Stable shock formation near plane-symmetry for nonlinear wave equations
|
|
- Vernon Garrett
- 5 years ago
- Views:
Transcription
1 Stable shock formation near plane-symmetry for nonlinear wave equations Willie WY Wong Michigan State University University of Minnesota 9 March
2 Based on: arxiv: , joint with Gustav Holzegel, Jonathan Luk, and Jared Speck arxiv: (to appear in JHDE), joint with Gustav Holzegel, Sergiu Klainerman, and Jared Speck 2
3 The set-up Cylindrical domain: R R T 3 (t,y, )=(x 0,x 1,x 2 ). Study behavior 2 t 2 y 2 = h(@ 2 (Der-QNLW) 2X i,j=0 1 p i p g (g 1 ) j = 0 (Geom-QNLW) 3
4 Derivative 2 t 2 y 2 = h(@ 2 Arises from Lagrangian field theories with Lagrangian density L(@ ) dvol Example: Irrotational Euler, with being the fluid potential. 4
5 Geometric QNLW where g( ) = 2X i,j=0 g = g( )= 1 p i p g (g 1 ) j =0 0 B@ O( ) CA 1 is a Lorentzian metric and g is the geometric wave operator for the metric g. 5
6 Conversion... In (Der-QNLW), of the equation we get that ~ solves g( ~) ~ = Q(@ ~,@~) where: g( ~) = diag( 1,1,1) + h( ~) ~ is treated as a scalar taking values in R 3 Q is quadratic form ~, with coe cients depending on ~ Key: Q is a strong null form 6
7 ... additional equations Our results apply to any equation that can be written in the form where Q is a strong null form. g( ) = Q(@,@ ) Strong null form: for every, the form Q is null relative to g( ). (Needs to hold even for cubic and higher terms!) Hence focus on (Geom-QNLW). 7
8 Di erent from Weak Null case Weak null 2 t 2 y 2 = h( 2. (Geom-QNLW) expands 2 t 2 y 2 = h( 2 + Q(@,@ )+h 0 The last term makes all the di erence when g( ). ( )g(0). Small data regime: define G LL = h 0 (0) (@ t y ) (@ t y ) measure failure of quadratic null condition. 8
9 Main results (abbreviated version) Let g be such that G LL, 0. Suppose solves (Geom-QNLW) with compactly supported smooth initial data ( 0 = t=0, 1 t t=0 ) satisfying the conditions 1. i = Small amplitude: 0 <. 3. Forward y 0 1 apple 0 and (g 1 ( )) (g 1 ( )) 01 1@ y 0 +(g 1 ( )) 11 (@ y 0 ) 2 =0. Then terminates in time (1 + O( )) 1 where := 1 2 supg LL@ y 0. in a shock singularity, Furthermore, the singularity formation is stable under su small, arbitrary perturbations. ciently 9
10 Remarks The first part of the theorem (singularity formation for -independent data) is essentially contained in the work of F. John from the 70s, which built on works of Lax and Glimm earlier. We will start with a quick review of this theory as applied to our simple setting. The second part of the theorem (stability) is new. The bulk of today s talk concerns this part. The conditions on small amplitude and forward propagating are not critical: they can be significantly relaxed and we assume them here for simplicity. 10
11 Part I: 1+1dimensional problem and importance of geometry 11
12 Reduction to 1+1dimensions Under the -independence assumption, (Geom-QNLW) reduces to the equation ḡ = 0 on R 2, where ḡ is the two-dimensional Lorentzian metric obtained by restricting to the t,y components of g dimensional geometric wave equations are conformally invariant. 12
13 Double null formulation ḡ( ) =0 v =0 where u,v are null/characteristic coordinates of R 2 relative to ḡ. u t y v t + y solve ḡ 1 (du,du)=ḡ 1 (dv,dv)=0. (A priori they depend non-locally on and hides the nonlinearity.) 13
14 Regular propagation Forward-propagating assumption =) = (u) is independent of v. Level sets of u are straight lines, along which is constant. Slope of these level sets determined at t = 0 by the initial data! 14
15 Shock formation Shock forms when two characteristics (level sets of u) intersect! This happens when wave speed decreases as y increases. Rate of change of wave speed governed by G y 0. 15
16 Time of existence When exactly does shock form? Let µ := (ḡ 1 (dt,du)) 1, the inverse foliation density of leaves of u. Shock forms when µ & 0. (We can re-parametrize u so that when t = 0, µ 1. ) 16
17 Time of existence The lines of constant u are geodesics of ḡ. Geodesic equation =) along lines of constant u, parametrized by t, we t µ = Ḡvv( )@ u. Here Ḡvv is the geometric version of G LL, and is equal to (1 + O( ))G LL. By propagation equation the RHS is constant along the level sets of u. Hence µ is linear in time. 17
18 Key structural conclusions u is the forward-going null coordinate. u are constant along level sets of u. µ, the inverse foliation density of u, evolves linearly in time t along level sets of u; its rate is governed by the (initial value) quantity G y 0. 18
19 Part II: structure of proof, heuristics, and obstructions 19
20 Structure of the 1+1proof In 3 steps 1. Rewrite equations in characteristic coordinates. (Here v u =0, v µ =0.) 2. Prove virtual global well-posedness in characteristic coordinates. (Trivial in ) 3. Show that initial data implies characteristic coordinates cannot be globally regular (µ! 0 in finite time) and hence solution exhibits shock. 20
