Stability and Instability of Standing Waves for the Nonlinear Fractional Schrödinger Equation. Shihui Zhu (joint with J. Zhang)
|
|
- Flora Wells
- 5 years ago
- Views:
Transcription
1 and of Standing Waves the Fractional Schrödinger Equation Shihui Zhu (joint with J. Zhang) Department of Mathematics, Sichuan Normal University & IMS, National University of Singapore
2 P1 iu t ( + k 2 ) s u + ( 1 x γ u 2 )u = 0, x R N, (1) u = u(t, x) : [0, T ) R N C, 0 < T + ; ( + k 2 ) s u = F 1 [( ξ 2 + k 2 ) s F[u](ξ)], s > 0; ( 1 x γ u 2 )(x) = u(y) 2 x y γ dy; 0 < γ < 4s. Boson star, fractional quantum mechanics etc.
3 Known P2 In particular, when N = 3, s = 1 2 Boson star equation. and γ = 1, Eq.(1) is the Fröhlich, Lenzmann, Lewin( ): well-posedness, orbital stability, singularity of blow-up solutions. Bao and Dong(2011): efficient and accurate numerical methods to compute the ground states and dynamics.
4 Known P3 For general 0 < s < 1, Laskin(2000,2002): expanding the Feynman path integral from the Brownian-like to the Lévylike quantum mechanical paths. Cho, Hwang, Hajaiej and Ozawa(2012): local well-posedness, existence of blow-up solutions. Guo, Huo, Huang( ): global well-posedness.
5 Known P4 Existence and stability of standing waves the classic nonlinear Schrödinger equation(i.e.s = 1) Strauss(1977): existence of standing waves. Cazenave and Lions(1982): orbital stability of standing waves. Weinstein(1983): strongly instability of standing waves. Existence and stability of standing waves the fourth-order nonlinear Schrödinger equation(i.e.s = 2) Zhu, Zhang and Yang(2010): existence of standing waves and ground state. Fibich, Baruch and Mandelbaum(2011): numerical studies standing waves. Levandosky(1998): orbital stability of standing waves.
6 Problems and Arguments P5 Problems 1. Existence of standing waves (1). 2. of these standing waves. Arguments Weinstein(1983): blow-up argument Gérard(1998) Hmidi & Keraani(2005): profile decomposition Cazenave and Lions(1982), Zhang(2000): variational argument
7 Local Well-posedness P6 Denote H s := {v S (R N ) (1 + ξ 2 ) s v(ξ) 2 dξ < + } and Impose E(v) := 1 2 v( ) s vdx 1 4 v( 1 x γ v 2 )vdx. u(0, x) = u 0 H s. (2) Proposition 1 (Cho, Hwang, Hajaiej and Ozawa, 2012) Let N 2, 0 < s < 1 and 0 < γ 2s. If u 0 H s, then! u(t, x) of (1)- (2) on I = [0, T ) such that u(t, x) C(I; H s ) C 1 (I; H s ). Moreover, all t I, u(t, x) satisfies the following conservation laws. (i) Conservation of mass u(t) 2 = u 0 2. (ii) Conservation of energy E(u(t)) = E(u 0)
8 Profile Decomposition P7 Proposition 2 Let N 2 and 0 < s < 1. If {v n} + n=1 is a bounded sequence in H s, then there exists a subsequence of {v n} + n=1 (still denoted {v n} + n=1 ), a family {xj n} + j=1 of sequences in RN and a family {V j } + j=1 of H s functions satisfying the following. (i) For every k j, x k n x j n + as n +. (ii) For every l 1 and every x R N, v n(x) can be decomposed as with lim l + v n(x) = Moreover, we have, as n +, l V j (x x j n) + vn(x) l j=1 lim sup vn l p = 0 every p (2, n + v n 2 2 = l V j vn l o(1), j=1 2N (N 2s) + ). l v n 2 Ḣ = V j 2 s Ḣ + vn l 2 s Ḣ + o(1). s j=1
9 Orbital of Standing Waves P8 Let k = 0, N 2 and M > 0. following variational problem For any 0 < γ < 2s, we define the d M := v(x) 2 v(y) 2 dxdy is the energy func- where E(v) := 1 2 ξ 2s v 2 dξ 1 4 tional. Define the set inf {v H s v 2 2 =M} E(v) (3) x y γ S M := {v H s v is the minimizer of the variational problem (3)}. From the Euler-Lagrange Theorem, we see that any v S M, there exists ω R such that (4) ( ) s v + ωv ( 1 x γ v 2 )v = 0, v H s.
10 Orbital of Standing Waves P9 Theorem 1 Let k = 0, N 2 and M > 0. Assume 0 < s < 1 and 0 < γ < 2s. Then arbitrary ε > 0, there exists δ > 0 such that any u 0 H s, if the initial data u 0 satisfies inf u 0 v H s < δ, v S M then the corresponding solution u(t, x) of the Cauchy problem (1)-(2) is such that inf u(t, x) v(x) H s < ε v S M all t > 0, where S M is defined in (4).
