The semi-geostrophic equations - a model for large-scale atmospheric flows
|
|
- Poppy Holt
- 5 years ago
- Views:
Transcription
1 The semi-geostrophic equations - a model for large-scale atmospheric flows Beatrice Pelloni, University of Reading with M. Cullen (Met Office), D. Gilbert, T. Kuna INI - MFE Dec 2013
2 Introduction - Motivation The behaviour of the atmosphere is described by the compressible Navier-Stokes equations ( geopotential Φ = gx 3. ) D t u + 2Ω u + Φ + 1 p = ν u, ρ ρ + (ρu) = 0, t D t θ = 0, p 1 κ = Rρθ. Notation: H typical vertical scale, L horizontal scale U typical horizontal velocity, T U L typical timescale From now on, assume no viscosity (ν = 0) - compressible Euler equations
3 Reductions to a model for large scale flows (1) Shallow atmosphere: H L << 1, This implies that the Coriolis force has no effect in the vertical direction 2Ω u = ( fu 2, fu 1, 0) = D t u + (fu 2, fu 1, g) + 1 ρ p = 0 f is variable (2) Hydrostatic balance: p x 3 = gρ In the resulting system, the vertical velocity can be resolved if either the Rossby number ε = U U fl or the Froude number η = NH are small small ε - rotation dominated flow small η - stratification dominated flow
4 Geostrophic balance (3) Geostrophic balance: For small Rossby number ε, the rotations terms dominate = 1 ρ 1p fu 2, 1 ρ 2p fu 1 Define the geostrophic velocity as the horizontal velocity satisfying the geostrophic balance exactly: v g = (v g 1, v g 2, 0) with v g 1 = 1 f ρ 2p, v g 2 = 1 f ρ 1p. There are different possible relative scalings of ε and η within this framework
5 ε = η << 1: the quasi-geostrophic system The scaling Froude=Rossby, i.e. H L = f N, gives rise, to first order in the approximation, to the quasi-geostrophic system: conservation of potential vorticity principle, but not valid on large scales (they require constant coefficients) Results by Bourgeois-Beale on global existence (at least for well prepared initial conditions) and validity as a reduction of NS - the scale analysis is rigorously justified.
6 ε η 2 << 1: the semigeostrophic system This is a second-order accurate approximation. Only horizontal momentum is approximated = equations are valid for large scale (f can be variable): D t v g (fu 2, fu 1 ) + 1 ρ ( 1p, 2 p) = 0 ρ + (ρu) = 0, t 1 p = ρfv g 2, 2p = ρfv g 1, 3p = ρg, D t θ = 0, p 1 κ = Rρθ. where D = t + u (energy is then conserved) First derived by Eliassen, then rediscovered by Bretherton and Hoskins as an approximate model for frontogenesis (John Methven s talk on Monday!)
7 Mathematical vs physical issues The QG system is a conservation law, but it is not valid on large scales. Also, it is not possible to extend results to variable geometry or variable coefficient cases. The system is valid on large scales, and is also a conservation law in the right variables. It is a more appropriate system to use to understand the predictability of the weather system on large scales, which requires that: the equations can be solved for given initial and boundary conditions the solutions, not necessarily smooth, are well defined at least for some time, ideally on the approximation timescale the scale analysis can be rigorously justified - there is a solution of the full NS system close to the semigeostrophic solution in an appropriate asymptotic sense
8 The semigeostrophic system - 3D incompressible case The geostrophic velocity is 2D and given by Assuming also (v g 1, v g 2 ) = ( 2p, 1 p) (geostrophic balance). 3 p = ρ (hydrostatic balance) (with Boussinesq approximation and all constants scaled to 1): D t (v g 1, v g 2 ) + ( 1p, 2 p) = (u 2, u 1 ) D t ρ = 0, ρ = 3 p, (v g 1, v g 2 ) = ( 2p, 1 p), u = 0, (D t = t + u denotes the lagrangian derivative) Boundary condition: u n = 0 on Ω. Unknowns: u = (u 1, u 2, u 3 ); v = (v 1 g, v 2 g, 0); p; ρ.
9 Dual variables Change to dual coordinates: X (t, x) = P = (v g 2 + x 1, v g 1 + x 2, ρ), where P(t, x) = p(t, x) (x x 2 2 ). ( Hoskins geostrophic variable change). Then D t X = J(X x), J = P(0, x) = P 0 (x); ( plus the conditions on u: u = 0; u n = 0 on Ω. Note: U = D t X = (v g 1, v g 2, 0) - the dual flow is purely geostrophic ),
10 Energy The geostrophic energy associated with the flow is { } 1 [ E = (v g 1 2 )2 + (v g 2 )2] + ρx 3 dx. Ω Cullen s stability principle: stable solutions are energy minimisers with respect to the rearrangements of particles in physical space that conserve absolute momentum and density In the dual variables X = P(x) = (v g 2 + x 1, v g 1 + x 2, ρ), { 1 [ E(X ) = (x1 X 1 ) 2 + (x 2 X 2 ) 2] } x 3 X 3 dx. 2 Ω In these variables the minimiser condition formally becomes P(t, x) = 1 2 (x x 2 2 ) + p(t, x) is a convex function
11 Minimising the dual energy = optimal transport problem The change of variable X = P(x) can be stated in terms of the measure ν on R 3 such that f ( P(x))dx = f (y)dν(y) f C c (R 3 ). Ω R 3 This is concisely written using the push-forward notation as ν = P#χ Ω ( weak form of the Monge-Ampère equation ν = det(d 2 P )). The minimisation of E(X ) can be phrased as the requirement that the change of variables X : Ω R 3 such that X #χ Ω = ν is the optimal transport map with respect to the cost function c(x, X ) = 1 2 [ (x1 X 1 ) 2 + (x 2 X 2 ) 2] x 3 X 3
12 Dual formulation Assume P convex (key assumption) - P is a well defined change of variables. Let P denote the Legendre transform of P: P (X ) = sup{x X P(x)} (X = P(x) x = P (X )) x Switch dependent/independent variable dual formulation (conservation law) ν = P#χ Ω ; t ν + (Uν) = 0; U = J(X P ); ν(0, X ) = ν 0 (X ) (with ν 0 = P 0 #χ Ω ).
