Wave operators with non-lipschitz coefficients: energy and observability estimates
|
|
- Colleen Crawford
- 6 years ago
- Views:
Transcription
1 Wave operators with non-lipschitz coefficients: energy and observability estimates Institut de Mathématiques de Jussieu-Paris Rive Gauche UNIVERSITÉ PARIS DIDEROT PARIS 7 JOURNÉE JEUNES CONTRÔLEURS 2014 Laboratoire Jacques-Louis Lions Université Pierre et Marie Curie Paris 6 Paris February 13, 2014
2 Contents of the talk Energy estimates Wave operators with non-regular coefficients The Cauchy problem: energy estimates (i Operators with non-lipschitz coefficients (ii Zygmund condition: a well-posedness result (iii The control problem: observability estimates (i Classical observability estimates (ii Estimates with loss (iii Remarks and ideas of the proof
3 THE CAUCHY PROBLEM: ENERGY ESTIMATES
4 General setting Energy estimates L u := 2 t u N j,k=1 j (a jk (t, x k u on a strip [0, T] R N, with N 0 < λ 0 ξ 2 a jk (t, x ξ j ξ k Λ 0 ξ 2 ξ R N \{0} j,k=1 Aim: studying the Cauchy problem (CP { Lu = f u t=0 = u 0, t u t=0 = u 1 in the Sobolev spaces framework
5 Classical result Energy estimates Hurd & Sattinger (1968 a jk (t, x { Lipschitz continuous in t only bounded with respect to x = well-posedness of (CP in H 1 L 2 More regularity in x = H s H s 1 Key: energy estimate with no loss of derivatives ( u(t, H s + t u(t, H s 1 sup 0 t T C s ( u(0, H s + t u(0, H s 1 + T 0 Lu(t, H s 1 dt
6 De Simon & Torelli (1974: a jk BV t Counterexamples: Hurd & Sattinger (1968: discontinuous coefficients Colombini, De Giorgi & Spagnolo (1979: Hölder coefficients General idea: lower regularity assumptions with respect to t suitable hypothesis on x to compensate it = H well-posedness, but eventually with a loss of derivatives in the energy estimates
7 Coefficients depending only on time Integral log-lipschitz condition 0 Colombini, De Giorgi & Spagnolo (1979 T τ a jk (t + τ a jk (t ( dt C 0 τ log τ Integral log-zygmund condition Tarama (2007 T τ ajk (t + τ + a jk (t τ 2a jk (t ( dt C0 τ log τ τ Theorem sup 0 t T ( u(t, H s δ + t u(t, H s 1 δ C s ( u(0, H s + t u(0, H s 1 + T 0 Lu(t, H s 1 δ dt
8 Remarks Energy estimates Proof: Approximation of the coefficients Fourier transform Linking dual variable and approximation parameter Hölder coefficients = solutions in Gevrey classes ( Colombini, De Giorgi & Spagnolo 1979 Counterexample to distributional solutions ( Colombini, De Giorgi & Spagnolo 1979
9 Coefficients depending on (t, x Pointwise log-lipschitz condition in all the variables Colombini & Lerner (1995 a jk (t +τ, x+y a jk (t, x ( C (τ + y log sup (t,x τ + y Log-Zygmund in time & log-lipschitz in space condition (LZ t sup ajk (t + τ, x + a jk (t τ, x 2a jk (t, x C0 τ log ( τ (LL x sup a jk (t, x + y a jk (t, x ( C 0 y log y scalar case x R: Colombini-Del Santo (2009 general case x R N : Colombini-Del Santo-F.-Métivier (2013
10 Theorem ( u(t, H s βt + t u(t, H s 1 βt sup 0 t T C s ( u(0, H s + t u(0, H s 1 + T 0 Lu(t, H s 1 βt dt loss of derivatives linearly increasing in time s ]0, 1[ Local in time estimates: T T β and C s depend only on L In particular a jk C b (RN x = H well-posedness, globally in time
11 Energy estimates with no loss of derivatives Lu(t, x := 2 t u N j,k=1 j (a jk (t, x k u Pointwise Zygmund condition in all the variables: a jk (t + τ, x + y + a jk (t τ, x y 2a jk (t, x C (τ + y sup (t,x Theorem (Colombini, Del Santo, F. & Métivier 2013 ( u(t H 1/2 + t u(t H 1/2 sup 0 t T C e λt ( u(0 H 1/2 + t u(0 H 1/2 + T 0 e λt Lu(t H 1/2dt