11 Sketch of the Proof of Theorem 1 P10 Key: The existence of minimizer of Variational Problem (3) i.e. Let N 2, 0 < γ < 2s and M > 0. Then, d M := inf E(v) = min E(v). {v H s v 2 2 =M} {v H s v 2 2 =M} Firstly, the variational problem (3) is well defined. Using the Hölder inequality with 1 = γ + 2s γ, we deduce that 2s 2s v(x) 2 x y dx γ C x s v(x + y) γ s 2 v 2s γ s 2 C v γ ṡ H s Thus, ( 1 v(x) x γ v(x) 2 ) v(x) 2 2 dx 2s γ s v 2. x y γ dx v(y) 2 2 C v γ ṡ H 4s γ s s v 2. From Young inequality, C(ε, γ, s, M) > 0 such that all 0 < ε < 1 2 E(v) 1 2 v 2 Ḣ C v 4s γ s s 2 v γ ṡ ( 1 H s 2 ε) v 2 Ḣ C(ε, γ, s, v 2). s (5)
12 Sketch of the Proof of Theorem 1 P11 Secondly, there exists C 0 > 0, d M C 0 < 0. Indeed, take v n = ρ N 2 n R(ρ nx), where ρ n > 0 and R is a function such that v n 2 2 = R 2 2 = M. We deduce that E(v n) = 1 2 ξ 2s v n 2 dξ 1 vn(x) 2 v n(y) 2 dxdy 4 x y γ ξ 2s R 2 dξ ργ n R(x) 2 R(y) 2 4 dxdy. = ρ2s n 2 x y γ Since 0 < γ < 2s, we can choose ρ n > 0 sufficiently small such that there exists C 0 > 0 such that E(v n) C 0 < 0.
13 Sketch of the Proof of Theorem 1 P12 Thirdly, we prove the existence of minimizers of (3). Take the minimizing sequence {v n} + n=1 Hs of (3) satisfying that as n +, E(v n) d M and v n 2 2 M. (6) Then, n large enough, E(v n) satisfies E(v n) < d M + 1. From (5), any 0 < ε < 1 and n 1 large enough, we deduce that 2 ( 1 2 ε) vn 2 Ḣ s d M C(ε, γ, s, M). Hence, {v n} + n=1 is bounded in Hs. Apply Proposition 2, we see that v n 2 2 = v n(x) = l V j (x x j n) + vn, l (7) j=1 l Vn j 2 2+ vn l 2 2+o(1), v n 2 Ḣ = s j=1 vn(x) 2 v n(y) 2 x y γ dxdy = l j=1 l Vn j 2 Ḣ + vn l 2 s Ḣ +o(1), s j=1 V j n (x) 2 V j n (y) 2 x y γ dxdy+ v l n (x) 2 x
14 Sketch of the Proof of Theorem 1 P13 Thus, we have E(v n) = l E(V j (x x j n)) + E(vn) l + o(1). (8) j=1 Take the scaling transmation V j ρ j = ρ jv j (x x j n) with ρ j = M V j (x x j n) 2 1. We have Vρ j j 2 2 = M. We deduce that as n + and l + d M E(v n) = l ( E(V j ρj ) + ρ2 j 1 ρ 2 4 j=1 j l d M ρ + inf 2 j 1 ρ 2 j j 1 4 j=1 d M + C 0 ( M V j ( V j (x x j n ) 2 V j (y x j n ) 2 l j=1 ) 1 + o(1), x y γ ) dxdy V j (x x j n ) 2 V j (y x j n ) 2 dxdy x y γ + E(vn) l + o(1 ) + vl n 2 2 M dm where C 0 > 0. Theree, there exists only one term V j0 0 in the decomposition (7) such that V j = M. V j 0 is the minimizer of (3).
15 Strong of Standing Waves P14 Theorem 2 Let k = 0 and N 2. Assume 0 < γ 2 = s < 1. Then, the ground state solitary waves e iωt Q(x) of the fractional nonlinear Schrödinger equation (1) are unstable in the following sense: For arbitrary ε > 0, there exist the radial initial data {u 0,n} + n=1 Hs 0 with s 0 = max{2s, 2s+1 2 } satisfying x u 0,n L 2, x u 0,n L 2, u 0,n Q H s < ε (9) and the corresponding solution {u n(t, x)} + n=1 blows up in the finite time, where Q is the ground state solution of ( ) s 1 Q + Q ( x 2s Q 2 )Q = 0. (10)
16 Sketch of the Proof of Theorem 2 P15 Proposition 3 Let N 2 and 0 < s < 1. Then, v 2 dx 2 N Proof. ( For all v H s, we have ) 1 vx( ) 1 s 2 ( xvdx v( ) s vdx) 1 2, v H s. v 2 dx = 2 N = 2 N 2 N 2 N F[ (xv)]f 1 [v]dξ F[v] ξ ξ F[v]dξ ξ s F[v] ξ 1 s ξ F[v] dξ ( v( ) s vdx ) 1 2 ( vx( ) 1 s xvdx ) 1 2.
17 Sketch of the Proof of Theorem 2 P16 Proposition 4 (see [Cho, Hwang, Kwon and Lee, 2012]) Let k = 0, N 2, 0 < s < 1 and γ = 2s. Assume that u 0 H s 0 with s 0 = max{2s, γ+1 2 } is radial symmetric, and x u 0 L 2 and x u 0 L 2. If u(t, x) is the solution of the Cauchy problem (1)-(2), then all t I (the maximal time interval), ux( ) 1 s udx is nonnegative and ux( ) 1 s xudx 2sE(u 0)t 2 + Ct + C.