13 Physical vs dual variables D t (v g 1, v g 2 ) + ( 1p, 2 p) = (u 2, u 1 ), v g = ( 2 p, 1 p, 0) D t ρ = 0, ρ = 3 p, P(t, x) = p(t, x) (x x 2 2 ), J = ( ) Physical space Dual space ( t + u )X = J(X x) t ν + (Uν) = 0 X = P ν = P#χ Ω ; U = J(X P ) X = X (x) = P x = x(x ) = P
14 The result of Benamou and Brenier Polar factorization theorem = statement on existence of a unique optimal transport map between χ Ω and ν, with respect to cost c(x, X ) = 1 2 x X 2 This map is given by the gradient of a convex function X = P minimisation of the geostrophic energy 1 E = 2 [(x 1 X 1 ) 2 + (x 2 X 2 ) 2 x 3 X 3 ]dx Ω Stable solution have to minimise E at each fixed time. = at each fixed t. ν determines a convex function P. Ω c(x, X (x))dx Moreover, if ν n ν weakly, then P n P in W 1,p (stability).
15 How does this solve the geostrophic system? The potential vorticity ν must also satisfy the transport equation t ν(t, X )+ (U(t, X )ν(t, X )) = 0, U(t, X ) = J(X P (t, X )) The issue in solving this transport equation is the lack of regularity of the velocity U(t, X ) = J(X P (t, X )). Polar factorization = U is not Lipschitz, only locally bounded, locally BV ( ψ W 1, ) In order to find a weak solution: time discretisation + regularisation + stability results. Use standard results to solve the sequence of regularised transport problem, plus the stability given by Brenier s theorem (Feldman talk) Note: do not get uniqueness
16 3D incompressible result in dual variables Theorem [BB]: P 0 (x) given convex function, such that ν 0 = P 0 #χ Ω is in L q (Ω) q > 1 (= compact support B(0, R 0 )). Then for all τ > 0, there exist ν(x, t) (L [0, τ), L q (R 3 )), supp ν B(0, R(τ)), P(t, X ) (L [0, τ), W 1, (Ω)), convex, P W 1,r (Ω)) for all r 1 P convex, locally bounded in t and X, with P (L [0, τ) C[0, T ), L (B(0, R(τ)))) U (L [0, τ), L loc BV loc(r 3 )) ν is a weak solution of the transport equation: For all Φ(t, X ) C c ([0, τ) R 3 ) [ t Φ + U Φ]νdXdt + ν 0 Φ(0, X )dx = 0
17 Generalisations Obvious aims: make sense of the solution in physical space (Feldman s talk) uniqueness of the solution inclusion of more general conditions (compressible equations, variable rotation rate, other boundary conditions,...) What has been done: Incompressible shallow water equations (2D) with a free boundary ( Cullen and Gangbo 2001) Solution of the compressible case in dual coordinates (Cullen and Maroofi 2003) Solution in physical rather than dual coordinates (Cullen and Feldman 2006)
18 A robust approach: the compressible dual result Energy: E(t) = Ω [ ] 1 2 v g 2 + Φ(x) + θ(t, x)p(t, x) κ 1 κ ρ(t, x)dx. The set-up is similar: in dual variables, it is the coupling of two problems 1. formulate the energy minimisation as an optimal transport problem at each fixed time t 2. solve (weakly) the transport equation for the dual density ν
19 The energy to be minimised can be written in dual variables as E ν (σ) = E(ν, σ)+k σ κ dx, E(ν, σ) = inf c(x, T (x))dx T #σ=ν Ω where σ = θρ is a physical density, and the cost is [ 1 2 (x 1 y 1 ) (x 2 y 2 ) 2 + Φ(x) ] c(x, y) = Ω y 3. Show that the optimal transport problem for E(ν, σ) has a unique solution T : Ω Λ, T = P with P convex Show that, for fixed ν, there exists a unique σ P ac (Ω) that minimizes the energy E ν (σ) - in addition σ W 1, (Ω) Uniformity estimates solutions to the transport problem for ν constructed by approximation+regularisation ( discrete time stepping procedure, as in Benamou-Brenier)
20 Extensions, some in progress, I want to discuss Alternative proof using Ambrosio-Gangbo Hamiltonian flows in spaces of probability measures 3D free boundary value problem for incompressible case Justification of the scaling reduction for sg Variable rate of rotation - in progress upstairs
21 1 - Alternative proof of the solvabiliy of the transport problem Very general approach (Ambrosio and Gangbo): study of hamiltonian ODEs in space P2 ac of probability measures - metric space with Wasserstein metric W 2 The differentiable structure of this space was developed following the work many people (Mc Cann, Otto, Ambrosio, Gangbo, Pacini). In particular one has a notion of (λ )displacement convexity. The notion of subdifferential mimicks the usual notion in convex analysis This is used to define rigourously Hamiltonian flows in the space of probability measures (Gangbo-Pacini).
22 Given a lower sc Hamiltonian H : P ac 2 R, and given the probability measure µ(t = 0), find a solution to the transport t µ (t) + (J H(µ (t) )µ (t) ), µ 0 given ( a path in probability space from µ 0 to µ (t) ) H denotes the element of minimal norm in the subdifferential Solutions exist under a growth condition and continuity conditions for the gradient of H - under additional convexity assumption, H is also constant along solutions.