12 Remarks Energy estimates Original statement for a complete operator Pu = 2 t u N j,k=1 j (a jk (t, x k u + B(t, x (t,x u + c(t, xu B L ([0, T] ; C θ (R N ( θ > 1/2 c L ([0, T] R N Global in time estimate Well-posedness of (CP in H 1/2 H 1/2 Well-posedness in H if a jk Z([0, T] C b (RN
13 Related results Energy estimates Tarama (2007: a jk = a jk (t Z([0, T] = no loss of derivatives in any H s H s 1 Cicognani & Colombini (2006 Modulus of continuity Loss of derivatives Lipschitz no loss intermediate arbitrarly small loss log-lipschitz finite loss β t
14 Zygmund functions Energy estimates Definition: f Z(R n if f L (R n and sup f (z + ζ + f (z ζ 2 f (z K 0 ζ z R n Basic properties Lip(R n Z(R n loglip(r n Condition on second derivatives: if f C 2 (R, then f (z + ζ + f (z ζ 2 f (z = ζ 2 f (φ z,ζ Z(R n B 1, (R n, f B 1, where ( := sup 2 ν ν f L ν N
15 Regularization in time Energy estimates f Z(R t, 0 < λ 0 f (t Λ 0 Approximation by convolution kernel: f ε (t := (ρ ε f (t = 1 ε Then: 0 < λ 0 f ε Λ 0 R f ε (t f (t C ε t f ε (t C log 2 t f ε (t C ε ( τ ρ f (t τ dτ ε ( ε
16 Littlewood-Paley Theory Littlewood-Paley decomposition Dyadic partition of unity in phase-space: χ 1 (ξ + supp χ 1 { ξ 1}, + ν=0 ψ ν (ξ 1 supp ψ ν { 2 ν 1 ξ 2 ν+1} = Operators: 1 := χ 1 (D x, ν := ψ ν (D x, S ν := Sobolev spaces = u S (R N, u = u H s + ν= 1 ν u ν 1 j= 1 ( 2 sν ν u L 2 l 2 ν 1 j
17 Paradifferential calculus with parameters Bony s paraproduct operator: a, u S (R N x T a u := ν 1 S ν 1 a ν u Regularization in space = well defined also if a(x α(t, x, ξ, rough in x Parameter γ 1 starting from high frequencies = α(t, x, ξ positive symbol = T α positive operator Symbolic calculus for Zygmund continuous symbols
18 Proof of the energy estimate (i a jk a jk,ε α ε (t, x, ξ, γ := j,k a jk,ε(t, x ξ j ξ k + γ 2 with ε = (γ 2 + ξ 2 1/2 (ii Approximation of the operator Lu = 2 t u + Re T αε u + Ru R : H s H s 1 for any s ]0, 1[ (iii Energy E(t := v(t 2 L 2 + w(t 2 L 2, with v(t, x := T α 1/4 ε w(t, x := T 1/4 α u ε t u T t (α u 1/4 ε
19 In particular, E(t t u(t 2 + u(t 2 H 1/2 H 1/2 ( if γ 1 large enough (iv Differentiation in time = cancellations Tarama s cancellation ( definition of E(t Paradifferential operator Re T αε (v Gronwall s inequality to conclude Remarks Cancellations only for s = 1/2 s 1/2 not clear
20 Classical observability estimates Main results Remarks and sketch of the proof THE CONTROL PROBLEM: OBSERVABILITY ESTIMATES
21 Setting Energy estimates Classical observability estimates Main results Remarks and sketch of the proof N = 1, coefficient just depending on x ω(x t 2 u x 2 u = 0 in [0, 1] [0, T] u(t, 0 = u(t, 1 = 0 in [0, T] u(0, x = u 0 (x, t u(0, x = u 1 (x in [0, 1] 0 < ω ω(x ω T > T (in our case, T ω L 1 x (0,1 Energy: E(t := = E(t E(0 on [0, T] 0 (ω(x u t (t, x 2 + u x (t, x 2 dx
22 Classical observability estimates Main results Remarks and sketch of the proof Internal observability: for any Ω := ]l 1, l 2 [ [0, 1], T l2 E(0 C (ω(x u t (t, x 2 + u x (t, x 2 dx dt 0 l 1 Boundary observability: Ω = {0, 1} (or a subset, T E(0 C ( u x (t, u x (t, 1 2 dt 0 Remarks: Observability Geometric Control Condition for Ω ( Bardos, Lebeau & Rauch 1992 ; Burq & Gérard 1997 N 2: (i Microlocal analysis = C 2 regularity (ii Carleman estimates = C 1 regularity ( Duyckaerts, Zhang, Zuazua 2008
23 Previous results Energy estimates Classical observability estimates Main results Remarks and sketch of the proof ω Lipschiz = observability estimates: E(0 C T Avellaneda, Bardos & Rauch (1992: 0 u x (t, 0 2 dt ω ε (x := ω(x/ε = lim ε 0 C ε = + Fernández-Cara & Zuazua (2002: ω BV(0, 1 = observability estimates Castro & Zuazua (2003: ω C s (0, 1 = NO observability estimates