18 Sketch of the Proof of Theorem 2 P17 Let {c n} + n=1 C\{0} be such that cn > 1 and lim cn = 1, and n + ρn = 1. We take the initial data {ρ n} + n=1 R+ be such that lim n + u 0,n = c nρ N 2 n Q(ρ nx), where Q is the ground state solution of (10). We see that all n 1, u 0,n 2 > Q 2 and and lim u 0,n 2 = lim cn Q 2 = Q 2 n + n + lim u 0,n Ḣs = lim cn n + n + ρs n Q Ḣs = Q Ḣs. Hence, from the Brézis-Lieb Lemma, ε > 0, u 0,n Q H s sufficiently large. On the other hand, we see that < ε as n is E(u 0,n ) = ( cn 2 c n 4 )ρ 2s n Q 2 2 Ḣ s C 0 < 0.
19 Sketch of the Proof of Theorem 2 P18 Then, u nx( ) 1 s xu ndx 2sC 0 t 2 + Ct + C, which implies that there exists 0 < T < + such that lim t T u nx( ) 1 s xu ndx = 0. Finally, using the conservation of mass and applying the inequality in Proposition 3 to u n, we see that all time t u 0,n 2 2 = un(t) ( N unx( ) 1 s xu ) ndx 1 2 ( u n( ) s u ndx) ( N unx( ) 1 s xu ) ndx 1 2 u n(t) Ḣs. Theree, there exists 0 < T < + such that lim un(t) t T Ḣ s = +.
20 Thanks!
Sharp blow-up criteria for the Davey-Stewartson system in R 3
Dynamics of PDE, Vol.8, No., 9-60, 011 Sharp blow-up criteria for the Davey-Stewartson system in R Jian Zhang Shihui Zhu Communicated by Y. Charles Li, received October 7, 010. Abstract. In this paper,
More informationBLOWUP THEORY FOR THE CRITICAL NONLINEAR SCHRÖDINGER EQUATIONS REVISITED
BLOWUP THEORY FOR THE CRITICAL NONLINEAR SCHRÖDINGER EQUATIONS REVISITED TAOUFIK HMIDI AND SAHBI KERAANI Abstract. In this note we prove a refined version of compactness lemma adapted to the blowup analysis
More informationOn the relation between scaling properties of functionals and existence of constrained minimizers
On the relation between scaling properties of functionals and existence of constrained minimizers Jacopo Bellazzini Dipartimento di Matematica Applicata U. Dini Università di Pisa January 11, 2011 J. Bellazzini
More informationUniqueness of ground state solutions of non-local equations in R N
Uniqueness of ground state solutions of non-local equations in R N Rupert L. Frank Department of Mathematics Princeton University Joint work with Enno Lenzmann and Luis Silvestre Uniqueness and non-degeneracy
More informationScattering for cubic-quintic nonlinear Schrödinger equation on R 3
Scattering for cubic-quintic nonlinear Schrödinger equation on R 3 Oana Pocovnicu Princeton University March 9th 2013 Joint work with R. Killip (UCLA), T. Oh (Princeton), M. Vişan (UCLA) SCAPDE UCLA 1
More informationarxiv: v3 [math.ap] 12 Mar 2014
arxiv:108.30v3 [math.ap] 1 Mar 014 ON FINITE TIME BLOWUP FOR THE MASS-CRITICAL HARTREE EQUATIONS YONGGEUN CHO DEPARTMENT OF MATHEMATICS, AND INSTITUTE OF PURE AND APPLIED MATHEMATICS, CHONBUK NATIONAL
More informationOn the Brezis and Mironescu conjecture concerning a Gagliardo-Nirenberg inequality for fractional Sobolev norms
On the Brezis and Mironescu conjecture concerning a Gagliardo-Nirenberg inequality for fractional Sobolev norms Vladimir Maz ya Tatyana Shaposhnikova Abstract We prove the Gagliardo-Nirenberg type inequality
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 informationMultisolitons for NLS
Multisolitons for NLS Stefan LE COZ Beijing 2007-07-03 Plan 1 Introduction 2 Existence of multi-solitons 3 (In)stability NLS (NLS) { iut + u + g( u 2 )u = 0 u t=0 = u 0 u : R t R d x C (A0) (regular) g
More informationAn introduction to semilinear elliptic equations
An introduction to semilinear elliptic equations Thierry Cazenave Laboratoire Jacques-Louis Lions UMR CNRS 7598 B.C. 187 Université Pierre et Marie Curie 4, place Jussieu 75252 Paris Cedex 05 France E-mail
More informationAn introduction to semilinear elliptic equations
An introduction to semilinear elliptic equations Thierry Cazenave Laboratoire Jacques-Louis Lions UMR CNRS 7598 B.C. 187 Université Pierre et Marie Curie 4, place Jussieu 75252 Paris Cedex 05 France E-mail
More informationarxiv:math/ v3 [math.ap] 29 Aug 2005
arxiv:math/0505456v3 [math.ap] 29 Aug 2005 Well-Posedness for Semi-Relativistic Hartree Equations of Critical Type Enno Lenzmann Department of Mathematics, ETH Zürich E-Mail: lenzmann@math.ethz.ch August
More informationOn critical systems involving fractional Laplacian
J. Math. Anal. Appl. 446 017 681 706 Contents lists available at ScienceDirect Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa On critical systems involving fractional Laplacian
More informationThe blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation
Annals of Mathematics, 6 005, 57 The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation By Frank Merle and Pierre Raphael Abstract We consider the critical
More informationExistence of a ground state and blow-up problem for a nonlinear Schrödinger equation with critical growth
Existence of a ground state and blow-up problem for a nonlinear Schrödinger equation with critical growth Takafumi Akahori, Slim Ibrahim, Hiroaki Kikuchi and Hayato Nawa 1 Introduction In this paper, we
More informationSome lecture notes for Math 6050E: PDEs, Fall 2016
Some lecture notes for Math 65E: PDEs, Fall 216 Tianling Jin December 1, 216 1 Variational methods We discuss an example of the use of variational methods in obtaining existence of solutions. Theorem 1.1.