21 What to do in higher dimensions? 1. Define a characteristic coordinate system so that the equations of motion appear to be perturbations of the case. 2. Prove, virtually globally, that smallness of the perturbation propagates under the equation. (Thereby proving virtual GWP.) (This is the hard part!) 3. Profit! v µ =small requires no modification.) 21
22 Coordinate choice Coordinate R R T by functions (t,u,!) 2 R R T. t is the same as the background rectilinear time.! is the angular variable. u is a null coordinate solving u t + y and g 1 (du,du) = 0. 22
23 Necessity of true characteristics One is tempted to simplify the construction of the characteristic coordinate function u by trying to replace it with either Projected null coordinates (that du restricted to constant surfaces are null in the induced geometry); The null coordinates for g(0) instead of g( ). Problem: In generic blows up at the shock point. When u is not a true characteristic, t to also blow-up, which would mean we become far from the perturbative regime that we want to be in! 23
24 Aside: why is one tempted? Solving the eikonal equation g 1 (du,du) = 0 loses derivatives... except in the case. Roughly speaking the eikonal equations can be re-written t where ru is R 3 -valued function representing rectangular derivatives of the coordinate function u. If we solve it as a transport equation this gives us regularity This is one derivative lower than g 1 ( )(du,du) = 0 may suggest. 24
25 Historical remark Importance of true characteristic geometry in shock problems first realized by S. Alinhac ca D. Christodoulou ca Both managed to deal with the derivative loss problem through essentially the same trick. (More on this later.) 25
26 Propagating smallness In use transport equation formulation. In higher dimensions, transport formulation (L 1 -L 1 estimates) loses derivative. Use energy estimate + Sobolev embedding. 26
27 Energy and anisotropy Problem: We are looking at shock formation! The solution is expected to be small. So g( ) expected to be close to Minkowskian. So volume forms are expected to be close to rectilinear. In particular: natural energy quantities will be similar to those of the linear wave equation, and is hence isotropic. However, expect solution to be anisotropic: derivatives tangent to u-level-sets are small, while transversal derivatives blow-up! 27
28 Solutions 1. To extract smallness: take derivatives of the equation in directions tangential to the u foliation. In the model case these derivatives are all vanishing. Hope to prove that in the perturbed case all these derivatives have very small energy. 2. To deal with the blow-up: instead of studying the natural energy (which blows up at shock formation), study the renormalized energy (µ times the natural energy). This quantity should remain bounded. 28
29 New problems 1. Commuting the equation with a derivative will generate error terms. Need to make sure the error terms don t themselves blow-up! To solve this require exposing the inherent null structures of the equations. (Here the strong null form assumption on the RHS plays a role.) 2. The the anisotropy of the expected evolution means that renormalized energy estimates loses coercivity near the blow-up time. To solve this use space-time integrated energy control that arises from the fact that the inverse foliation density is expected to be approximately monotonic along constant u slices. 29
30 Part III: aspects of the proof of higher dimensional stability theorem 30
31 Virtual Global Existence Prolonged time of existence in the wave zone compared to direct local existence argument. Standard local existence by energy method: solution exists at least up to time Expect that shock formation within time (1 + O( )) 1, where = 1 2 supg LL@ y 0. Will prove that as long as µ doesn t tend to zero, well-posedness holds up to time
32 Coordinates Let u be a solution to the eikonal equation g 1 (du,du) = 0 such that at t = 0 we u = 0 and µ = 1. Fix t to be the rectilinear time. Fix! so that at t = 0,! =. And furthermore that the lines {! =! 0,u = u 0 } are null geodesics. 32
33 Transformation map Let x µ, µ 2{0,1,2} be the rectilinear coordinates. Define the Jacobian components X µ t t (x µ ), X µ! (x µ ). (Don t need the u version: that can be solved from the metric and X µ t.) Recall µ = g 1 (dt,du) 1. Note t X µ! X µ t. Let g u! = g(@ u,@! ) and g!! = g(@!,@!). 33
34 Decomposition of metric In the coordinate system, we have! 2 g = 2µ du dt + µ 2 du 2 g + g u!!! du +d!. g!! And where (g 1 t t R + R = 1 u! g g!!! µ u. Let R = u. 34
35 The system: the main unknowns The function. The Jacobian components X t = µg 1 (du,dx ) and X! The inverse foliation density µ 35