24 The Zygmund case Energy estimates Classical observability estimates Main results Remarks and sketch of the proof ω(x 2 t u 2 x u = 0 in [0, 1] [0, T] u(t, 0 = u(t, 1 = 0 in [0, T] u(0, x = u 0 (x, t u(0, x = u 1 (x in [0, 1] 0 < ω ω(x ω, ω Z : ω(x + y + ω(x y 2 ω(x dx K y T > T := 2 ω L 1 x (0,1 Theorem ( F. & Zuazua 2013 T u 0 2 H0 1(Ω + u 1 2 L 2 (Ω C x u(t, 0 2 dt 0
25 with loss Classical observability estimates Main results Remarks and sketch of the proof (LL Integral log-lipschitz condition: ω(x + y ω(x dx C y log( y (LZ Integral log-zygmund condition: ω(x + y + ω(x y 2ω(x dx C y log( y Theorem ( F. & Zuazua 2013 T u 0 2 H0 1(Ω + u 1 2 L 2 (Ω C t m x u(t, 0 2 dt 0 Right-hand side finite for smooth enough data
26 About the first variation Classical observability estimates Main results Remarks and sketch of the proof Modulus of continuity ω Lipschitz = no loss ω between Lipschitz and log-lipschitz = arbitrarly small loss = m = 1 ω log-lipschitz = finite loss ω worse than log-lipschitz = infinite loss
27 Proof of observability Energy estimates Classical observability estimates Main results Remarks and sketch of the proof Sidewise energy estimates (i Sidewise energy: F(x := 1 2 T (T /2x (T /2x ( ω(x t u(t, x 2 + x u(t, x 2 dt In particular, F(0 = (ω(0/2 T 0 xu(t, 0 2 dt (ii Zygmund log-zygmund Tarama (2007 log-lipschitz Colombini-De Giorgi-Spagnolo (1979 ( thanks to finite propagation speed (iii integration in space
28 On the counterexamples Classical observability estimates Main results Remarks and sketch of the proof Castro & Zuazua (2003: ω C s (0, 1 = NO observability estimates Proof: counterexample ( ideas from Colombini & Spagnolo (1989 Fractal partition of [0, 1] Construction of the oscillating coefficient: more and more oscillations for x 0 energy decreasing for x 0 energy exponentially concentrated inside the subintervals = energy too small at x = 0 Construction ok also for x 1 and for internal observability
29 Remarks Energy estimates Classical observability estimates Main results Remarks and sketch of the proof Z B 1 1, = W1,1 Z Example by Tarama (2007 = BV Z Controllability results ω(x t 2 y x 2 y = 0 in [0, 1] [0, T] y(t, 0 = f (t, y(t, 1 = 0 in [0, T] y(0, x = y 0 (x, t y(0, x = y 1 (x in [0, 1] (i ω Z = f L 2 (0, T (ii ω LZ LL = f H m (0, T
30 THANK YOU!
The well-posedness issue in Sobolev spaces for hyperbolic systems with Zygmund-type coefficients. 1 Università di Pisa
The well-posedness issue in Sobolev spaces for hyperbolic systems with Zygmund-type coefficients Ferruccio Colombini 1, Daniele Del Santo 2, Francesco Fanelli 3, Guy Métivier 4 1 Università di Pisa Dipartimento
More informationTIME-DEPENDENT LOSS OF DERIVATIVES FOR HYPERBOLIC OPERATORS WITH NON REGULAR COEFFICIENTS. Ferruccio Colombini. Daniele Del Santo.
TIME-DEPENDENT LOSS OF DERIVATIVES FOR HYPERBOLIC OPERATORS WITH NON REGULAR COEFFICIENTS Ferruccio Colombini Università di Pisa colombini@dm.unipi.it Daniele Del Santo Università di Trieste delsanto@units.it
More informationA note on complete hyperbolic operators with log- Zygmund coefficients
A note on complete hyperbolic operators with log- Zygmund coefficients Ferruccio Colombini, Daniele Del Santo, Francesco Fanelli and Guy Métivier Abstract. The present paper is the continuation of the
More informationarxiv: v1 [math.ap] 6 May 2013
A WELL-POSEDNESS RESULT FOR HYPERBOLIC OPERATORS WITH ZYGMUND COEFFICIENTS Ferruccio Colombini Università di Pisa colombini@dm.unipi.it Daniele Del Santo Università di Trieste delsanto@units.it arxiv:1305.1292v1
More informationTIME-DEPENDENT LOSS OF DERIVATIVES FOR HYPERBOLIC OPERATORS WITH NON-REGULAR COEFFICIENTS. Ferruccio Colombini. Daniele Del Santo.