More informationWell-Posedness and Adiabatic Limit for Quantum Zakharov System
Well-Posedness and Adiabatic Limit for Quantum Zakharov System Yung-Fu Fang (joint work with Tsai-Jung Chen, Jun-Ichi Segata, Hsi-Wei Shih, Kuan-Hsiang Wang, Tsung-fang Wu) Department of Mathematics National
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 informationGlobal well-posedness for semi-linear Wave and Schrödinger equations. Slim Ibrahim
Global well-posedness for semi-linear Wave and Schrödinger equations Slim Ibrahim McMaster University, Hamilton ON University of Calgary, April 27th, 2006 1 1 Introduction Nonlinear Wave equation: ( 2
More informationLaplace s Equation. Chapter Mean Value Formulas
Chapter 1 Laplace s Equation Let be an open set in R n. A function u C 2 () is called harmonic in if it satisfies Laplace s equation n (1.1) u := D ii u = 0 in. i=1 A function u C 2 () is called subharmonic
More informationSolutions with prescribed mass for nonlinear Schrödinger equations
Solutions with prescribed mass for nonlinear Schrödinger equations Dario Pierotti Dipartimento di Matematica, Politecnico di Milano (ITALY) Varese - September 17, 2015 Work in progress with Gianmaria Verzini
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 informationSCATTERING FOR THE NON-RADIAL 3D CUBIC NONLINEAR SCHRÖDINGER EQUATION
SCATTERING FOR THE NON-RADIAL 3D CUBIC NONLINEAR SCHRÖDINGER EQUATION THOMAS DUYCKAERTS, JUSTIN HOLMER, AND SVETLANA ROUDENKO Abstract. Scattering of radial H 1 solutions to the 3D focusing cubic nonlinear
More informationThe Brézis-Nirenberg Result for the Fractional Elliptic Problem with Singular Potential
arxiv:1705.08387v1 [math.ap] 23 May 2017 The Brézis-Nirenberg Result for the Fractional Elliptic Problem with Singular Potential Lingyu Jin, Lang Li and Shaomei Fang Department of Mathematics, South China
More informationPresenter: Noriyoshi Fukaya
Y. Martel, F. Merle, and T.-P. Tsai, Stability and Asymptotic Stability in the Energy Space of the Sum of N Solitons for Subcritical gkdv Equations, Comm. Math. Phys. 31 (00), 347-373. Presenter: Noriyoshi
More informationNOTES ON SCHAUDER ESTIMATES. r 2 x y 2
NOTES ON SCHAUDER ESTIMATES CRISTIAN E GUTIÉRREZ JULY 26, 2005 Lemma 1 If u f in B r y), then ux) u + r2 x y 2 B r y) B r y) f, x B r y) Proof Let gx) = ux) Br y) u r2 x y 2 Br y) f We have g = u + Br
More informationExponential Energy Decay for the Kadomtsev-Petviashvili (KP-II) equation
São Paulo Journal of Mathematical Sciences 5, (11), 135 148 Exponential Energy Decay for the Kadomtsev-Petviashvili (KP-II) equation Diogo A. Gomes Department of Mathematics, CAMGSD, IST 149 1 Av. Rovisco
More informationEnergy transfer model and large periodic boundary value problem for the quintic NLS
Energy transfer model and large periodic boundary value problem for the quintic NS Hideo Takaoka Department of Mathematics, Kobe University 1 ntroduction This note is based on a talk given at the conference
More informationINFINITELY MANY SOLUTIONS FOR A CLASS OF SUBLINEAR SCHRÖDINGER EQUATIONS
Journal of Applied Analysis and Computation Volume 8, Number 5, October 018, 1475 1493 Website:http://jaac-online.com/ DOI:10.11948/018.1475 INFINITELY MANY SOLUTIONS FOR A CLASS OF SUBLINEAR SCHRÖDINGER
More informationLECTURE NOTES : INTRODUCTION TO DISPERSIVE PARTIAL DIFFERENTIAL EQUATIONS
LECTURE NOTES : INTRODUCTION TO DISPERSIVE PARTIAL DIFFERENTIAL EQUATIONS NIKOLAOS TZIRAKIS Abstract. The aim of this manuscript is to provide a short and accessible introduction to the modern theory of
More informationNonlinear Schrödinger Equation BAOXIANG WANG. Talk at Tsinghua University 2012,3,16. School of Mathematical Sciences, Peking University.