36 The system: the main evolution equations Geodesic equation (transport): (G is Wave t µ = 1 2 G tt R( 1 ) 2 µ(g tt + G Rt )@ t t Xt = 1 1 X 2 g!g + O(1)@ t (Geod2)!! 0=µ g t [µ@ t +2 R( )] + µ 2!! 1 t g!! g!! R( ) + terms involving t (Wave) 36
37 The system: the main evolution equations In the -independent case, (Geod1) is unchanged (except that the RHS is constant in t). (Geod2) reduces t Xt = 0. (And Xt 0 = X2 t u level sets are flat.) = 0.) (This means that (Wave) becomes t pg!! [@ t +2 R( )]. 37
38 Commutation structure Observe that ]=0, t, R] [@!, R] This means that if we commute (Geod1), (Geod2), and (Wave) with u-tangential t,@!, we will not pick up extra R derivative terms. 38
39 Commutation structure This implies we can close a bootstrap argument using L 1 estimates for derivatives of R. L 2 estimates for derivatives of involving no more than three involving no more than one R. They are proved by Commuting (Geod1), (Geod2) t,@!, and up to two R and estimating as transport equation. Commuting (Wave) with t,@!, and doing energy estimates. 39
40 Energy estimates Let P denote a string t derivatives. (Wave) gives t [µ@ t P +2 R(P )] + µ 2!!P 1 t P g!! g!! R( ) + other error terms We focus on the one error term (in red), which can potentially lose derivatives, to illustrate the strategy. Some other error terms are similarly bad, but we ignore them for this presentation. 40
41 Energy estimates Multiply (Wave) t P + 2[µ@ t P + R(P )] and integrate by parts in a region {(t,u,!):t 2 [0,T),T <2 1,u 2 [0,U],U apple u } (By assumption the initial data has compact support, make the support contained in u 2 [0,u ].) u = 0 boundary gives 0 contribution by finite speed of propagation t = 0 boundary is initial data 41
42 Energy estimates t = T energy: E T,U (P )= Z U Z apple µ 2 (1 + 2µ)(@ tp ) 2 +2µ@ t P RP 0! + 2( RP ) 2 + µ 2 (1 + 2µ)(@!P ) 2 g!! p g!! d! du u = U flux: F T,U (P )= Z T 0 Z! " (1 + µ)(@ t P ) 2 + µ (@!P ) 2 # pg!! d! dt g!! 42
43 Energy estimates E T,U + F T,U = initial data + Z T Z U Z 0 0! Where Bulk includes, for example, Bulk p g!! d! du dt. From µ 2!!P, the t µ g!! (@! P ) 2. From 1 t P g!! g!! R, the term 1 t P g!! g!! ( R )( RP ). 43
44 Energy degeneracy When µ 1, E only controls RP, while F only t P. What P? Use the bulk term! 44
45 Point of no return Lemma µ< 1 4 tµ apple 1 4. So the t µ g!! (@! P ) 2 gives control of the time-integrated angular derivatives, which is su cient to close energy estimates. 45
46 Heuristic justification of lemma Recall that µ(t = 0) = 1. And we study the interval t 2 [0,2 1 ]. Furthermore we expect µ to be almost linear. So we expect that for µ to decay by 3/4 within time at most 2 1 means that the minimum slope t µ = 3 8 > 1 4. We leave some room for error terms. 46
47 Derivative / µ loss As mentioned before, the coe cients of (Wave) depend on u in a way that could potentially lead to derivative loss. We can see it t P g!!. g!! = g( ) X!X! So the term with the worst regularity t P g!! is To t X! we 2 t X t = 1 2 g X!@ t P X!. 1 g!! X!G 2!!
48 Derivatve / µ loss So the worst term t P g!! looks like (after some computations) Z t G tt@ 2!!P. This requires one higher degree of regularity than that appears in the energy (which is of ). 48
49 Derivative / µ dichotomy Solution: At non-top-level: control by Gronwall into higher energy. At top-level: use the equation! Replace 1 2!! = 1 t[µ@ t +2 R( )] +... So the problematic term becomes t P g!! ( R )( RP ) 2 g!! = 1 Gtt R (µ@ 4µ t P +2 RP )( RP )+l.o.t.(w/ t-integral) {z t µ 49
50 Top level End result: top-order energy estimate of the form E T,U (P ) E 0,U (P ) apple C Z T 0 sup t µ µ E s,u (P )ds +... Important: The number C is a universal structure constant, independent of the precise form of g. (In fact it is apple 12.) Assuming µ only depends on t and forgetting other error terms then we get E T,U (P ) apple µ C E 0,U (P ). Top level energy can blow up! 50
51 Below top level Suppose we know that E T,U (@! P the term that loses derivatives, ) apple µ C Data. We observe that Z t G tt@ 2!!P, can be controlled, after a L 2 spatial integration, by Z t µ C/2 1 2 Data. 0 51
52 Below top level Since we expect µ to be approximately linear, we in fact have Z t µ C/2 1/2 µ 1/2 C/2. 0 Noting that we have another time integration ( R t 0 G tt@ t P g!! ), we conclude finally that E T,U (@! P ) apple µ C =) E T,U (P ) apple µ C+2. 52
53 Descent scheme Sacrifice the top C/2 derivatives: their corresponding energies are allowed to blow up. Below that in the medium range we prove only energy estimates with no µ-related blow-up. Below that in the low range we gain (via Sobolev) pointwise estimates. 53
54 Thank you! 54
Shock Formation in Small-Data Solutions to 3D Quasilinear Wave Equations
arxiv:407.6320v [math.ap] 23 Jul 204 Shock Formation in Small-Data Solutions to 3D Quasilinear Wave Equations JARED SPECK July 24, 204 Massachusetts Institute of Technology, Department of Mathematics.