TIME-DEPENDENT LOSS OF DERIVATIVES FOR HYPERBOLIC OPERATORS WITH NON-REGULAR COEFFICIENTS Ferruccio Colombini Università di Pisa colombini@dm.unipi.it Daniele Del Santo Università di Trieste delsanto@units.it
More informationdoi: /j.jde
doi: 10.1016/j.jde.016.08.019 On Second Order Hyperbolic Equations with Coefficients Degenerating at Infinity and the Loss of Derivatives and Decays Tamotu Kinoshita Institute of Mathematics, University
More informationControllability of linear PDEs (I): The wave equation
Controllability of linear PDEs (I): The wave equation M. González-Burgos IMUS, Universidad de Sevilla Doc Course, Course 2, Sevilla, 2018 Contents 1 Introduction. Statement of the problem 2 Distributed
More informationHilbert Uniqueness Method and regularity
Hilbert Uniqueness Method and regularity Sylvain Ervedoza 1 Joint work with Enrique Zuazua 2 1 Institut de Mathématiques de Toulouse & CNRS 2 Basque Center for Applied Mathematics Institut Henri Poincaré
More informationLocal null controllability of the N-dimensional Navier-Stokes system with N-1 scalar controls in an arbitrary control domain
Local null controllability of the N-dimensional Navier-Stokes system with N-1 scalar controls in an arbitrary control domain Nicolás Carreño Université Pierre et Marie Curie-Paris 6 UMR 7598 Laboratoire
More informationParaproducts and the bilinear Calderón-Zygmund theory
Paraproducts and the bilinear Calderón-Zygmund theory Diego Maldonado Department of Mathematics Kansas State University Manhattan, KS 66506 12th New Mexico Analysis Seminar April 23-25, 2009 Outline of
More informationThe heat equation. Paris-Sud, Orsay, December 06
Paris-Sud, Orsay, December 06 The heat equation Enrique Zuazua Universidad Autónoma 28049 Madrid, Spain enrique.zuazua@uam.es http://www.uam.es/enrique.zuazua Plan: 3.- The heat equation: 3.1 Preliminaries
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 informationInégalités spectrales pour le contrôle des EDP linéaires : groupe de Schrödinger contre semigroupe de la chaleur.
Inégalités spectrales pour le contrôle des EDP linéaires : groupe de Schrödinger contre semigroupe de la chaleur. Luc Miller Université Paris Ouest Nanterre La Défense, France Pde s, Dispersion, Scattering
More informationConservation law equations : problem set
Conservation law equations : problem set Luis Silvestre For Isaac Neal and Elia Portnoy in the 2018 summer bootcamp 1 Method of characteristics For the problems in this section, assume that the solutions
More informationStability of nonlinear locally damped partial differential equations: the continuous and discretized problems. Part II
. Stability of nonlinear locally damped partial differential equations: the continuous and discretized problems. Part II Fatiha Alabau-Boussouira 1 Emmanuel Trélat 2 1 Univ. de Lorraine, LMAM 2 Univ. Paris
More informationA sufficient condition for observability of waves by measurable subsets
A sufficient condition for observability of waves by measurable subsets Emmanuel Humbert Yannick Privat Emmanuel Trélat Abstract We consider the wave equation on a closed Riemannian manifold (M, g). Given
More informationMicrolocal analysis and inverse problems Lecture 3 : Carleman estimates
Microlocal analysis and inverse problems ecture 3 : Carleman estimates David Dos Santos Ferreira AGA Université de Paris 13 Monday May 16 Instituto de Ciencias Matemáticas, Madrid David Dos Santos Ferreira
More informationControl of Waves: Theory and Numerics
BCAM, October, 2010 Control of Waves: Theory and Numerics Enrique Zuazua BCAM Basque Center for Applied Mathematics E-48160 Derio - Basque Country - Spain zuazua@bcamath.org www.bcamath.org/zuazua THE
More informationSobolev Spaces. Chapter 10
Chapter 1 Sobolev Spaces We now define spaces H 1,p (R n ), known as Sobolev spaces. For u to belong to H 1,p (R n ), we require that u L p (R n ) and that u have weak derivatives of first order in L p
More informationNonlinear stabilization via a linear observability
via a linear observability Kaïs Ammari Department of Mathematics University of Monastir Joint work with Fathia Alabau-Boussouira Collocated feedback stabilization Outline 1 Introduction and main result
More informationObservability and measurable sets
Observability and measurable sets Luis Escauriaza UPV/EHU Luis Escauriaza (UPV/EHU) Observability and measurable sets 1 / 41 Overview Interior: Given T > 0 and D Ω (0, T ), to find N = N(Ω, D, T ) > 0
More informationHyperbolic inverse problems and exact controllability
Hyperbolic inverse problems and exact controllability Lauri Oksanen University College London An inverse initial source problem Let M R n be a compact domain with smooth strictly convex boundary, and let
More informationLittlewood-Paley theory
Chapitre 6 Littlewood-Paley theory Introduction The purpose of this chapter is the introduction by this theory which is nothing but a precise way of counting derivatives using the localization in the frequency
More informationA quantitative Fattorini-Hautus test: the minimal null control time problem in the parabolic setting
A quantitative Fattorini-Hautus test: the minimal null control time problem in the parabolic setting Morgan MORANCEY I2M, Aix-Marseille Université August 2017 "Controllability of parabolic equations :
More informationOn the local existence for an active scalar equation in critical regularity setting
