Statistics University
More informationPoint Vortex Dynamics in Two Dimensions
Spring School on Fluid Mechanics and Geophysics of Environmental Hazards 9 April to May, 9 Point Vortex Dynamics in Two Dimensions Ruth Musgrave, Mostafa Moghaddami, Victor Avsarkisov, Ruoqian Wang, Wei
More informationSeminorms and locally convex spaces
(April 23, 2014) Seminorms and locally convex spaces Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/fun/notes 2012-13/07b seminorms.pdf]
More informationMagnetostatics. Lecture 23: Electromagnetic Theory. Professor D. K. Ghosh, Physics Department, I.I.T., Bombay
Magnetostatics Lecture 23: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay Magnetostatics Up until now, we have been discussing electrostatics, which deals with physics
More informationOn Fluid Maxwell Equations
On Fluid Maxwell Equations Tsutomu Kambe, (Former Professor, Physics), University of Tokyo, Abstract Fluid mechanics is a field theory of Newtonian mechanics of Galilean symmetry, concerned with fluid
More informationAngular momentum preserving CFD on general grids
B. Després LJLL-Paris VI Thanks CEA and ANR Chrome Angular momentum preserving CFD on general grids collaboration Emmanuel Labourasse (CEA) B. Després LJLL-Paris VI Thanks CEA and ANR Chrome collaboration
More informationNotes for 858D: Mathematical Aspects of Fluid Mechanics
Notes for 858D: Mathematical Aspects of Fluid Mechanics Jacob Bedrossian (with lots of notes written also by Vlad Vicol September 25, 218 Contents 1 Preface and disclaimer 2 2 Transport and ideal, incompressible
More informationWeek 6 Notes, Math 865, Tanveer
Week 6 Notes, Math 865, Tanveer. Energy Methods for Euler and Navier-Stokes Equation We will consider this week basic energy estimates. These are estimates on the L 2 spatial norms of the solution u(x,
More information4 Divergence theorem and its consequences
Tel Aviv University, 205/6 Analysis-IV 65 4 Divergence theorem and its consequences 4a Divergence and flux................. 65 4b Piecewise smooth case............... 67 4c Divergence of gradient: Laplacian........
More informationThe Inviscid Limit and Boundary Layers for Navier-Stokes flows
The Inviscid Limit and Boundary Layers for Navier-Stokes flows Yasunori Maekawa Department of Mathematics Graduate School of Science Kyoto University Kitashirakawa Oiwake-cho, Sakyo-ku Kyoto 606-8502,
More informationTwo-Body Problem. Central Potential. 1D Motion
Two-Body Problem. Central Potential. D Motion The simplest non-trivial dynamical problem is the problem of two particles. The equations of motion read. m r = F 2, () We already know that the center of
More informationKirchhoff s Elliptical Vortex
1 Figure 1. An elliptical vortex oriented at an angle φ with respect to the positive x axis. Kirchhoff s Elliptical Vortex In the atmospheric and oceanic context, two-dimensional (height-independent) vortices
More informationCourse Description for Real Analysis, Math 156
Course Description for Real Analysis, Math 156 In this class, we will study elliptic PDE, Fourier analysis, and dispersive PDE. Here is a quick summary of the topics will study study. They re described
More information[#1] R 3 bracket for the spherical pendulum
.. Holm Tuesday 11 January 2011 Solutions to MSc Enhanced Coursework for MA16 1 M3/4A16 MSc Enhanced Coursework arryl Holm Solutions Tuesday 11 January 2011 [#1] R 3 bracket for the spherical pendulum
More informationImproved estimates for the Ginzburg-Landau equation: the elliptic case
Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) Vol. IV (2005), 319-355 Improved estimates for the Ginzburg-Landau equation: the elliptic case FABRICE BETHUEL, GIANDOMENICO ORLANDI AND DIDIER SMETS Abstract.
More informationanalysis for transport equations and applications
Multi-scale analysis for transport equations and applications Mihaï BOSTAN, Aurélie FINOT University of Aix-Marseille, FRANCE mihai.bostan@univ-amu.fr Numerical methods for kinetic equations Strasbourg
More informationSOLAR MHD Lecture 2 Plan
SOLAR MHD Lecture Plan Magnetostatic Equilibrium ü Structure of Magnetic Flux Tubes ü Force-free fields Waves in a homogenous magnetized medium ü Linearized wave equation ü Alfvén wave ü Magnetoacoustic
More informationTHE STOKES SYSTEM R.E. SHOWALTER
THE STOKES SYSTEM R.E. SHOWALTER Contents 1. Stokes System 1 Stokes System 2 2. The Weak Solution of the Stokes System 3 3. The Strong Solution 4 4. The Normal Trace 6 5. The Mixed Problem 7 6. The Navier-Stokes
More informationOn the local well-posedness of compressible viscous flows with bounded density
On the local well-posedness of compressible viscous flows with bounded density Marius Paicu University of Bordeaux joint work with Raphaël Danchin and Francesco Fanelli Mathflows 2018, Porquerolles September
More informationVelocities in Quantum Mechanics
Velocities in Quantum Mechanics Toshiki Shimbori and Tsunehiro Kobayashi Institute of Physics, University of Tsukuba Ibaraki 305-8571, Japan Department of General Education for the Hearing Impaired, Tsukuba
More informationExistence and Continuation for Euler Equations
Existence and Continuation for Euler Equations David Driver Zhuo Min Harold Lim 22 March, 214 Contents 1 Introduction 1 1.1 Physical Background................................... 1 1.2 Notation and Conventions................................
More informationFundamentals of Fluid Dynamics: Ideal Flow Theory & Basic Aerodynamics
Fundamentals of Fluid Dynamics: Ideal Flow Theory & Basic Aerodynamics Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI (after: D.J. ACHESON s Elementary Fluid Dynamics ) bluebox.ippt.pan.pl/
More informationA Ginzburg-Landau approach to dislocations. Marcello Ponsiglione Sapienza Università di Roma
Marcello Ponsiglione Sapienza Università di Roma Description of a dislocation line A deformed crystal C can be described by The displacement function u : C R 3. The strain function β = u. A dislocation
More informationRenormalized Energy with Vortices Pinning Effect
Renormalized Energy with Vortices Pinning Effect Shijin Ding Department of Mathematics South China Normal University Guangzhou, Guangdong 5063, China Abstract. This paper is a successor of the previous
More informationDynamics of a rigid body in a two dimensional incompressible perfect fluid and the zero-radius limit
.. Dynamics of a rigid body in a two dimensional incompressible perfect fluid and the zero-radius limit Franck Sueur Abstract In this survey we report some recent results on the dynamics of a rigid body
More informationASYMPTOTIC STRUCTURE FOR SOLUTIONS OF THE NAVIER STOKES EQUATIONS. Tian Ma. Shouhong Wang
DISCRETE AND CONTINUOUS Website: http://aimsciences.org DYNAMICAL SYSTEMS Volume 11, Number 1, July 004 pp. 189 04 ASYMPTOTIC STRUCTURE FOR SOLUTIONS OF THE NAVIER STOKES EQUATIONS Tian Ma Department of
More informationLaplace s Equation. Chapter Mean Value Formulas
Chapter 1 Laplace s Equation Let be an open set in R n. A function u C 2 () is called harmonic in if it satisfies Laplace s equation n (1.1) u := D ii u = 0 in. i=1 A function u C 2 () is called subharmonic
More information