More informationFINITE-TIME DEGENERATION OF HYPERBOLICITY WITHOUT BLOWUP FOR QUASILINEAR WAVE EQUATIONS
FINITE-TIME DEGENERATION OF HYPERBOLICITY WITHOUT BLOWUP FOR QUASILINEAR WAVE EQUATIONS JARED SPECK Abstract. In 3D, we study the Cauchy problem for the wave equation 2 t Ψ + (1 + Ψ) P Ψ = 0 for P {1,
More informationEnergy method for wave equations
Energy method for wave equations Willie Wong Based on commit 5dfb7e5 of 2017-11-06 13:29 Abstract We give an elementary discussion of the energy method (and particularly the vector field method) in the
More informationFORMATION OF TRAPPED SURFACES II. 1. Introduction
FORMATION OF TRAPPED SURFACES II SERGIU KLAINERMAN, JONATHAN LUK, AND IGOR RODNIANSKI 1. Introduction. Geometry of a null hypersurface As in [?] we consider a region D = D(u, u ) of a vacuum spacetime
More informationCausality, hyperbolicity, and shock formation in Lovelock theories
Causality, hyperbolicity, and shock formation in Lovelock theories Harvey Reall DAMTP, Cambridge University HSR, N. Tanahashi and B. Way, arxiv:1406.3379, 1409.3874 G. Papallo, HSR arxiv:1508.05303 Lovelock
More informationat time t, in dimension d. The index i varies in a countable set I. We call configuration the family, denoted generically by Φ: U (x i (t) x j (t))
Notations In this chapter we investigate infinite systems of interacting particles subject to Newtonian dynamics Each particle is characterized by its position an velocity x i t, v i t R d R d at time
More informationTHE BONDI-SACHS FORMALISM JEFF WINICOUR UNIVERSITY OF PITTSBURGH. Scholarpedia 11(12):33528 (2016) with Thomas Mädler
THE BONDI-SACHS FORMALISM JEFF WINICOUR UNIVERSITY OF PITTSBURGH Scholarpedia 11(12):33528 (2016) with Thomas Mädler NULL HYPERSURFACES u = const Normal co-vector @ u is null g @ u @ u =0 Normal vector
More informationSHOCK BOUNDARIES IN 2D QUASILINEAR WAVE EQUATIONS. 1. Introduction
SHOCK BOUNDARIES IN 2D QUASILINEAR WAVE EQUATIONS JULIAN CHAIDEZ In this paper we study shock formation in 2+1 quasilinear waves solving g µ (@ )@ µ @ =, under the assumption of radial symmetry and small
More informationIntroduction to Algebraic and Geometric Topology Week 14
Introduction to Algebraic and Geometric Topology Week 14 Domingo Toledo University of Utah Fall 2016 Computations in coordinates I Recall smooth surface S = {f (x, y, z) =0} R 3, I rf 6= 0 on S, I Chart
More informationz x = f x (x, y, a, b), z y = f y (x, y, a, b). F(x, y, z, z x, z y ) = 0. This is a PDE for the unknown function of two independent variables.
Chapter 2 First order PDE 2.1 How and Why First order PDE appear? 2.1.1 Physical origins Conservation laws form one of the two fundamental parts of any mathematical model of Continuum Mechanics. These
More informationRegularization by noise in infinite dimensions
Regularization by noise in infinite dimensions Franco Flandoli, University of Pisa King s College 2017 Franco Flandoli, University of Pisa () Regularization by noise King s College 2017 1 / 33 Plan of
More informationExact Solutions of the Einstein Equations
Notes from phz 6607, Special and General Relativity University of Florida, Fall 2004, Detweiler Exact Solutions of the Einstein Equations These notes are not a substitute in any manner for class lectures.
More informationCurved spacetime tells matter how to move
Curved spacetime tells matter how to move Continuous matter, stress energy tensor Perfect fluid: T 1st law of Thermodynamics Relativistic Euler equation Compare with Newton =( c 2 + + p)u u /c 2 + pg j
More informationGeometric Methods in Hyperbolic PDEs
Geometric Methods in Hyperbolic PDEs Jared Speck jspeck@math.princeton.edu Department of Mathematics Princeton University January 24, 2011 Unifying mathematical themes Many physical phenomena are modeled
More informationTHE WAVE EQUATION. d = 1: D Alembert s formula We begin with the initial value problem in 1 space dimension { u = utt u xx = 0, in (0, ) R, (2)
THE WAVE EQUATION () The free wave equation takes the form u := ( t x )u = 0, u : R t R d x R In the literature, the operator := t x is called the D Alembertian on R +d. Later we shall also consider the
More informationLocality of Energy Transfer
(E) Locality of Energy Transfer See T & L, Section 8.2; U. Frisch, Section 7.3 The Essence of the Matter We have seen that energy is transferred from scales >`to scales
More informationCalculus: Several Variables Lecture 27
alculus: Several Variables Lecture 27 Instructor: Maksim Maydanskiy Lecture 27 Plan 1. Work integrals over a curve continued. (15.4) Work integral and circulation. Example by inspection. omputation via
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 informationWhat happens at the horizon of an extreme black hole?
What happens at the horizon of an extreme black hole? Harvey Reall DAMTP, Cambridge University Lucietti and HSR arxiv:1208.1437 Lucietti, Murata, HSR and Tanahashi arxiv:1212.2557 Murata, HSR and Tanahashi,
More informationGlobal classical solutions to the spherically symmetric Nordström-Vlasov system
Global classical solutions to the spherically symmetric Nordström-Vlasov system Håkan Andréasson, Simone Calogero Department of Mathematics, Chalmers University of Technology, S-4196 Göteborg, Sweden Gerhard
More informationRIGIDITY OF STATIONARY BLACK HOLES WITH SMALL ANGULAR MOMENTUM ON THE HORIZON
RIGIDITY OF STATIONARY BLACK HOLES WITH SMALL ANGULAR MOMENTUM ON THE HORIZON S. ALEXAKIS, A. D. IONESCU, AND S. KLAINERMAN Abstract. We prove a black hole rigidity result for slowly rotating stationary
More informationThe Ricci Flow Approach to 3-Manifold Topology. John Lott
The Ricci Flow Approach to 3-Manifold Topology John Lott Two-dimensional topology All of the compact surfaces that anyone has ever seen : These are all of the compact connected oriented surfaces without
More informationOrbital Motion in Schwarzschild Geometry
Physics 4 Lecture 29 Orbital Motion in Schwarzschild Geometry Lecture 29 Physics 4 Classical Mechanics II November 9th, 2007 We have seen, through the study of the weak field solutions of Einstein s equation
More informationStability and Instability of Black Holes
Stability and Instability of Black Holes Stefanos Aretakis September 24, 2013 General relativity is a successful theory of gravitation. Objects of study: (4-dimensional) Lorentzian manifolds (M, g) which
More informationShock Waves in Plane Symmetric Spacetimes
Communications in Partial Differential Equations, 33: 2020 2039, 2008 Copyright Taylor & Francis Group, LLC ISSN 0360-5302 print/1532-4133 online DOI: 10.1080/03605300802421948 Shock Waves in Plane Symmetric
More information3 = arccos. A a and b are parallel, B a and b are perpendicular, C a and b are normalized, or D this is always true.