On the local existence for an active scalar equation in critical regularity setting Walter Rusin Department of Mathematics, Oklahoma State University, Stillwater, OK 7478 Fei Wang Department of Mathematics,
More informationRome - May 12th Université Paris-Diderot - Laboratoire Jacques-Louis Lions. Mean field games equations with quadratic
Université Paris-Diderot - Laboratoire Jacques-Louis Lions Rome - May 12th 2011 Hamiltonian MFG Hamiltonian on the domain [0, T ] Ω, Ω standing for (0, 1) d : (HJB) (K) t u + σ2 2 u + 1 2 u 2 = f (x, m)
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 informationarxiv: v2 [math.ap] 30 Jan 2015
LOCAL WELL-POSEDNESS FOR THE HALL-MHD EQUATIONS WITH FRACTIONAL MAGNETIC DIFFUSION DONGHO CHAE 1, RENHUI WAN 2 AND JIAHONG WU 3 arxiv:144.486v2 [math.ap] 3 Jan 215 Abstract. The Hall-magnetohydrodynamics
More informationNeighboring feasible trajectories in infinite dimension
Neighboring feasible trajectories in infinite dimension Marco Mazzola Université Pierre et Marie Curie (Paris 6) H. Frankowska and E. M. Marchini Control of State Constrained Dynamical Systems Padova,
More informationDuality of multiparameter Hardy spaces H p on spaces of homogeneous type
Duality of multiparameter Hardy spaces H p on spaces of homogeneous type Yongsheng Han, Ji Li, and Guozhen Lu Department of Mathematics Vanderbilt University Nashville, TN Internet Analysis Seminar 2012
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 informationDetermination of singular time-dependent coefficients for wave equations from full and partial data
Determination of singular time-dependent coefficients for wave equations from full and partial data Guanghui Hu, Yavar Kian To cite this version: Guanghui Hu, Yavar Kian. Determination of singular time-dependent
More informationVelocity averaging a general framework
Outline Velocity averaging a general framework Martin Lazar BCAM ERC-NUMERIWAVES Seminar May 15, 2013 Joint work with D. Mitrović, University of Montenegro, Montenegro Outline Outline 1 2 L p, p >= 2 setting
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 informationInégalités de dispersion via le semi-groupe de la chaleur
Inégalités de dispersion via le semi-groupe de la chaleur Valentin Samoyeau, Advisor: Frédéric Bernicot. Laboratoire de Mathématiques Jean Leray, Université de Nantes January 28, 2016 1 Introduction Schrödinger
More information8 Singular Integral Operators and L p -Regularity Theory
8 Singular Integral Operators and L p -Regularity Theory 8. Motivation See hand-written notes! 8.2 Mikhlin Multiplier Theorem Recall that the Fourier transformation F and the inverse Fourier transformation
More informationA local estimate from Radon transform and stability of Inverse EIT with partial data
A local estimate from Radon transform and stability of Inverse EIT with partial data Alberto Ruiz Universidad Autónoma de Madrid ge Joint work with P. Caro (U. Helsinki) and D. Dos Santos Ferreira (Paris
More informationA new class of pseudodifferential operators with mixed homogenities
A new class of pseudodifferential operators with mixed homogenities Po-Lam Yung University of Oxford Jan 20, 2014 Introduction Given a smooth distribution of hyperplanes on R N (or more generally on a
More informationJUHA KINNUNEN. Harmonic Analysis
JUHA KINNUNEN Harmonic Analysis Department of Mathematics and Systems Analysis, Aalto University 27 Contents Calderón-Zygmund decomposition. Dyadic subcubes of a cube.........................2 Dyadic cubes
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 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 informationReal Analysis Problems
Real Analysis Problems Cristian E. Gutiérrez September 14, 29 1 1 CONTINUITY 1 Continuity Problem 1.1 Let r n be the sequence of rational numbers and Prove that f(x) = 1. f is continuous on the irrationals.
More informationOn a class of pseudodifferential operators with mixed homogeneities
On a class of pseudodifferential operators with mixed homogeneities Po-Lam Yung University of Oxford July 25, 2014 Introduction Joint work with E. Stein (and an outgrowth of work of Nagel-Ricci-Stein-Wainger,
More informationOn the optimality of some observability inequalities for plate systems with potentials
On the optimality of some observability inequalities for plate systems with potentials Xiaoyu Fu, Xu Zhang and Enrique Zuazua 3 School of Mathematics, Sichuan University, Chengdu, China Yangtze Center
More informationControl from an Interior Hypersurface
Control from an Interior Hypersurface Matthieu Léautaud École Polytechnique Joint with Jeffrey Galkowski Murramarang, microlocal analysis on the beach March, 23. 2018 Outline General questions Eigenfunctions
More informationMixed exterior Laplace s problem
Mixed exterior Laplace s problem Chérif Amrouche, Florian Bonzom Laboratoire de mathématiques appliquées, CNRS UMR 5142, Université de Pau et des Pays de l Adour, IPRA, Avenue de l Université, 64000 Pau
More informationRelation between Distributional and Leray-Hopf Solutions to the Navier-Stokes Equations
Relation between Distributional and Leray-Hopf Solutions to the Navier-Stokes Equations Giovanni P. Galdi Department of Mechanical Engineering & Materials Science and Department of Mathematics University
More informationStabilization of the wave equation with localized Kelvin-Voigt damping
Stabilization of the wave equation with localized Kelvin-Voigt damping Louis Tebou Florida International University Miami SEARCDE University of Memphis October 11-12, 2014 Louis Tebou (FIU, Miami) Stabilization...