23 Hamiltonian ODEs Let H : P ac 2 R be lsc, and satisfy (H1): H(µ)(z) C(1 + z ) for a.e. z (H2): for µ n µ narrowly, there exists a subsequence and L 2 functions w k = H(µ nk ), w = H(µ) a.e. such that w k w in L 2. Theorem [AG] (absolutely continuous case) Given a Hamiltonian H as above, there exists a Hamiltonian flow µ t starting from a given µ 0 P2 ac, whose velocity field coincides with H(µ t ) for a.e. t, i.e. a solution of t µ (t) + (J H(µ (t) )µ (t) ) = 0, µ (0) = µ 0 t µ t is a Lipschitz map, and the support of the measures is controlled. If H is λ-convex, the Hamiltonian is constant along trajectories.
24 Alternative proof of dual space result Incompressible: coupling of two problems 1. find P convex: P#χ Ω = ν optimal transport problem at each fixed time t 2. evolution equation for ν Dual equations: U = J(X P (X )) t ν + (Uν) = 0, ν = T #σ Use the result of Ambrosio and Gangbo by proving directly that the semigeostrophic energy is subdifferentiable, (-2)-convex and lower s.c. computing explicitly H and showing that for fixed t H = X P (X )
25 2 - Free boundary value problem Consider the 3D incompressible semigeostrophic system in a domain Ω h (t) = Ω 2 [0, h(t, x 1, x 2 )], Ω 2 R 2. where h describe the (free) top boundary (rigid bottom at x 3 = 0). Incompressibility: Ω h (t) = 1 for all t < τ h L 1 L 2 (Ω 2 ) Boundary conditions: u n = 0 x Ω h (t) \ {x 3 = h}, { h h t h + u 1 x 1 + u 2 x 2 = u 3, x Ω h (t) : x 3 = h(t, x 1, x 2 ), p(t, x 1, x 2, h(x 1, x 2 )) = p h,
26 Dual variables problem x Ω h (t) T(t, x) = (x 1 + u g 2, x 2 u g 1, ρ) Λ R3, Potential density ν = T#σ h P ac (Λ), with σ h = χ Ωh Cullen s stability principle Given ν P 2 ac(λ), a stable solution corresponds to the following minimal value for the energy: E(t, ν) = inf σ h H E ν(h) H P ac (R 3 ) is an appropriate subset of P ac (R 3 ), and E ν (h) = inf T: T#σ h =ν R 3 [ 1 ( x1 T x 2 T 2 2) ] x 3 T 3 σ h (x) dx 2
27 Lagrangian dual form ν + (νw) = 0, t in [0, τ) Λ, w(t, y) = J(y T 1 (t, y)), in [0, τ) Λ, T(t, ) = P(t, ), t [0, τ), P#σ h = ν, σ h minimises E ν(t, ) ( ) over H, t [0, τ). plus initial conditions h(0, ) = h 0 ( ) W 1, (Ω 2 ), ν(0, ) = ν 0 ( ) L r (Λ 0 ), r (1, ), P 0 (x) W 1, (Ω h0 ) : P 0 #σ h0 = ν 0. Unknowns: h(t, x 1, x 2 ) and P(t, x). Then ν = P#σ h ; p(t, x) = P(t, x) 1 2 (x x 2 2)
28 Tools for the proof Same idea as the proof for the problem in a fixed domain: 1 Prove that the minimising problem has a unique solution for P and h - optimal transport problem but with T = P depending on h = minimisation in both T and h Uses strict convexity with respect to the usual linear structure of L 1, plus h > 0, ρ > 0. 2 Show that the Hamiltonian satisfies the continuity and growth conditions required by Ambrosio-Gangbo result so that the transport problem with velocity J H(ν) can be solved. Then show that J H(ν) = w
29 3 - Justification of the scaling reduction Quantifying the relation with the solutions of Euler (or NS) is crucial for the rigorous justification of the scaling reduction. QG: : Bourgeois-Beale: existence of global solutions in physical space (periodic or Neumann conditions) Based on refined energy estimates (control of velocity gradients) - basis is fundamental work of Temam on Euler = Existence of Euler solutions close to QG solution to the same order as that of approximation Brenier-Cullen results - periodic 2d - but order of approximation is not optimal ( ε instead of ε). Applies only to smooth solutions (that exist only locally) and periodic BC
30 2D result is based on the Bregman functional η P (t, y, y ɛ ) = P (t, y ɛ ) P (t, y) ( P )(y ɛ y) and the (brute force) estimation of the related energy functional [ e(t) = ε 2 + v ɛ ] (t, x) v(t, x) + η P (t, y, v ɛ ) dx. 2 B Recently extended to 3D periodic problem: Consider a periodic box in R 3. Let (y ɛ, v ɛ, P ɛ ) and (y = P, v) be smooth local solutions to Euler and 3D respectively on a finite time interval [0, T ]. Assume further that P has a smooth convex extension. Then the L 2 distance between y ɛ and y stays uniformly of order ɛ as ɛ 0, provided it does at t = 0 and the initial velocity v ɛ (0, x) is uniformly bounded in L 2. Exploiting Lagrangian trajectories in a clever way, the optimal order of approximation can be obtained (work in progress).