Math 210-101 Test #1 Sept. 16 th, 2016 Name: Answer Key Be sure to show your work! 1. (20 points) Vector Basics: Let v = 1, 2,, w = 1, 2, 2, and u = 2, 1, 1. (a) Find the area of a parallelogram spanned
More information8.821 String Theory Fall 2008
MIT OpenCourseWare http://ocw.mit.edu 8.8 String Theory Fall 008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 8.8 F008 Lecture 0: CFTs in D > Lecturer:
More informationORDINARY DIFFERENTIAL EQUATIONS
ORDINARY DIFFERENTIAL EQUATIONS GABRIEL NAGY Mathematics Department, Michigan State University, East Lansing, MI, 48824. JANUARY 3, 25 Summary. This is an introduction to ordinary differential equations.
More informationGlobal stability problems in General Relativity
Global stability problems in General Relativity Peter Hintz with András Vasy Murramarang March 21, 2018 Einstein vacuum equations Ric(g) + Λg = 0. g: Lorentzian metric (+ ) on 4-manifold M Λ R: cosmological
More informationNULL CONDITION FOR SEMILINEAR WAVE EQUATION WITH VARIABLE COEFFICIENTS. Fabio Catalano
Serdica Math J 25 (999), 32-34 NULL CONDITION FOR SEMILINEAR WAVE EQUATION WITH VARIABLE COEFFICIENTS Fabio Catalano Communicated by V Petkov Abstract In this work we analyse the nonlinear Cauchy problem
More informationThe stability of Kerr-de Sitter black holes
The stability of Kerr-de Sitter black holes András Vasy (joint work with Peter Hintz) July 2018, Montréal This talk is about the stability of Kerr-de Sitter (KdS) black holes, which are certain Lorentzian
More informationThe harmonic map flow
Chapter 2 The harmonic map flow 2.1 Definition of the flow The harmonic map flow was introduced by Eells-Sampson in 1964; their work could be considered the start of the field of geometric flows. The flow
More informationThe linear stability of the Schwarzschild solution to gravitational perturbations in the generalised wave gauge
The linear stability of the Schwarzschild solution to gravitational perturbations in the generalised wave gauge Imperial College London Mathematical Relativity Seminar, Université Pierre et Marie Curie,
More informationSeveral forms of the equations of motion
Chapter 6 Several forms of the equations of motion 6.1 The Navier-Stokes equations Under the assumption of a Newtonian stress-rate-of-strain constitutive equation and a linear, thermally conductive medium,
More informationDYNAMICAL FORMATION OF BLACK HOLES DUE TO THE CONDENSATION OF MATTER FIELD
DYNAMICAL FORMATION OF BLACK HOLES DUE TO THE CONDENSATION OF MATTER FIELD PIN YU Abstract. The purpose of the paper is to understand a mechanism of evolutionary formation of trapped surfaces when there
More informationTHEODORE VORONOV DIFFERENTIAL GEOMETRY. Spring 2009
[under construction] 8 Parallel transport 8.1 Equation of parallel transport Consider a vector bundle E B. We would like to compare vectors belonging to fibers over different points. Recall that this was
More informationSingularity formation in black hole interiors
Singularity formation in black hole interiors Grigorios Fournodavlos DPMMS, University of Cambridge Heraklion, Crete, 16 May 2018 Outline The Einstein equations Examples Initial value problem Large time
More informationMyths, Facts and Dreams in General Relativity
Princeton university November, 2010 MYTHS (Common Misconceptions) MYTHS (Common Misconceptions) 1 Analysts prove superfluous existence results. MYTHS (Common Misconceptions) 1 Analysts prove superfluous
More information8 Symmetries and the Hamiltonian
8 Symmetries and the Hamiltonian Throughout the discussion of black hole thermodynamics, we have always assumed energy = M. Now we will introduce the Hamiltonian formulation of GR and show how to define
More informationEuclidean Spaces. Euclidean Spaces. Chapter 10 -S&B
Chapter 10 -S&B The Real Line: every real number is represented by exactly one point on the line. The plane (i.e., consumption bundles): Pairs of numbers have a geometric representation Cartesian plane
More informationFefferman Graham Expansions for Asymptotically Schroedinger Space-Times
Copenhagen, February 20, 2012 p. 1/16 Fefferman Graham Expansions for Asymptotically Schroedinger Space-Times Jelle Hartong Niels Bohr Institute Nordic String Theory Meeting Copenhagen, February 20, 2012
More informationPeriodic Sinks and Observable Chaos
Periodic Sinks and Observable Chaos Systems of Study: Let M = S 1 R. T a,b,l : M M is a three-parameter family of maps defined by where θ S 1, r R. θ 1 = a+θ +Lsin2πθ +r r 1 = br +blsin2πθ Outline of Contents:
More informationBondi-Sachs Formulation of General Relativity (GR) and the Vertices of the Null Cones
Bondi-Sachs Formulation of General Relativity (GR) and the Vertices of the Null Cones Thomas Mädler Observatoire de Paris/Meudon, Max Planck Institut for Astrophysics Sept 10, 2012 - IAP seminar Astrophysical
More informationIDEAL CLASSES AND RELATIVE INTEGERS
IDEAL CLASSES AND RELATIVE INTEGERS KEITH CONRAD The ring of integers of a number field is free as a Z-module. It is a module not just over Z, but also over any intermediate ring of integers. That is,
More informationConstrained Optimization
1 / 22 Constrained Optimization ME598/494 Lecture Max Yi Ren Department of Mechanical Engineering, Arizona State University March 30, 2015 2 / 22 1. Equality constraints only 1.1 Reduced gradient 1.2 Lagrange
More informationAerothermodynamics of high speed flows
Aerothermodynamics of high speed flows AERO 0033 1 Lecture 6: D potential flow, method of characteristics Thierry Magin, Greg Dimitriadis, and Johan Boutet Thierry.Magin@vki.ac.be Aeronautics and Aerospace
More informationDiffraction by Edges. András Vasy (with Richard Melrose and Jared Wunsch)
Diffraction by Edges András Vasy (with Richard Melrose and Jared Wunsch) Cambridge, July 2006 Consider the wave equation Pu = 0, Pu = D 2 t u gu, on manifolds with corners M; here g 0 the Laplacian, D
More informationÜbungen zu RT2 SS (4) Show that (any) contraction of a (p, q) - tensor results in a (p 1, q 1) - tensor.