Talk at Tsinghua University 2012,3,16 Nonlinear Schrödinger Equation BAOXIANG WANG School of Mathematical Sciences, Peking University 1 1 33 1. Schrödinger E. Schrödinger (1887-1961) E. Schrödinger, (1887,
More informationSCATTERING FOR THE TWO-DIMENSIONAL NLS WITH EXPONENTIAL NONLINEARITY
SCATTERING FOR THE TWO-DIMENSIONAL NLS WITH EXPONENTIAL NONLINEARITY S. IBRAHIM, M. MAJDOUB, N. MASMOUDI, AND K. NAKANISHI Abstract. We investigate existence and asymptotic completeness of the wave operators
More informationA CONVEX-CONCAVE ELLIPTIC PROBLEM WITH A PARAMETER ON THE BOUNDARY CONDITION
A CONVEX-CONCAVE ELLIPTIC PROBLEM WITH A PARAMETER ON THE BOUNDARY CONDITION JORGE GARCÍA-MELIÁN, JULIO D. ROSSI AND JOSÉ C. SABINA DE LIS Abstract. In this paper we study existence and multiplicity of
More informationRegularity of Weak Solution to Parabolic Fractional p-laplacian
Regularity of Weak Solution to Parabolic Fractional p-laplacian Lan Tang at BCAM Seminar July 18th, 2012 Table of contents 1 1. Introduction 1.1. Background 1.2. Some Classical Results for Local Case 2
More informationGROUND STATES FOR FRACTIONAL KIRCHHOFF EQUATIONS WITH CRITICAL NONLINEARITY IN LOW DIMENSION
GROUND STATES FOR FRACTIONAL KIRCHHOFF EQUATIONS WITH CRITICAL NONLINEARITY IN LOW DIMENSION ZHISU LIU, MARCO SQUASSINA, AND JIANJUN ZHANG Abstract. We study the existence of ground states to a nonlinear
More informationA Variational Analysis of a Gauged Nonlinear Schrödinger Equation
A Variational Analysis of a Gauged Nonlinear Schrödinger Equation Alessio Pomponio, joint work with David Ruiz Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari Variational and Topological
More informationM ath. Res. Lett. 15 (2008), no. 6, c International Press 2008 SCATTERING FOR THE NON-RADIAL 3D CUBIC NONLINEAR SCHRÖDINGER EQUATION
M ath. Res. Lett. 15 (2008), no. 6, 1233 1250 c International Press 2008 SCATTERING FOR THE NON-RADIAL 3D CUBIC NONLINEAR SCHRÖDINGER EQUATION Thomas Duyckaerts, Justin Holmer, and Svetlana Roudenko Abstract.
More informationThe Singular Diffusion Equation with Boundary Degeneracy
The Singular Diffusion Equation with Boundary Degeneracy QINGMEI XIE Jimei University School of Sciences Xiamen, 36 China xieqingmei@63com HUASHUI ZHAN Jimei University School of Sciences Xiamen, 36 China
More informationRecent developments on the global behavior to critical nonlinear dispersive equations. Carlos E. Kenig
Recent developments on the global behavior to critical nonlinear dispersive equations Carlos E. Kenig In the last 25 years or so, there has been considerable interest in the study of non-linear partial
More informationNecessary Conditions and Sufficient Conditions for Global Existence in the Nonlinear Schrödinger Equation
Necessary Conditions and Sufficient Conditions for Global Existence in the Nonlinear Schrödinger Equation Pascal Bégout aboratoire Jacques-ouis ions Université Pierre et Marie Curie Boîte Courrier 187,
More informationEXISTENCE OF NONTRIVIAL SOLUTIONS FOR A QUASILINEAR SCHRÖDINGER EQUATIONS WITH SIGN-CHANGING POTENTIAL
Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 05, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE OF NONTRIVIAL
More informationVariational Theory of Solitons for a Higher Order Generalized Camassa-Holm Equation
International Journal of Mathematical Analysis Vol. 11, 2017, no. 21, 1007-1018 HIKAI Ltd, www.m-hikari.com https://doi.org/10.12988/ijma.2017.710141 Variational Theory of Solitons for a Higher Order Generalized
More informationOn Chern-Simons-Schrödinger equations including a vortex point
On Chern-Simons-Schrödinger equations including a vortex point Alessio Pomponio Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari Workshop in Nonlinear PDEs Brussels, September 7
More informationarxiv: v1 [math.oc] 12 Nov 2018
EXTERNAL OPTIMAL CONTROL OF NONLOCAL PDES HARBIR ANTIL, RATNA KHATRI, AND MAHAMADI WARMA arxiv:1811.04515v1 [math.oc] 12 Nov 2018 Abstract. Very recently Warma [35] has shown that for nonlocal PDEs associated
More informationFractional Cahn-Hilliard equation
Fractional Cahn-Hilliard equation Goro Akagi Mathematical Institute, Tohoku University Abstract In this note, we review a recent joint work with G. Schimperna and A. Segatti (University of Pavia, Italy)
More informationOn the minimum of certain functional related to the Schrödinger equation
Electronic Journal of Qualitative Theory of Differential Equations 2013, No. 8, 1-21; http://www.math.u-szeged.hu/ejqtde/ On the minimum of certain functional related to the Schrödinger equation Artūras
More informationSolutions to the Nonlinear Schrödinger Equation in Hyperbolic Space
Solutions to the Nonlinear Schrödinger Equation in Hyperbolic Space SPUR Final Paper, Summer 2014 Peter Kleinhenz Mentor: Chenjie Fan Project suggested by Gigliola Staffilani July 30th, 2014 Abstract In
More informationSharp energy estimates and 1D symmetry for nonlinear equations involving fractional Laplacians