31 Conclusions - what mathematics has to offer Relatively technical mathematical analysis can confirm rigorously the validity of models reduced on the basis of physical considerations.... Existence of solutions - the model is a useful representation of reality Smooth or weak solutions? Reality is not very smooth... (difference between QG and ) Global(=long time) existence predictability, on large scales, at least on certain (long) time intervals With special thanks to Mike Cullen
On Lagrangian solutions for the semi-geostrophic system with singular initial data
On Lagrangian solutions for the semi-geostrophic system with singular initial data Mikhail Feldman Department of Mathematics University of Wisconsin-Madison Madison, WI 5376, USA feldman@math.wisc.edu
More informationOn Lagrangian solutions for the semi-geostrophic system with singular initial data
On Lagrangian solutions for the semi-geostrophic system with singular initial data Mikhail Feldman Department of Mathematics University of Wisconsin-Madison Madison, WI 5376, USA feldman@math.wisc.edu
More informationSobolev regularity for the Monge-Ampère equation, with application to the semigeostrophic equations
Sobolev regularity for the Monge-Ampère equation, with application to the semigeostrophic equations Alessio Figalli Abstract In this note we review some recent results on the Sobolev regularity of solutions
More informationLarge-scale atmospheric circulation, semi-geostrophic motion and Lagrangian particle methods
Large-scale atmospheric circulation, semi-geostrophic motion and Lagrangian particle methods Colin Cotter (Imperial College London) & Sebastian Reich (Universität Potsdam) Outline 1. Hydrostatic and semi-geostrophic
More informationDynamic and Stochastic Brenier Transport via Hopf-Lax formulae on Was
Dynamic and Stochastic Brenier Transport via Hopf-Lax formulae on Wasserstein Space With many discussions with Yann Brenier and Wilfrid Gangbo Brenierfest, IHP, January 9-13, 2017 ain points of the
More informationComputational challenges in Numerical Weather Prediction
Computational challenges in Numerical Weather Prediction Mike Cullen Oxford 15 September 2008 Contents This presentation covers the following areas Historical background Current challenges Why does it
More informationRearrangement vs Convection:
Rearrangement vs Convection: GR MOA, FREJUS 31 AOUT-3 SEPT 2009 Yann Brenier brenier@math.unice.fr CNRS FR2800 Université de Nice Outline 1. A toy-model for (very fast) convection based on rearrangement
More informationConvex discretization of functionals involving the Monge-Ampère operator
Convex discretization of functionals involving the Monge-Ampère operator Quentin Mérigot CNRS / Université Paris-Dauphine Joint work with J.D. Benamou, G. Carlier and É. Oudet GeMeCod Conference October
More informationSEMI-GEOSTROPHIC SYSTEM WITH VARIABLE CORIOLIS PARAMETER
SEMI-GEOSTROPHIC SYSTEM WITH VARIABLE CORIOLIS PARAMETER JINGRUI CHENG, MICHAEL CULLEN, AND MIKHAIL FELDMAN Abstract. We prove short time existence and uniqueness of smooth solutions ( in C k+2,α with
More informationEstimation of State Noise for the Ensemble Kalman filter algorithm for 2D shallow water equations.
Estimation of State Noise for the Ensemble Kalman filter algorithm for 2D shallow water equations. May 6, 2009 Motivation Constitutive Equations EnKF algorithm Some results Method Navier Stokes equations
More informationCLASSICAL SOLUTIONS TO SEMI-GEOSTROPHIC SYSTEM WITH VARIABLE CORIOLIS PARAMETER
CLASSICAL SOLUTIONS TO SEMI-GEOSTROPHIC SYSTEM WITH VARIABLE CORIOLIS PARAMETER JINGRUI CHENG, MICHAEL CULLEN, AND MIKHAIL FELDMAN Abstract. We prove short time existence and uniqueness of smooth solutions
More informationLagrangian discretization of incompressibility, congestion and linear diffusion
Lagrangian discretization of incompressibility, congestion and linear diffusion Quentin Mérigot Joint works with (1) Thomas Gallouët (2) Filippo Santambrogio Federico Stra (3) Hugo Leclerc Seminaire du
More informationGradient Flows: Qualitative Properties & Numerical Schemes
Gradient Flows: Qualitative Properties & Numerical Schemes J. A. Carrillo Imperial College London RICAM, December 2014 Outline 1 Gradient Flows Models Gradient flows Evolving diffeomorphisms 2 Numerical
More informationConvex discretization of functionals involving the Monge-Ampère operator
Convex discretization of functionals involving the Monge-Ampère operator Quentin Mérigot CNRS / Université Paris-Dauphine Joint work with J.D. Benamou, G. Carlier and É. Oudet Workshop on Optimal Transport
More informationJoint work with Nguyen Hoang (Univ. Concepción, Chile) Padova, Italy, May 2018
EXTENDED EULER-LAGRANGE AND HAMILTONIAN CONDITIONS IN OPTIMAL CONTROL OF SWEEPING PROCESSES WITH CONTROLLED MOVING SETS BORIS MORDUKHOVICH Wayne State University Talk given at the conference Optimization,
More informationRecent results for the 3D Quasi-Geostrophic Equation
Recent results for the 3D Quasi-Geostrophic Equation Alexis F. Vasseur Joint work with Marjolaine Puel (U. of Nice, France) and Matt Novack (UT Austin) The University of Texas at Austin Transport Phenomena
More informationUna aproximación no local de un modelo para la formación de pilas de arena
Cabo de Gata-2007 p. 1/2 Una aproximación no local de un modelo para la formación de pilas de arena F. Andreu, J.M. Mazón, J. Rossi and J. Toledo Cabo de Gata-2007 p. 2/2 OUTLINE The sandpile model of
More informationA few words about the MTW tensor
A few words about the Ma-Trudinger-Wang tensor Université Nice - Sophia Antipolis & Institut Universitaire de France Salah Baouendi Memorial Conference (Tunis, March 2014) The Ma-Trudinger-Wang tensor
More informationOptimal Transportation. Nonlinear Partial Differential Equations
Optimal Transportation and Nonlinear Partial Differential Equations Neil S. Trudinger Centre of Mathematics and its Applications Australian National University 26th Brazilian Mathematical Colloquium 2007
More informationViscous capillary fluids in fast rotation
Viscous capillary fluids in fast rotation Centro di Ricerca Matematica Ennio De Giorgi SCUOLA NORMALE SUPERIORE BCAM BASQUE CENTER FOR APPLIED MATHEMATICS BCAM Scientific Seminar Bilbao May 19, 2015 Contents
More informationIntroduction to optimal transport
Introduction to optimal transport Nicola Gigli May 20, 2011 Content Formulation of the transport problem The notions of c-convexity and c-cyclical monotonicity The dual problem Optimal maps: Brenier s
More informationAn introduction to Mathematical Theory of Control
An introduction to Mathematical Theory of Control Vasile Staicu University of Aveiro UNICA, May 2018 Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018
More informationPARTIAL REGULARITY OF BRENIER SOLUTIONS OF THE MONGE-AMPÈRE EQUATION
PARTIAL REGULARITY OF BRENIER SOLUTIONS OF THE MONGE-AMPÈRE EQUATION ALESSIO FIGALLI AND YOUNG-HEON KIM Abstract. Given Ω, Λ R n two bounded open sets, and f and g two probability densities concentrated
More informationFrom Euler to Monge and vice versa. Y.B. Lecture 2: From Euler to Vlasov through Monge-Ampère and Kantorovich.