Übungen zu RT2 SS 2010 (1) Show that the tensor field g µν (x) = η µν is invariant under Poincaré transformations, i.e. x µ x µ = L µ νx ν + c µ, where L µ ν is a constant matrix subject to L µ ρl ν ση
More informationSelf-intersections of Closed Parametrized Minimal Surfaces in Generic Riemannian Manifolds
Self-intersections of Closed Parametrized Minimal Surfaces in Generic Riemannian Manifolds John Douglas Moore Department of Mathematics University of California Santa Barbara, CA, USA 93106 e-mail: moore@math.ucsb.edu
More informationA scaling limit from Euler to Navier-Stokes equations with random perturbation
A scaling limit from Euler to Navier-Stokes equations with random perturbation Franco Flandoli, Scuola Normale Superiore of Pisa Newton Institute, October 208 Newton Institute, October 208 / Subject of
More informationThe Skorokhod reflection problem for functions with discontinuities (contractive case)
The Skorokhod reflection problem for functions with discontinuities (contractive case) TAKIS KONSTANTOPOULOS Univ. of Texas at Austin Revised March 1999 Abstract Basic properties of the Skorokhod reflection
More informationGraviton contributions to the graviton self-energy at one loop order during inflation
Graviton contributions to the graviton self-energy at one loop order during inflation PEDRO J. MORA DEPARTMENT OF PHYSICS UNIVERSITY OF FLORIDA PASI2012 1. Description of my thesis problem. i. Graviton
More informationLecture 2: Isoperimetric methods for the curve-shortening flow and for the Ricci flow on surfaces
Lecture 2: Isoperimetric methods for the curve-shortening flow and for the Ricci flow on surfaces Ben Andrews Mathematical Sciences Institute, Australian National University Winter School of Geometric
More informationThin airfoil theory. Chapter Compressible potential flow The full potential equation
hapter 4 Thin airfoil theory 4. ompressible potential flow 4.. The full potential equation In compressible flow, both the lift and drag of a thin airfoil can be determined to a reasonable level of accuracy
More informationInverse problems for hyperbolic PDEs
Inverse problems for hyperbolic PDEs Lauri Oksanen University College London Example: inverse problem for the wave equation Let c be a smooth function on Ω R n and consider the wave equation t 2 u c 2
More informationUmbilic cylinders in General Relativity or the very weird path of trapped photons
Umbilic cylinders in General Relativity or the very weird path of trapped photons Carla Cederbaum Universität Tübingen European Women in Mathematics @ Schloss Rauischholzhausen 2015 Carla Cederbaum (Tübingen)
More informationA REMARK ON AN EQUATION OF WAVE MAPS TYPE WITH VARIABLE COEFFICIENTS
A REMARK ON AN EQUATION OF WAVE MAPS TYPE WITH VARIABLE COEFFICIENTS DAN-ANDREI GEBA Abstract. We obtain a sharp local well-posedness result for an equation of wave maps type with variable coefficients.
More informationLinear Algebra, Summer 2011, pt. 2
Linear Algebra, Summer 2, pt. 2 June 8, 2 Contents Inverses. 2 Vector Spaces. 3 2. Examples of vector spaces..................... 3 2.2 The column space......................... 6 2.3 The null space...........................