Sharp energy estimates and 1D symmetry for nonlinear equations involving fractional Laplacians Eleonora Cinti Università degli Studi di Bologna - Universitat Politécnica de Catalunya (joint work with Xavier
More informationComm. Nonlin. Sci. and Numer. Simul., 12, (2007),
Comm. Nonlin. Sci. and Numer. Simul., 12, (2007), 1390-1394. 1 A Schrödinger singular perturbation problem A.G. Ramm Mathematics Department, Kansas State University, Manhattan, KS 66506-2602, USA ramm@math.ksu.edu
More informationGROUND STATES FOR SUPERLINEAR FRACTIONAL SCHRÖDINGER EQUATIONS IN R N
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 41, 2016, 745 756 GROUND STATES FOR SUPERLINEAR FRACTIONAL SCHRÖDINGER EQUATIONS IN Vincenzo Ambrosio Università degli Studi di Napoli Federico
More informationScattering theory for nonlinear Schrödinger equation with inverse square potential
Scattering theory for nonlinear Schrödinger equation with inverse square potential Université Nice Sophia-Antipolis Based on joint work with: Changxing Miao (IAPCM) and Junyong Zhang (BIT) February -6,
More informationNonlinear Schrödinger equation with combined power-type nonlinearities and harmonic potential
Appl. Math. Mech. -Engl. Ed. 31(4), 521 528 (2010) DOI 10.1007/s10483-010-0412-7 c Shanghai University and Springer-Verlag Berlin Heidelberg 2010 Applied Mathematics and Mechanics (English Edition) Nonlinear
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 informationNon-radial solutions to a bi-harmonic equation with negative exponent
Non-radial solutions to a bi-harmonic equation with negative exponent Ali Hyder Department of Mathematics, University of British Columbia, Vancouver BC V6TZ2, Canada ali.hyder@math.ubc.ca Juncheng Wei
More informationFrom the N-body problem to the cubic NLS equation
From the N-body problem to the cubic NLS equation François Golse Université Paris 7 & Laboratoire J.-L. Lions golse@math.jussieu.fr Los Alamos CNLS, January 26th, 2005 Formal derivation by N.N. Bogolyubov
More informationAbstract In this paper, we consider bang-bang property for a kind of timevarying. time optimal control problem of null controllable heat equation.
JOTA manuscript No. (will be inserted by the editor) The Bang-Bang Property of Time-Varying Optimal Time Control for Null Controllable Heat Equation Dong-Hui Yang Bao-Zhu Guo Weihua Gui Chunhua Yang Received:
More informationMath The Laplacian. 1 Green s Identities, Fundamental Solution
Math. 209 The Laplacian Green s Identities, Fundamental Solution Let be a bounded open set in R n, n 2, with smooth boundary. The fact that the boundary is smooth means that at each point x the external
More informationEffective dynamics of many-body quantum systems
Effective dynamics of many-body quantum systems László Erdős University of Munich Grenoble, May 30, 2006 A l occassion de soixantiéme anniversaire de Yves Colin de Verdiére Joint with B. Schlein and H.-T.
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 informationFriedrich symmetric systems
viii CHAPTER 8 Friedrich symmetric systems In this chapter, we describe a theory due to Friedrich [13] for positive symmetric systems, which gives the existence and uniqueness of weak solutions of boundary
More informationGlobal unbounded solutions of the Fujita equation in the intermediate range
Global unbounded solutions of the Fujita equation in the intermediate range Peter Poláčik School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA Eiji Yanagida Department of Mathematics,
More informationExistence and stability of solitary-wave solutions to nonlocal equations
Existence and stability of solitary-wave solutions to nonlocal equations Mathias Nikolai Arnesen Norwegian University of Science and Technology September 22nd, Trondheim The equations u t + f (u) x (Lu)
More informationInvariant measures and the soliton resolution conjecture
Invariant measures and the soliton resolution conjecture Stanford University The focusing nonlinear Schrödinger equation A complex-valued function u of two variables x and t, where x R d is the space variable
More informationAppearance of Anomalous Singularities in. a Semilinear Parabolic Equation. (Tokyo Institute of Technology) with Shota Sato (Tohoku University)
Appearance of Anomalous Singularities in a Semilinear Parabolic Equation Eiji Yanagida (Tokyo Institute of Technology) with Shota Sato (Tohoku University) 4th Euro-Japanese Workshop on Blow-up, September
More informationGROUND STATE SOLUTIONS FOR CHOQUARD TYPE EQUATIONS WITH A SINGULAR POTENTIAL. 1. Introduction In this article, we study the Choquard type equation
Electronic Journal of Differential Equations, Vol. 2017 (2017), o. 52, pp. 1 14. ISS: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu GROUD STATE SOLUTIOS FOR CHOQUARD TYPE EQUATIOS
More informationThe Almost Everywhere Convergence of Eigenfunction Expansions of Schrödinger Operator in L p Classes.