From Euler to Monge and vice versa. Y.B. Lecture 2: From Euler to Vlasov through Monge-Ampère and Kantorovich. Yann Brenier, Mikaela Iacobelli, Filippo Santambrogio, Paris, Durham, Lyon. MFO SEMINAR 1842,
More informationLogarithmic Sobolev Inequalities
Logarithmic Sobolev Inequalities M. Ledoux Institut de Mathématiques de Toulouse, France logarithmic Sobolev inequalities what they are, some history analytic, geometric, optimal transportation proofs
More informationA DIFFUSIVE MODEL FOR MACROSCOPIC CROWD MOTION WITH DENSITY CONSTRAINTS
A DIFFUSIVE MODEL FOR MACROSCOPIC CROWD MOTION WITH DENSITY CONSTRAINTS Abstract. In the spirit of the macroscopic crowd motion models with hard congestion (i.e. a strong density constraint ρ ) introduced
More informationVariational approach to mean field games with density constraints
1 / 18 Variational approach to mean field games with density constraints Alpár Richárd Mészáros LMO, Université Paris-Sud (based on ongoing joint works with F. Santambrogio, P. Cardaliaguet and F. J. Silva)
More informationThe optimal partial transport problem
The optimal partial transport problem Alessio Figalli Abstract Given two densities f and g, we consider the problem of transporting a fraction m [0, min{ f L 1, g L 1}] of the mass of f onto g minimizing
More informationInformation geometry for bivariate distribution control
Information geometry for bivariate distribution control C.T.J.Dodson + Hong Wang Mathematics + Control Systems Centre, University of Manchester Institute of Science and Technology Optimal control of stochastic
More informationFrom a Mesoscopic to a Macroscopic Description of Fluid-Particle Interaction
From a Mesoscopic to a Macroscopic Description of Fluid-Particle Interaction Carnegie Mellon University Center for Nonlinear Analysis Working Group, October 2016 Outline 1 Physical Framework 2 3 Free Energy
More informationGlobal well-posedness of the primitive equations of oceanic and atmospheric dynamics
Global well-posedness of the primitive equations of oceanic and atmospheric dynamics Jinkai Li Department of Mathematics The Chinese University of Hong Kong Dynamics of Small Scales in Fluids ICERM, Feb
More informationKramers formula for chemical reactions in the context of Wasserstein gradient flows. Michael Herrmann. Mathematical Institute, University of Oxford
eport no. OxPDE-/8 Kramers formula for chemical reactions in the context of Wasserstein gradient flows by Michael Herrmann Mathematical Institute, University of Oxford & Barbara Niethammer Mathematical
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 informationCandidates must show on each answer book the type of calculator used. Log Tables, Statistical Tables and Graph Paper are available on request.