More informationFinite-dimensional spaces. C n is the space of n-tuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product
Chapter 4 Hilbert Spaces 4.1 Inner Product Spaces Inner Product Space. A complex vector space E is called an inner product space (or a pre-hilbert space, or a unitary space) if there is a mapping (, )
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 informationHARMONIC ANALYSIS. Date:
HARMONIC ANALYSIS Contents. Introduction 2. Hardy-Littlewood maximal function 3. Approximation by convolution 4. Muckenhaupt weights 4.. Calderón-Zygmund decomposition 5. Fourier transform 6. BMO (bounded
More informationAN EXAMPLE OF FUNCTIONAL WHICH IS WEAKLY LOWER SEMICONTINUOUS ON W 1,p FOR EVERY p > 2 BUT NOT ON H0
AN EXAMPLE OF FUNCTIONAL WHICH IS WEAKLY LOWER SEMICONTINUOUS ON W,p FOR EVERY p > BUT NOT ON H FERNANDO FARRONI, RAFFAELLA GIOVA AND FRANÇOIS MURAT Abstract. In this note we give an example of functional
More informationStrichartz Estimates for the Schrödinger Equation in Exterior Domains
Strichartz Estimates for the Schrödinger Equation in University of New Mexico May 14, 2010 Joint work with: Hart Smith (University of Washington) Christopher Sogge (Johns Hopkins University) The Schrödinger
More informationGlobal Carleman inequalities and theoretical and numerical control results for systems governed by PDEs
Global Carleman inequalities and theoretical and numerical control results for systems governed by PDEs Enrique FERNÁNDEZ-CARA Dpto. E.D.A.N. - Univ. of Sevilla joint work with A. MÜNCH Lab. Mathématiques,
More informationPartial Differential Equations, 2nd Edition, L.C.Evans Chapter 5 Sobolev Spaces
Partial Differential Equations, nd Edition, L.C.Evans Chapter 5 Sobolev Spaces Shih-Hsin Chen, Yung-Hsiang Huang 7.8.3 Abstract In these exercises always denote an open set of with smooth boundary. As
More informationWeighted Littlewood-Paley inequalities
Nicolaus Copernicus University, Toruń, Poland 3-7 June 2013, Herrnhut, Germany for every interval I in R let S I denote the partial sum operator, i.e., (S I f ) = χ I f (f L2 (R)), for an arbitrary family
More informationOn the bang-bang property of time optimal controls for infinite dimensional linear systems
On the bang-bang property of time optimal controls for infinite dimensional linear systems Marius Tucsnak Université de Lorraine Paris, 6 janvier 2012 Notation and problem statement (I) Notation: X (the
More informationStrichartz Estimates in Domains
Department of Mathematics Johns Hopkins University April 15, 2010 Wave equation on Riemannian manifold (M, g) Cauchy problem: 2 t u(t, x) gu(t, x) =0 u(0, x) =f (x), t u(0, x) =g(x) Strichartz estimates:
More informationRESULTS ON FOURIER MULTIPLIERS
RESULTS ON FOURIER MULTIPLIERS ERIC THOMA Abstract. The problem of giving necessary and sufficient conditions for Fourier multipliers to be bounded on L p spaces does not have a satisfactory answer for
More informationGLOBAL REGULARITY RESULTS FOR THE CLIMATE MODEL WITH FRACTIONAL DISSIPATION
GOBA REGUARITY RESUTS FOR THE CIMATE MODE WITH FRACTIONA DISSIPATION BO-QING DONG 1, WENJUAN WANG 1, JIAHONG WU AND HUI ZHANG 3 Abstract. This paper studies the global well-posedness problem on a tropical
More informationVANISHING VISCOSITY IN THE PLANE FOR NONDECAYING VELOCITY AND VORTICITY
VANISHING VISCOSITY IN THE PLANE FOR NONDECAYING VELOCITY AND VORTICITY ELAINE COZZI Abstract. Assuming that initial velocity and initial vorticity are bounded in the plane, we show that on a sufficiently
More informationThis article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and
This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution
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 informationLocal Well-Posedness for the Hall-MHD Equations with Fractional Magnetic Diffusion
J. Math. Fluid Mech. 17 (15), 67 638 c 15 Springer Basel 14-698/15/467-1 DOI 1.17/s1-15--9 Journal of Mathematical Fluid Mechanics Local Well-Posedness for the Hall-MHD Equations with Fractional Magnetic
More informationExistence of minimizers for the pure displacement problem in nonlinear elasticity
Existence of minimizers for the pure displacement problem in nonlinear elasticity Cristinel Mardare Université Pierre et Marie Curie - Paris 6, Laboratoire Jacques-Louis Lions, Paris, F-75005 France Abstract
More informationRegularity and Decay Estimates of the Navier-Stokes Equations
Regularity and Decay Estimates of the Navier-Stokes Equations Hantaek Bae Ulsan National Institute of Science and Technology (UNIST), Korea Recent Advances in Hydrodynamics, 216.6.9 Joint work with Eitan
More informationNonlinear Control Systems
Nonlinear Control Systems António Pedro Aguiar pedro@isr.ist.utl.pt 3. Fundamental properties IST-DEEC PhD Course http://users.isr.ist.utl.pt/%7epedro/ncs2012/ 2012 1 Example Consider the system ẋ = f
More informationT (1) and T (b) Theorems on Product Spaces