More informationCOMPLEX ALGEBRAIC SURFACES CLASS 15
COMPLEX ALGEBRAIC SURFACES CLASS 15 RAVI VAKIL CONTENTS 1. Every smooth cubic has 27 lines, and is the blow-up of P 2 at 6 points 1 1.1. An alternate approach 4 2. Castelnuovo s Theorem 5 We now know that
More informationAbstract. Jacobi curves are far going generalizations of the spaces of \Jacobi
Principal Invariants of Jacobi Curves Andrei Agrachev 1 and Igor Zelenko 2 1 S.I.S.S.A., Via Beirut 2-4, 34013 Trieste, Italy and Steklov Mathematical Institute, ul. Gubkina 8, 117966 Moscow, Russia; email:
More informationPDEs, part 1: Introduction and elliptic PDEs
PDEs, part 1: Introduction and elliptic PDEs Anna-Karin Tornberg Mathematical Models, Analysis and Simulation Fall semester, 2013 Partial di erential equations The solution depends on several variables,
More informationVarious lecture notes for
Various lecture notes for 18311. R. R. Rosales (MIT, Math. Dept., 2-337) April 12, 2013 Abstract Notes, both complete and/or incomplete, for MIT s 18.311 (Principles of Applied Mathematics). These notes
More informationNull Cones to Infinity, Curvature Flux, and Bondi Mass
Null Cones to Infinity, Curvature Flux, and Bondi Mass Arick Shao (joint work with Spyros Alexakis) University of Toronto May 22, 2013 Arick Shao (University of Toronto) Null Cones to Infinity May 22,
More information,, rectilinear,, spherical,, cylindrical. (6.1)
Lecture 6 Review of Vectors Physics in more than one dimension (See Chapter 3 in Boas, but we try to take a more general approach and in a slightly different order) Recall that in the previous two lectures
More informationLecture XIV: Global structure, acceleration, and the initial singularity
Lecture XIV: Global structure, acceleration, and the initial singularity Christopher M. Hirata Caltech M/C 350-17, Pasadena CA 91125, USA (Dated: December 5, 2012) I. OVERVIEW In this lecture, we will
More information1 Some general theory for 2nd order linear nonhomogeneous
Math 175 Honors ODE I Spring, 013 Notes 5 1 Some general theory for nd order linear nonhomogeneous equations 1.1 General form of the solution Suppose that p; q; and g are continuous on an interval I; and
More informationThe Stability of the Irrotational Euler-Einstein System with a Positive Cosmological Constant
The Stability of the Irrotational Euler-Einstein System with a Positive Cosmological Constant Jared Speck & Igor Rodnianski jspeck@math.princeton.edu University of Cambridge & Princeton University October
More informationIt is convenient to introduce some notation for this type of problems. I will write this as. max u (x 1 ; x 2 ) subj. to. p 1 x 1 + p 2 x 2 m ;
4 Calculus Review 4.1 The Utility Maimization Problem As a motivating eample, consider the problem facing a consumer that needs to allocate a given budget over two commodities sold at (linear) prices p
More informationChapter 14. Basics of The Differential Geometry of Surfaces. Introduction. Parameterized Surfaces. The First... Home Page. Title Page.
Chapter 14 Basics of The Differential Geometry of Surfaces Page 649 of 681 14.1. Almost all of the material presented in this chapter is based on lectures given by Eugenio Calabi in an upper undergraduate
More informationInstability of extreme black holes
Instability of extreme black holes James Lucietti University of Edinburgh EMPG seminar, 31 Oct 2012 Based on: J.L., H. Reall arxiv:1208.1437 Extreme black holes Extreme black holes do not emit Hawking
More informationCausal Dissipation for the Relativistic Fluid Dynamics of Ideal Gases
Causal Dissipation for the Relativistic Fluid Dynamics of Ideal Gases Heinrich Freistühler and Blake Temple Proceedings of the Royal Society-A May 2017 Culmination of a 15 year project: In this we propose:
More informationPhysics 351 Wednesday, February 14, 2018
Physics 351 Wednesday, February 14, 2018 HW4 due Friday. For HW help, Bill is in DRL 3N6 Wed 4 7pm. Grace is in DRL 2C2 Thu 5:30 8:30pm. Respond at pollev.com/phys351 or text PHYS351 to 37607 once to join,
More informationOverview of the proof of the Bounded L 2 Curvature Conjecture. Sergiu Klainerman Igor Rodnianski Jeremie Szeftel
Overview of the proof of the Bounded L 2 Curvature Conjecture Sergiu Klainerman Igor Rodnianski Jeremie Szeftel Department of Mathematics, Princeton University, Princeton NJ 8544 E-mail address: seri@math.princeton.edu
More informationPhysics 212: Statistical mechanics II Lecture XI
Physics 212: Statistical mechanics II Lecture XI The main result of the last lecture was a calculation of the averaged magnetization in mean-field theory in Fourier space when the spin at the origin is
More informationContents (I): The model: The physical scenario and the derivation of the model: Muskat and the confined Hele-Shaw cell problems.
Contents (I): The model: The physical scenario and the derivation of the model: Muskat and the confined Hele-Shaw cell problems. First ideas and results: Existence and uniqueness But, are the boundaries
More information9 Symmetries of AdS 3
9 Symmetries of AdS 3 This section consists entirely of exercises. If you are not doing the exercises, then read through them anyway, since this material will be used later in the course. The main goal
More informationHamiltonian partial differential equations and Painlevé transcendents
The 6th TIMS-OCAMI-WASEDA Joint International Workshop on Integrable Systems and Mathematical Physics March 22-26, 2014 Hamiltonian partial differential equations and Painlevé transcendents Boris DUBROVIN
More informationGlobal Existence of Large BV Solutions in a Model of Granular Flow
This article was downloaded by: [Pennsylvania State University] On: 08 February 2012, At: 09:55 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered
More informationHyperbolic Geometric Flow
Hyperbolic Geometric Flow Kefeng Liu CMS and UCLA August 20, 2007, Dong Conference Page 1 of 51 Outline: Joint works with D. Kong and W. Dai Motivation Hyperbolic geometric flow Local existence and nonlinear
More information1 Introduction: connections and fiber bundles
[under construction] 1 Introduction: connections and fiber bundles Two main concepts of differential geometry are those of a covariant derivative and of a fiber bundle (in particular, a vector bundle).
More informationNumerical Methods for Inverse Kinematics
Numerical Methods for Inverse Kinematics Niels Joubert, UC Berkeley, CS184 2008-11-25 Inverse Kinematics is used to pose models by specifying endpoints of segments rather than individual joint angles.