UNIVERSITY OF EAST ANGLIA School of Mathematics Spring Semester Examination 2004 FLUID DYNAMICS Time allowed: 3 hours Attempt Question 1 and FOUR other questions. Candidates must show on each answer book
More informationBranched transport limit of the Ginzburg-Landau functional
Branched transport limit of the Ginzburg-Landau functional Michael Goldman CNRS, LJLL, Paris 7 Joint work with S. Conti, F. Otto and S. Serfaty Introduction Superconductivity was first observed by Onnes
More informationOn a Class of Multidimensional Optimal Transportation Problems
Journal of Convex Analysis Volume 10 (2003), No. 2, 517 529 On a Class of Multidimensional Optimal Transportation Problems G. Carlier Université Bordeaux 1, MAB, UMR CNRS 5466, France and Université Bordeaux
More informationUniqueness of the solution to the Vlasov-Poisson system with bounded density
Uniqueness of the solution to the Vlasov-Poisson system with bounded density Grégoire Loeper December 16, 2005 Abstract In this note, we show uniqueness of weak solutions to the Vlasov- Poisson system
More informationMath background. Physics. Simulation. Related phenomena. Frontiers in graphics. Rigid fluids
Fluid dynamics Math background Physics Simulation Related phenomena Frontiers in graphics Rigid fluids Fields Domain Ω R2 Scalar field f :Ω R Vector field f : Ω R2 Types of derivatives Derivatives measure
More informationOn the well-posedness of the Prandtl boundary layer equation
On the well-posedness of the Prandtl boundary layer equation Vlad Vicol Department of Mathematics, The University of Chicago Incompressible Fluids, Turbulence and Mixing In honor of Peter Constantin s
More informationarxiv: v1 [math.ap] 10 Apr 2013
QUASI-STATIC EVOLUTION AND CONGESTED CROWD TRANSPORT DAMON ALEXANDER, INWON KIM, AND YAO YAO arxiv:1304.3072v1 [math.ap] 10 Apr 2013 Abstract. We consider the relationship between Hele-Shaw evolution with
More informationGradient Flow. Chang Liu. April 24, Tsinghua University. Chang Liu (THU) Gradient Flow April 24, / 91
Gradient Flow Chang Liu Tsinghua University April 24, 2017 Chang Liu (THU) Gradient Flow April 24, 2017 1 / 91 Contents 1 Introduction 2 Gradient flow in the Euclidean space Variants of Gradient Flow in
More informationDecay profiles of a linear artificial viscosity system
Decay profiles of a linear artificial viscosity system Gene Wayne, Ryan Goh and Roland Welter Boston University rwelter@bu.edu July 2, 2018 This research was supported by funding from the NSF. Roland Welter
More informationThe Shallow Water Equations
If you have not already done so, you are strongly encouraged to read the companion file on the non-divergent barotropic vorticity equation, before proceeding to this shallow water case. We do not repeat
More informationMinimal time mean field games
based on joint works with Samer Dweik and Filippo Santambrogio PGMO Days 2017 Session Mean Field Games and applications EDF Lab Paris-Saclay November 14th, 2017 LMO, Université Paris-Sud Université Paris-Saclay
More informationLocal semiconvexity of Kantorovich potentials on non-compact manifolds
Local semiconvexity of Kantorovich potentials on non-compact manifolds Alessio Figalli, Nicola Gigli Abstract We prove that any Kantorovich potential for the cost function c = d / on a Riemannian manifold
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 informationOptimal transport for Seismic Imaging
Optimal transport for Seismic Imaging Bjorn Engquist In collaboration with Brittany Froese and Yunan Yang ICERM Workshop - Recent Advances in Seismic Modeling and Inversion: From Analysis to Applications,
More informationPAPER 333 FLUID DYNAMICS OF CLIMATE
MATHEMATICAL TRIPOS Part III Wednesday, 1 June, 2016 1:30 pm to 4:30 pm Draft 21 June, 2016 PAPER 333 FLUID DYNAMICS OF CLIMATE Attempt no more than THREE questions. There are FOUR questions in total.
More informationThe 2D Magnetohydrodynamic Equations with Partial Dissipation. Oklahoma State University
The 2D Magnetohydrodynamic Equations with Partial Dissipation Jiahong Wu Oklahoma State University IPAM Workshop Mathematical Analysis of Turbulence IPAM, UCLA, September 29-October 3, 2014 1 / 112 Outline
More informationFluid Dynamics Exercises and questions for the course
Fluid Dynamics Exercises and questions for the course January 15, 2014 A two dimensional flow field characterised by the following velocity components in polar coordinates is called a free vortex: u r
More informationOptimal Transport for Applied Mathematicians
Optimal Transport for Applied Mathematicians Calculus of Variations, PDEs and Modelling Filippo Santambrogio 1 1 Laboratoire de Mathématiques d Orsay, Université Paris Sud, 91405 Orsay cedex, France filippo.santambrogio@math.u-psud.fr,
More informationAtmospheric dynamics and meteorology
Atmospheric dynamics and meteorology B. Legras, http://www.lmd.ens.fr/legras III Frontogenesis (pre requisite: quasi-geostrophic equation, baroclinic instability in the Eady and Phillips models ) Recommended
More informationOn uniqueness of weak solutions to transport equation with non-smooth velocity field
On uniqueness of weak solutions to transport equation with non-smooth velocity field Paolo Bonicatto Abstract Given a bounded, autonomous vector field b: R d R d, we study the uniqueness of bounded solutions
More informationMinimal Surface equations non-solvability strongly convex functional further regularity Consider minimal surface equation.
Lecture 7 Minimal Surface equations non-solvability strongly convex functional further regularity Consider minimal surface equation div + u = ϕ on ) = 0 in The solution is a critical point or the minimizer
More informationGFD 2012 Lecture 1: Dynamics of Coherent Structures and their Impact on Transport and Predictability
GFD 2012 Lecture 1: Dynamics of Coherent Structures and their Impact on Transport and Predictability Jeffrey B. Weiss; notes by Duncan Hewitt and Pedram Hassanzadeh June 18, 2012 1 Introduction 1.1 What
More informationPartitioned Methods for Multifield Problems
C Partitioned Methods for Multifield Problems Joachim Rang, 6.7.2016 6.7.2016 Joachim Rang Partitioned Methods for Multifield Problems Seite 1 C One-dimensional piston problem fixed wall Fluid flexible
More informationMartingale optimal transport with Monge s cost function
Martingale optimal transport with Monge s cost function Martin Klimmek, joint work with David Hobson (Warwick) klimmek@maths.ox.ac.uk Crete July, 2013 ca. 1780 c(x, y) = x y No splitting, transport along
More informationChapter 2. The continuous equations
Chapter. The continuous equations Fig. 1.: Schematic of a forecast with slowly varying weather-related variations and superimposed high frequency Lamb waves. Note that even though the forecast of the slow
More information2.5 Shallow water equations, quasigeostrophic filtering, and filtering of inertia-gravity waves
Chapter. The continuous equations φ=gh Φ=gH φ s =gh s Fig..5: Schematic of the shallow water model, a hydrostatic, incompressible fluid with a rigid bottom h s (x,y), a free surface h(x,y,t), and horizontal
More informationRelaxation methods and finite element schemes for the equations of visco-elastodynamics. Chiara Simeoni
Relaxation methods and finite element schemes for the equations of visco-elastodynamics Chiara Simeoni Department of Information Engineering, Computer Science and Mathematics University of L Aquila (Italy)
More informationExistence and uniqueness results for the MFG system
Existence and uniqueness results for the MFG system P. Cardaliaguet Paris-Dauphine Based on Lasry-Lions (2006) and Lions lectures at Collège de France IMA Special Short Course Mean Field Games", November
More informationPropagation d interfaces avec termes non locaux
Propagation d interfaces avec termes non locaux P. Cardaliaguet Univ. Brest Janvier 2008 Joint works with G. Barles (Tours), O. Alvarez (Rouen), O. Ley (Tours), R. Monneau (CERMICS), A. Monteillet (Brest).