T (1) and T (b) Theorems on Product Spaces Yumeng Ou Department of Mathematics Brown University Providence RI yumeng ou@brown.edu June 2, 2014 Yumeng Ou (BROWN) T (1) and T (b) Theorems on Product Spaces
More informationBlow-up on manifolds with symmetry for the nonlinear Schröding
Blow-up on manifolds with symmetry for the nonlinear Schrödinger equation March, 27 2013 Université de Nice Euclidean L 2 -critical theory Consider the one dimensional equation i t u + u = u 4 u, t > 0,
More informationAnalisis para la ecuacion de Boltzmann Soluciones y Approximaciones
Analisis para la ecuacion de Boltzmann Soluciones y Approximaciones Irene M. Gamba Department of Mathematics and ICES The University of Texas at Austin Buenos Aires, June 2012 Collaborators: R. Alonso,
More informationInfinite dimensional controllability
Infinite dimensional controllability Olivier Glass Contents 0 Glossary 1 1 Definition of the subject and its importance 1 2 Introduction 2 3 First definitions and examples 2 4 Linear systems 6 5 Nonlinear
More informationMinimal time issues for the observability of Grushin-type equations
Intro Proofs Further Minimal time issues for the observability of Grushin-type equations Karine Beauchard (1) Jérémi Dardé (2) (2) (1) ENS Rennes (2) Institut de Mathématiques de Toulouse GT Contrôle LJLL
More informationGLOBAL WELL-POSEDNESS FOR NONLINEAR NONLOCAL CAUCHY PROBLEMS ARISING IN ELASTICITY
Electronic Journal of Differential Equations, Vol. 2017 (2017), No. 55, pp. 1 7. ISSN: 1072-6691. UL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu GLOBAL WELL-POSEDNESS FO NONLINEA NONLOCAL
More informationMicro-local analysis in Fourier Lebesgue and modulation spaces.
Micro-local analysis in Fourier Lebesgue and modulation spaces. Stevan Pilipović University of Novi Sad Nagoya, September 30, 2009 (Novi Sad) Nagoya, September 30, 2009 1 / 52 Introduction We introduce
More informationAsymptotic Behavior of a Hyperbolic-parabolic Coupled System Arising in Fluid-structure Interaction
International Series of Numerical Mathematics, Vol. 154, 445 455 c 2006 Birkhäuser Verlag Basel/Switzerland Asymptotic Behavior of a Hyperbolic-parabolic Coupled System Arising in Fluid-structure Interaction
More informationGlobal well-posedness and decay for the viscous surface wave problem without surface tension
Global well-posedness and decay for the viscous surface wave problem without surface tension Ian Tice (joint work with Yan Guo) Université Paris-Est Créteil Laboratoire d Analyse et de Mathématiques Appliquées
More informationThe Schrödinger propagator for scattering metrics
The Schrödinger propagator for scattering metrics Andrew Hassell (Australian National University) joint work with Jared Wunsch (Northwestern) MSRI, May 5-9, 2003 http://arxiv.org/math.ap/0301341 1 Schrödinger
More informationTD M1 EDP 2018 no 2 Elliptic equations: regularity, maximum principle
TD M EDP 08 no Elliptic equations: regularity, maximum principle Estimates in the sup-norm I Let be an open bounded subset of R d of class C. Let A = (a ij ) be a symmetric matrix of functions of class
More informationJordan Journal of Mathematics and Statistics (JJMS) 9(1), 2016, pp BOUNDEDNESS OF COMMUTATORS ON HERZ-TYPE HARDY SPACES WITH VARIABLE EXPONENT
Jordan Journal of Mathematics and Statistics (JJMS 9(1, 2016, pp 17-30 BOUNDEDNESS OF COMMUTATORS ON HERZ-TYPE HARDY SPACES WITH VARIABLE EXPONENT WANG HONGBIN Abstract. In this paper, we obtain the boundedness
More informationNew phenomena for the null controllability of parabolic systems: Minim
New phenomena for the null controllability of parabolic systems F.Ammar Khodja, M. González-Burgos & L. de Teresa Aix-Marseille Université, CNRS, Centrale Marseille, l2m, UMR 7373, Marseille, France assia.benabdallah@univ-amu.fr
More informationZygmund s Fourier restriction theorem and Bernstein s inequality
Zygmund s Fourier restriction theorem and Bernstein s inequality Jordan Bell jordanbell@gmailcom Department of Mathematics, University of Toronto February 13, 2015 1 Zygmund s restriction theorem Write
More informationThe Wave Equation: Control and Numerics
The Wave Equation: Control and Numerics Sylvain Ervedoza and Enrique Zuazua Abstract In these Notes we make a self-contained presentation of the theory that has been developed recently for the numerical
More informationDISPERSIVE ESTIMATES FOR WAVE EQUATIONS WITH ROUGH COEFFICIENTS
DISPERSIVE ESTIMATES FOR WAVE EQUATIONS WITH ROUGH COEFFICIENTS DANIEL TATARU AND DAN-ANDREI GEBA Abstract. We obtain a multiscale wave packet representation for the fundamental solution of the wave equation
More informationGlobal solutions for the cubic non linear wave equation
Global solutions for the cubic non linear wave equation Nicolas Burq Université Paris-Sud, Laboratoire de Mathématiques d Orsay, CNRS, UMR 8628, FRANCE and Ecole Normale Supérieure Oxford sept 12th, 2012
More information3 (Due ). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure?