More informationΩ Ω /ω. To these, one wants to add a fourth condition that arises from physics, what is known as the anomaly cancellation, namely that
String theory and balanced metrics One of the main motivations for considering balanced metrics, in addition to the considerations already mentioned, has to do with the theory of what are known as heterotic
More informationStability and Instability of Extremal Black Holes
Stability and Instability of Extremal Black Holes Stefanos Aretakis Department of Pure Mathematics and Mathematical Statistics, University of Cambridge s.aretakis@dpmms.cam.ac.uk December 13, 2011 MIT
More informationRay equations of a weak shock in a hyperbolic system of conservation laws in multi-dimensions
Proc. Indian Acad. Sci. (Math. Sci.) Vol. 126, No. 2, May 2016, pp. 199 206. c Indian Academy of Sciences Ray equations of a weak in a hyperbolic system of conservation laws in multi-dimensions PHOOLAN
More informationIntroduction to Poincare Conjecture and the Hamilton-Perelman program
Introduction to Poincare Conjecture and the Hamilton-Perelman program David Glickenstein Math 538, Spring 2009 January 20, 2009 1 Introduction This lecture is mostly taken from Tao s lecture 2. In this
More informationarxiv: v3 [math.ap] 10 Oct 2014
THE BOUNDED L 2 CURVATURE CONJECTURE arxiv:1204.1767v3 [math.ap] 10 Oct 2014 SERGIU KLAINERMAN, IGOR RODNIANSKI, AND JEREMIE SZEFTEL Abstract. This is the main paper in a sequence in which we give a complete
More informationA LOWER BOUND ON BLOWUP RATES FOR THE 3D INCOMPRESSIBLE EULER EQUATION AND A SINGLE EXPONENTIAL BEALE-KATO-MAJDA ESTIMATE. 1.
A LOWER BOUND ON BLOWUP RATES FOR THE 3D INCOMPRESSIBLE EULER EQUATION AND A SINGLE EXPONENTIAL BEALE-KATO-MAJDA ESTIMATE THOMAS CHEN AND NATAŠA PAVLOVIĆ Abstract. We prove a Beale-Kato-Majda criterion
More informationWeek 1. 1 The relativistic point particle. 1.1 Classical dynamics. Reading material from the books. Zwiebach, Chapter 5 and chapter 11
Week 1 1 The relativistic point particle Reading material from the books Zwiebach, Chapter 5 and chapter 11 Polchinski, Chapter 1 Becker, Becker, Schwartz, Chapter 2 1.1 Classical dynamics The first thing
More informationAsymptotic Behavior of Marginally Trapped Tubes
Asymptotic Behavior of Marginally Trapped Tubes Catherine Williams January 29, 2009 Preliminaries general relativity General relativity says that spacetime is described by a Lorentzian 4-manifold (M, g)
More informationNotation : M a closed (= compact boundaryless) orientable 3-dimensional manifold
Overview John Lott Notation : M a closed (= compact boundaryless) orientable 3-dimensional manifold Conjecture. (Poincaré, 1904) If M is simply connected then it is diffeomorphic to the three-sphere S
More informationHyperbolic Geometric Flow
Hyperbolic Geometric Flow Kefeng Liu Zhejiang University UCLA Page 1 of 41 Outline Introduction Hyperbolic geometric flow Local existence and nonlinear stability Wave character of metrics and curvatures
More informationDECOUPLING LECTURE 6
18.118 DECOUPLING LECTURE 6 INSTRUCTOR: LARRY GUTH TRANSCRIBED BY DONGHAO WANG We begin by recalling basic settings of multi-linear restriction problem. Suppose Σ i,, Σ n are some C 2 hyper-surfaces in
More informationMath 433 Outline for Final Examination
Math 433 Outline for Final Examination Richard Koch May 3, 5 Curves From the chapter on curves, you should know. the formula for arc length of a curve;. the definition of T (s), N(s), B(s), and κ(s) for
More informationHow to solve quasi linear first order PDE. v(u, x) Du(x) = f(u, x) on U IR n, (1)
How to solve quasi linear first order PDE A quasi linear PDE is an equation of the form subject to the initial condition v(u, x) Du(x) = f(u, x) on U IR n, (1) u = g on Γ, (2) where Γ is a hypersurface
More informationDonoghue, Golowich, Holstein Chapter 4, 6
1 Week 7: Non linear sigma models and pion lagrangians Reading material from the books Burgess-Moore, Chapter 9.3 Donoghue, Golowich, Holstein Chapter 4, 6 Weinberg, Chap. 19 1 Goldstone boson lagrangians
More informationHolographic Entanglement Entropy
Motivation Time-dependent Multi-region Summary Holographic entanglement entropy for time dependent states and disconnected regions Durham University INT08: From Strings to Things, April 3, 2008 VH, M.
More informationThe Non-commutative S matrix
The Suvrat Raju Harish-Chandra Research Institute 9 Dec 2008 (work in progress) CONTEMPORARY HISTORY In the past few years, S-matrix techniques have seen a revival. (Bern et al., Britto et al., Arkani-Hamed
More informationRANDOM PROPERTIES BENOIT PAUSADER
RANDOM PROPERTIES BENOIT PAUSADER. Quasilinear problems In general, one consider the following trichotomy for nonlinear PDEs: A semilinear problem is a problem where the highest-order terms appears linearly
More informationTable of contents. d 2 y dx 2, As the equation is linear, these quantities can only be involved in the following manner:
M ath 0 1 E S 1 W inter 0 1 0 Last Updated: January, 01 0 Solving Second Order Linear ODEs Disclaimer: This lecture note tries to provide an alternative approach to the material in Sections 4. 4. 7 and
More information