More informationPairwise Comparison Dynamics for Games with Continuous Strategy Space
Pairwise Comparison Dynamics for Games with Continuous Strategy Space Man-Wah Cheung https://sites.google.com/site/jennymwcheung University of Wisconsin Madison Department of Economics Nov 5, 2013 Evolutionary
More informationGreedy Control. Enrique Zuazua 1 2
Greedy Control Enrique Zuazua 1 2 DeustoTech - Bilbao, Basque Country, Spain Universidad Autónoma de Madrid, Spain Visiting Fellow of LJLL-UPMC, Paris enrique.zuazua@deusto.es http://enzuazua.net X ENAMA,
More informationOverview of the Numerics of the ECMWF. Atmospheric Forecast Model
Overview of the Numerics of the Atmospheric Forecast Model M. Hortal Seminar 6 Sept 2004 Slide 1 Characteristics of the model Hydrostatic shallow-atmosphere approimation Pressure-based hybrid vertical
More informationThe reconstruction problem for the Euler-Poisson system in cosmology
The reconstruction problem for the Euler-Poisson system in cosmology Grégoire LOEPER February, 008 arxiv:math/0306430v [math.ap] 7 Apr 005 Abstract The motion of a continuum of matter subject to gravitational
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 informationA regularised particle method for linear and nonlinear diffusion
1/25 A regularised particle method for linear and nonlinear diffusion Francesco Patacchini Department of Mathematical Sciences, Carnegie Mellon University Joint work with J. A. Carrillo (Imperial College
More informationComputing High Frequency Waves By the Level Set Method
Computing High Frequency Waves By the Level Set Method Hailiang Liu Department of Mathematics Iowa State University Collaborators: Li-Tien Cheng (UCSD), Stanley Osher (UCLA) Shi Jin (UW-Madison), Richard
More informationUnderstanding inertial instability on the f-plane with complete Coriolis force
Understanding inertial instability on the f-plane with complete Coriolis force Abstract Vladimir Zeitlin Laboratory of Dynamical Meteorology, University P. and M. Curie and Ecole Normale Supérieure, Paris,
More informationColumnar Clouds and Internal Waves
Columnar Clouds and Internal Waves Daniel Ruprecht 1, Andrew Majda 2, Rupert Klein 1 1 Mathematik & Informatik, Freie Universität Berlin 2 Courant Institute, NYU Tropical Meteorology Workshop, Banff April
More informationSeismic imaging and optimal transport
Seismic imaging and optimal transport Bjorn Engquist In collaboration with Brittany Froese, Sergey Fomel and Yunan Yang Brenier60, Calculus of Variations and Optimal Transportation, Paris, January 10-13,
More informationLow Froude Number Limit of the Rotating Shallow Water and Euler Equations
Low Froude Number Limit of the Rotating Shallow Water and Euler Equations Kung-Chien Wu Department of Pure Mathematics and Mathematical Statistics University of Cambridge, Wilberforce Road Cambridge, CB3
More informationNon-Differentiable Embedding of Lagrangian structures
Non-Differentiable Embedding of Lagrangian structures Isabelle Greff Joint work with J. Cresson Université de Pau et des Pays de l Adour CNAM, Paris, April, 22nd 2010 Position of the problem 1. Example
More informationOn the regularity of solutions of optimal transportation problems
On the regularity of solutions of optimal transportation problems Grégoire Loeper April 25, 2008 Abstract We give a necessary and sufficient condition on the cost function so that the map solution of Monge
More informationApplications of the compensated compactness method on hyperbolic conservation systems
Applications of the compensated compactness method on hyperbolic conservation systems Yunguang Lu Department of Mathematics National University of Colombia e-mail:ylu@unal.edu.co ALAMMI 2009 In this talk,
More informationThe incompressible Navier-Stokes equations in vacuum
The incompressible, Université Paris-Est Créteil Piotr Bogus law Mucha, Warsaw University Journées Jeunes EDPistes 218, Institut Elie Cartan, Université de Lorraine March 23th, 218 Incompressible 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 informationOn some singular limits for an atmosphere flow
On some singular limits for an atmosphere flow Donatella Donatelli Dipartimento di Ingegneria e Scienze dell Informazione e Matematica Università degli Studi dell Aquila 67100 L Aquila, Italy donatella.donatelli@univaq.it
More informationSpaces with Ricci curvature bounded from below
Spaces with Ricci curvature bounded from below Nicola Gigli February 23, 2015 Topics 1) On the definition of spaces with Ricci curvature bounded from below 2) Analytic properties of RCD(K, N) spaces 3)
More informationOn Fluid-Particle Interaction
Complex Fluids On Fluid-Particle Interaction Women in Applied Mathematics University of Crete, May 2-5, 2011 Konstantina Trivisa Complex Fluids 1 Model 1. On the Doi Model: Rod-like Molecules Colaborator:
More informationP(E t, Ω)dt, (2) 4t has an advantage with respect. to the compactly supported mollifiers, i.e., the function W (t)f satisfies a semigroup law:
Introduction Functions of bounded variation, usually denoted by BV, have had and have an important role in several problems of calculus of variations. The main features that make BV functions suitable