MA 645-4A (Real Analysis), Dr. Chernov Homework assignment 1 (Due ). Show that the open disk x 2 + y 2 < 1 is a countable union of planar elementary sets. Show that the closed disk x 2 + y 2 1 is a countable
More informationPartial regularity for suitable weak solutions to Navier-Stokes equations
Partial regularity for suitable weak solutions to Navier-Stokes equations Yanqing Wang Capital Normal University Joint work with: Quansen Jiu, Gang Wu Contents 1 What is the partial regularity? 2 Review
More informationON WEAK SOLUTION OF A HYPERBOLIC DIFFERENTIAL INCLUSION WITH NONMONOTONE DISCONTINUOUS NONLINEAR TERM
Internat. J. Math. & Math. Sci. Vol. 22, No. 3 (999 587 595 S 6-72 9922587-2 Electronic Publishing House ON WEAK SOLUTION OF A HYPERBOLIC DIFFERENTIAL INCLUSION WITH NONMONOTONE DISCONTINUOUS NONLINEAR
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 informationGLOBAL WELL-POSEDNESS IN SPATIALLY CRITICAL BESOV SPACE FOR THE BOLTZMANN EQUATION
GLOBAL WELL-POSEDNESS IN SPATIALLY CRITICAL BESOV SPACE FOR THE BOLTZMANN EQUATION RENJUN DUAN, SHUANGQIAN LIU, AND JIANG XU Abstract. The unique global strong solution in the Chemin-Lerner type space
More informationB. Vedel Joint work with P.Abry (ENS Lyon), S.Roux (ENS Lyon), M. Clausel LJK, Grenoble),
Hyperbolic wavelet analysis of textures : global regularity and multifractal formalism B. Vedel Joint work with P.Abry (ENS Lyon), S.Roux (ENS Lyon), M. Clausel LJK, Grenoble), S.Jaffard(Créteil) Harmonic
More informationCUTOFF RESOLVENT ESTIMATES AND THE SEMILINEAR SCHRÖDINGER EQUATION
CUTOFF RESOLVENT ESTIMATES AND THE SEMILINEAR SCHRÖDINGER EQUATION HANS CHRISTIANSON Abstract. This paper shows how abstract resolvent estimates imply local smoothing for solutions to the Schrödinger equation.
More information2tdt 1 y = t2 + C y = which implies C = 1 and the solution is y = 1
Lectures - Week 11 General First Order ODEs & Numerical Methods for IVPs In general, nonlinear problems are much more difficult to solve than linear ones. Unfortunately many phenomena exhibit nonlinear
More informationRégularité des équations de Hamilton-Jacobi du premier ordre et applications aux jeux à champ moyen
Régularité des équations de Hamilton-Jacobi du premier ordre et applications aux jeux à champ moyen Daniela Tonon en collaboration avec P. Cardaliaguet et A. Porretta CEREMADE, Université Paris-Dauphine,
More informationNonlinear and Nonlocal Degenerate Diffusions on Bounded Domains
Nonlinear and Nonlocal Degenerate Diffusions on Bounded Domains Matteo Bonforte Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco 28049 Madrid, Spain matteo.bonforte@uam.es
More informationSome uniqueness results for the determination of forces acting over Germain-Lagrange plates
Some uniqueness results for the determination of forces acting over Germain-Lagrange plates Three different techniques A. Kawano Escola Politecnica da Universidade de Sao Paulo A. Kawano (Poli-USP) Germain-Lagrange
More informationThe Gaussian free field, Gibbs measures and NLS on planar domains
The Gaussian free field, Gibbs measures and on planar domains N. Burq, joint with L. Thomann (Nantes) and N. Tzvetkov (Cergy) Université Paris Sud, Laboratoire de Mathématiques d Orsay, CNRS UMR 8628 LAGA,
More information+ 2x sin x. f(b i ) f(a i ) < ɛ. i=1. i=1
Appendix To understand weak derivatives and distributional derivatives in the simplest context of functions of a single variable, we describe without proof some results from real analysis (see [7] and
More informationA generalised Ladyzhenskaya inequality and a coupled parabolic-elliptic problem
A generalised Ladyzhenskaya inequality and a coupled parabolic-elliptic problem Dave McCormick joint work with James Robinson and José Rodrigo Mathematics and Statistics Centre for Doctoral Training University
More informationA local estimate from Radon transform and stability of Inverse EIT with partial data
A local estimate from Radon transform and stability of Inverse EIT with partial data Alberto Ruiz Universidad Autónoma de Madrid U. California, Irvine.June 2012 w/ P. Caro (U. Helsinki) and D. Dos Santos
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 information