Malaysian Journal of Mathematical Sciences (S) April : 9 36 (7) Special Issue: The nd International Conference and Workshop on Mathematical Analysis (ICWOMA 6) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES
More informationGROUND-STATES FOR THE LIQUID DROP AND TFDW MODELS WITH LONG-RANGE ATTRACTION
GROUND-STATES FOR THE LIQUID DROP AND TFDW MODELS WITH LONG-RANGE ATTRACTION STAN ALAMA, LIA BRONSARD, RUSTUM CHOKSI, AND IHSAN TOPALOGLU Abstract. We prove that both the liquid drop model in with an attractive
More informationPara el cumpleaños del egregio profesor Ireneo Peral
On two coupled nonlinear Schrödinger equations Para el cumpleaños del egregio profesor Ireneo Peral Dipartimento di Matematica Sapienza Università di Roma Salamanca 13.02.2007 Coauthors Luca Fanelli (Sapienza
More informationEXISTENCE OF SOLUTIONS TO ASYMPTOTICALLY PERIODIC SCHRÖDINGER EQUATIONS
Electronic Journal of Differential Equations, Vol. 017 (017), No. 15, pp. 1 7. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS TO ASYMPTOTICALLY PERIODIC
More informationInstitut für Mathematik
RHEINISCH-WESTFÄLISCHE TECHNISCHE HOCHSCHULE AACHEN Institut für Mathematik Travelling Wave Solutions of the Heat Equation in Three Dimensional Cylinders with Non-Linear Dissipation on the Boundary by
More informationBielefeld Course on Nonlinear Waves - June 29, Department of Mathematics University of North Carolina, Chapel Hill. Solitons on Manifolds
Joint work (on various projects) with Pierre Albin (UIUC), Hans Christianson (UNC), Jason Metcalfe (UNC), Michael Taylor (UNC), Laurent Thomann (Nantes) Department of Mathematics University of North Carolina,
More informationConstruction of concentrating bubbles for the energy-critical wave equation
Construction of concentrating bubbles for the energy-critical wave equation Jacek Jendrej University of Chicago University of North Carolina February 13th 2017 Jacek Jendrej Energy-critical wave equations
More informationu( x) = g( y) ds y ( 1 ) U solves u = 0 in U; u = 0 on U. ( 3)
M ath 5 2 7 Fall 2 0 0 9 L ecture 4 ( S ep. 6, 2 0 0 9 ) Properties and Estimates of Laplace s and Poisson s Equations In our last lecture we derived the formulas for the solutions of Poisson s equation
More informationEXISTENCE OF SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS WITH UNBOUNDED POTENTIAL. 1. Introduction In this article, we consider the Kirchhoff type problem
Electronic Journal of Differential Equations, Vol. 207 (207), No. 84, pp. 2. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS
More informationStationary Kirchhoff equations with powers by Emmanuel Hebey (Université de Cergy-Pontoise)
Stationary Kirchhoff equations with powers by Emmanuel Hebey (Université de Cergy-Pontoise) Lectures at the Riemann center at Varese, at the SNS Pise, at Paris 13 and at the university of Nice. June 2017
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 informationGLOBAL EXISTENCE AND ENERGY DECAY OF SOLUTIONS TO A PETROVSKY EQUATION WITH GENERAL NONLINEAR DISSIPATION AND SOURCE TERM
Georgian Mathematical Journal Volume 3 (26), Number 3, 397 4 GLOBAL EXITENCE AND ENERGY DECAY OF OLUTION TO A PETROVKY EQUATION WITH GENERAL NONLINEAR DIIPATION AND OURCE TERM NOUR-EDDINE AMROUN AND ABBE
More informationSolution Sheet 3. Solution Consider. with the metric. We also define a subset. and thus for any x, y X 0
Solution Sheet Throughout this sheet denotes a domain of R n with sufficiently smooth boundary. 1. Let 1 p
More informationScattering for the NLS equation
Scattering for the NLS equation joint work with Thierry Cazenave (UPMC) Ivan Naumkin Université Nice Sophia Antipolis February 2, 2017 Introduction. Consider the nonlinear Schrödinger equation with the
More informationORBITAL STABILITY OF SOLITARY WAVES FOR A 2D-BOUSSINESQ SYSTEM
Electronic Journal of Differential Equations, Vol. 05 05, No. 76, pp. 7. ISSN: 07-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu ORBITAL STABILITY OF SOLITARY
More informationGlobal Well-Posedness for Energy Critical Fourth-Order Schrödinger Equations in the Radial Case
Dynamics of PDE, Vol.4, No.3, 197-5, 007 Global Well-Posedness for Energy Critical Fourth-Order Schrödinger Equations in the Radial Case Benoit Pausader Communicated by Terence Tao, received May 31, 007,
More informationIMPROVED SOBOLEV EMBEDDINGS, PROFILE DECOMPOSITION, AND CONCENTRATION-COMPACTNESS FOR FRACTIONAL SOBOLEV SPACES
IMPROVED SOBOLEV EMBEDDINGS, PROFILE DECOMPOSITION, AND CONCENTRATION-COMPACTNESS FOR FRACTIONAL SOBOLEV SPACES GIAMPIERO PALATUCCI AND ADRIANO PISANTE Abstract. We obtain an improved Sobolev inequality
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 information2 A Model, Harmonic Map, Problem
ELLIPTIC SYSTEMS JOHN E. HUTCHINSON Department of Mathematics School of Mathematical Sciences, A.N.U. 1 Introduction Elliptic equations model the behaviour of scalar quantities u, such as temperature or
More informationOPTIMAL POTENTIALS FOR SCHRÖDINGER OPERATORS. 1. Introduction In this paper we consider optimization problems of the form. min F (V ) : V V, (1.