More informationLECTURE 3: DISCRETE GRADIENT FLOWS
LECTURE 3: DISCRETE GRADIENT FLOWS Department of Mathematics and Institute for Physical Science and Technology University of Maryland, USA Tutorial: Numerical Methods for FBPs Free Boundary Problems and
More informationHyperbolic Systems of Conservation Laws
Hyperbolic Systems of Conservation Laws III - Uniqueness and continuous dependence and viscous approximations Alberto Bressan Mathematics Department, Penn State University http://www.math.psu.edu/bressan/
More informationOptimal Transport: A Crash Course
Optimal Transport: A Crash Course Soheil Kolouri and Gustavo K. Rohde HRL Laboratories, University of Virginia Introduction What is Optimal Transport? The optimal transport problem seeks the most efficient
More informationAPPENDIX B. The primitive equations
APPENDIX B The primitive equations The physical and mathematical basis of all methods of dynamical atmospheric prediction lies in the principles of conservation of momentum, mass, and energy. Applied to
More informationSAMPLE CHAPTERS UNESCO EOLSS WAVES IN THE OCEANS. Wolfgang Fennel Institut für Ostseeforschung Warnemünde (IOW) an der Universität Rostock,Germany
WAVES IN THE OCEANS Wolfgang Fennel Institut für Ostseeforschung Warnemünde (IOW) an der Universität Rostock,Germany Keywords: Wind waves, dispersion, internal waves, inertial oscillations, inertial waves,
More informationLecture 1: Introduction to Linear and Non-Linear Waves
Lecture 1: Introduction to Linear and Non-Linear Waves Lecturer: Harvey Segur. Write-up: Michael Bates June 15, 2009 1 Introduction to Water Waves 1.1 Motivation and Basic Properties There are many types
More informationApproximation of fluid-structure interaction problems with Lagrange multiplier
Approximation of fluid-structure interaction problems with Lagrange multiplier Daniele Boffi Dipartimento di Matematica F. Casorati, Università di Pavia http://www-dimat.unipv.it/boffi May 30, 2016 Outline
More informationConvex Optimization. Ofer Meshi. Lecture 6: Lower Bounds Constrained Optimization
Convex Optimization Ofer Meshi Lecture 6: Lower Bounds Constrained Optimization Lower Bounds Some upper bounds: #iter μ 2 M #iter 2 M #iter L L μ 2 Oracle/ops GD κ log 1/ε M x # ε L # x # L # ε # με f
More informationOn Pressure Stabilization Method and Projection Method for Unsteady Navier-Stokes Equations 1
On Pressure Stabilization Method and Projection Method for Unsteady Navier-Stokes Equations 1 Jie Shen Department of Mathematics, Penn State University University Park, PA 1682 Abstract. We present some
More informationCVaR and Examples of Deviation Risk Measures
CVaR and Examples of Deviation Risk Measures Jakub Černý Department of Probability and Mathematical Statistics Stochastic Modelling in Economics and Finance November 10, 2014 1 / 25 Contents CVaR - Dual
More informationPartial Differential Equations
Part II Partial Differential Equations Year 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2015 Paper 4, Section II 29E Partial Differential Equations 72 (a) Show that the Cauchy problem for u(x,
More informationOn Asymptotic Variational Wave Equations
On Asymptotic Variational Wave Equations Alberto Bressan 1, Ping Zhang 2, and Yuxi Zheng 1 1 Department of Mathematics, Penn State University, PA 1682. E-mail: bressan@math.psu.edu; yzheng@math.psu.edu
More informationOn a Suitable Weak Solution of the Navier Stokes Equation with the Generalized Impermeability Boundary Conditions
Proceedings of the 3rd IASME/WSEAS Int. Conf. on FLUID DYNAMICS & AERODYNAMICS, Corfu, Greece, August -, 5 pp36-41 On a Suitable Weak Solution of the Navier Stokes Equation with the Generalized Impermeability
More informationModel equations for planetary and synoptic scale atmospheric motions associated with different background stratification
Model equations for planetary and synoptic scale atmospheric motions associated with different background stratification Stamen Dolaptchiev & Rupert Klein Potsdam Institute for Climate Impact Research
More informationBounded uniformly continuous functions
Bounded uniformly continuous functions Objectives. To study the basic properties of the C -algebra of the bounded uniformly continuous functions on some metric space. Requirements. Basic concepts of analysis:
More informationTD 1: Hilbert Spaces and Applications
Université Paris-Dauphine Functional Analysis and PDEs Master MMD-MA 2017/2018 Generalities TD 1: Hilbert Spaces and Applications Exercise 1 (Generalized Parallelogram law). Let (H,, ) be a Hilbert space.
More informationBALANCED FLOW: EXAMPLES (PHH lecture 3) Potential Vorticity in the real atmosphere. Potential temperature θ. Rossby Ertel potential vorticity
BALANCED FLOW: EXAMPLES (PHH lecture 3) Potential Vorticity in the real atmosphere Need to introduce a new measure of the buoyancy Potential temperature θ In a compressible fluid, the relevant measure
More informationPartial Differential Equations, 2nd Edition, L.C.Evans The Calculus of Variations
Partial Differential Equations, 2nd Edition, L.C.Evans Chapter 8 The Calculus of Variations Yung-Hsiang Huang 2018.03.25 Notation: denotes a bounded smooth, open subset of R n. All given functions are
More information