OPTIMAL POTENTIALS FOR SCHRÖDINGER OPERATORS G. BUTTAZZO, A. GEROLIN, B. RUFFINI, AND B. VELICHKOV Abstract. We consider the Schrödinger operator + V (x) on H 0 (), where is a given domain of R d. Our
More informationREMARKS ON THE VANISHING OBSTACLE LIMIT FOR A 3D VISCOUS INCOMPRESSIBLE FLUID
REMARKS ON THE VANISHING OBSTACLE LIMIT FOR A 3D VISCOUS INCOMPRESSIBLE FLUID DRAGOŞ IFTIMIE AND JAMES P. KELLIHER Abstract. In [Math. Ann. 336 (2006), 449-489] the authors consider the two dimensional
More informationOn semilinear elliptic equations with measure data
On semilinear elliptic equations with measure data Andrzej Rozkosz (joint work with T. Klimsiak) Nicolaus Copernicus University (Toruń, Poland) Controlled Deterministic and Stochastic Systems Iasi, July
More informationGLOBAL WELL-POSEDNESS FOR THE NONLINEAR SCHRÖDINGER EQUATION WITH DERIVATIVE IN ENERGY SPACE
GLOBAL WELL-POSEDNESS FO THE NONLINEA SCHÖDINGE EQUATION WITH DEIVATIVE IN ENEGY SPACE YIFEI WU arxiv:131.7166v2 [math.ap] 17 Apr 214 Abstract. In this paper, we prove that there exists some small ε >,
More informationUNIQUENESS OF POSITIVE SOLUTION TO SOME COUPLED COOPERATIVE VARIATIONAL ELLIPTIC SYSTEMS
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9947(XX)0000-0 UNIQUENESS OF POSITIVE SOLUTION TO SOME COUPLED COOPERATIVE VARIATIONAL ELLIPTIC SYSTEMS YULIAN
More informationWELL-POSEDNESS OF WEAK SOLUTIONS TO ELECTRORHEOLOGICAL FLUID EQUATIONS WITH DEGENERACY ON THE BOUNDARY
Electronic Journal of Differential Equations, Vol. 2017 (2017), No. 13, pp. 1 15. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu WELL-POSEDNESS OF WEAK SOLUTIONS TO ELECTRORHEOLOGICAL
More informationThe Navier Stokes Equations for Incompressible Flows: Solution Properties at Potential Blow Up Times
The Navier Stokes Equations for Incompressible Flows: Solution Properties at Potential Blow Up Times Jens Lorenz Department of Mathematics and Statistics, UNM, Albuquerque, NM 873 Paulo Zingano Dept. De
More informationElliptic PDEs of 2nd Order, Gilbarg and Trudinger
Elliptic PDEs of 2nd Order, Gilbarg and Trudinger Chapter 2 Laplace Equation Yung-Hsiang Huang 207.07.07. Mimic the proof for Theorem 3.. 2. Proof. I think we should assume u C 2 (Ω Γ). Let W be an open
More informationHigher order asymptotic analysis of the nonlinear Klein- Gordon equation in the non-relativistic limit regime
Higher order asymptotic analysis of the nonlinear Klein- Gordon equation in the non-relativistic limit regime Yong Lu and Zhifei Zhang Abstract In this paper, we study the asymptotic behavior of the Klein-Gordon
More informationModeling, Analysis and Simulation for Degenerate Dipolar Quantum Gas
Modeling, Analysis and Simulation for Degenerate Dipolar Quantum Gas Weizhu Bao Department of Mathematics National University of Singapore Email: matbaowz@nus.edu.sg URL: http://www.math.nus.edu.sg/~bao
More informationMulti-dimensional Stochastic Singular Control Via Dynkin Game and Dirichlet Form
Multi-dimensional Stochastic Singular Control Via Dynkin Game and Dirichlet Form Yipeng Yang * Under the supervision of Dr. Michael Taksar Department of Mathematics University of Missouri-Columbia Oct
More informationStability of solitary waves for nonlinear Schrödinger equations with inhomogeneous nonlinearities
Physica D 175 3) 96 18 Stability of solitary waves for nonlinear Schrödinger equations with inhomogeneous nonlinearities Gadi Fibich a, Xiao-Ping Wang b, a School of Mathematical Sciences, Tel Aviv University,
More informationThe infinity-laplacian and its properties
U.U.D.M. Project Report 2017:40 Julia Landström Examensarbete i matematik, 15 hp Handledare: Kaj Nyström Examinator: Martin Herschend December 2017 Department of Mathematics Uppsala University Department
More informationDerivation of Mean-Field Dynamics for Fermions
IST Austria Mathematical Many Body Theory and its Applications Bilbao, June 17, 2016 Derivation = show that solution of microscopic eq. is close to solution of effective eq. (with fewer degrees of freedom)
More informationExistence of ground states for fourth-order wave equations
Accepted Manuscript Existence of ground states for fourth-order wave equations Paschalis Karageorgis, P.J. McKenna PII: S0362-546X(10)00167-7 DOI: 10.1016/j.na.2010.03.025 Reference: NA 8226 To appear
More informationRegularity of the p-poisson equation in the plane
Regularity of the p-poisson equation in the plane Erik Lindgren Peter Lindqvist Department of Mathematical Sciences Norwegian University of Science and Technology NO-7491 Trondheim, Norway Abstract We
More information