Energy transfer model and large periodic boundary value problem for the quintic NLS

Size: px
Start display at page:

Download "Energy transfer model and large periodic boundary value problem for the quintic NLS"

Transcription

1 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 on Nonlinear Wave and Dispersive Equations, Kyoto University. We consider a dynamics of mass density û(t, ξ and energy exchanges between a linear oscillator and a nonlinear interaction for the following defocusing quintic NS equation: i t u + xu = u 4 u, t R, x T = [0, π], (1.1 where u = u(t, x : R T C is a complex-valued function and the spatial domain T is taken to be a torus of length π > 0, i.e., we assume the periodic boundary condition: u(t, x + π = u(t, x. The aim is to understand the dynamics of mass density û(t, ξ, namely, (i how the wave energy is exchanged to another, (ii provide a demonstration of the conservative energy exchange of solutions between the modes initially excited. The equation (1.1 possesses at least two conservation laws, the mass M[u](t and energy E[u](t: M[u](t = u(t, E[u](t = T 1 xu(t, x u(t, x 6 dx = E[u](0. These quantities impose the constraints on a dynamics of mass density of solutions. We expect the following conjecture. This work was supported by JSPS KAKENH Grant Number

2 Conjecture 1.1. On bounded domain case: if the nonlinear Schrödinger equation is not integrable, then there exists a time global smooth solution u(t and some Sobolev exponent s > 1 such that lim t u(t H s =. Remark 1.1. The above conjecture means that the mass is shifted to high frequencies. n fact, on R domain case, the local well-posedness for (1.1 was proved by Cazenave and Weissler [3] for data in (see also [8] and [1]. Notice that the time of existence time depends on the position of data and not only on its size. One can also prove the global well-posedness in provided that the initial data in is sufficiently small by using the above conservation laws. t is should be noted that the equation (1.1 is left invariant by the scaling u u λ ; u(t, x u λ (t, x = λ 1/ u(λ t, λx, λ > 0, which preserves the homogeneous Sobolev norm Ḣs (R with s = 0. The global existence result for any data in was proved by Dodson [6]. n [6], the deep results on scattering behavior of solutions were also obtained. On the other hand, the associated focusing nonlinear Schrödinger equation (the minus sign applies to the nonlinear term has a finite time blow-up solution [9]. Past works on the periodic boundary domain When the spatial dimension is the two-dimensional torus T, Bourgain [] considered the cubic nonlinear Schrödinger equation in the defocusing case and obtained the apriori estimate of solutions i t u + u = u u, u(t H s t (s 1+ for u(0 H s, s 4. n [5], Colliander, Keel, Staffilani, Takaoka and Tao constructed the solution satisfying that for any s > 1, K 1, 0 < σ < 1 there exist a solution u(t and a time T > 0 such that u(0 H s σ and u(t H s K. Observe that the cubic nonlinear Schrödinger equation in two spatial dimensions is known as an example of invariant under the -scaling: u u λ = λu(λ t, λx (λ > 0. On the other hand, the quintic nonlinear Schrödinger equation in one dimension obeys scale invariance under the -scaling. n [10], Grébert and. Thomann examined the dynamics exhibited by the following Cauchy problem i t u + xu = ν u 4 u, (t, x R T, (.1 for ν > 0. More precisely, they proved the following theorem.

3 Theorem.1 (Grébert and Thomann [10]. et k Z\{0} and n Z. A is a set of the form A = {a, a 1, b, b 1 } where a = n, a 1 = n + 3k, b = n + 4k, b 1 = n + k. There exist T > 0, λ 0 > 0 and a T -periodic function K : R (0, 1 which satisfies K (0 1/4 and K (T 3/4 so that if 0 < ν < ν 0, there exists a solution to (.1 satisfying for all 0 t ν 3/ u(t, x = u j (te ijx + ν 1/4 q 1 (t, x + ν 3/ tq (t, x, with j A u a1 (t = u a (t = K (νt, u b1 (t = u b (t = 1 K (νt, and where for all s R, q 1 (t, H s (Tz C s for all t R +, and q (t, H s (T C s for all 0 t ν 3/. We now define the wavenumber set consisting of nonlinear resonance interactions in the equation (1.1. Definition.1 (Resonance interaction set. et n Z and k Z\{0} be fixed. For j Z, we set α 1,j, α 3,j, α,j and α 4,j as follows: With α m,j, we set α 1,j π = n + 3k + j, α 3,j π = n + j, α,j π = n + k + j, α 4,j π = n + 4k + j. for 1 m 4, and C = 4 m=1a m. A m = {α m,j j Z, 0 j }, Here we consider (1.1 on T instead of (.1 on T, and obtain the following theorem. Theorem.. et n Z, k Z\{0}, s 1 and large integer > 0 be given. There exist a smooth global solution u(t and a time T = O( 1/ s.t. where 4 u(t, x = u Am (t, x + e(t, x, m=1 u A1 (t = u A 3 (t = 1 K(t, K(t sin(arctan t 4 3 for t 1/, where a constant c j s are independent of. u A (t = u A 4 (t = 1 + K(t, sup e(t, H s 1 t T. 1/ 3

4 Remark.1. Putting n = 0, s 0, k s 5 +ε, we have that there exists a solution u(t s.t. for some time t 0 > 0, u(t 0 H s u( t 0 H s k s (1 + 4s 3s K(t 0. f s > 1, then u( t 0 H s < u(t 0 H s (energy does not decrease. As opposites, we can construct solution whose energy will not increase for a long time. 3 Proof of Theorem. The proof of Theorem. consists of three steps: 1. construct resonant sets,. construct finite dimensional model with initial replacement, 3. construct approximation lemma (error estimates. 3.1 Resonant sets We first set nonlinear resonant interaction sets as follows. Definition 3.1. We say the set {(ξ 1, ξ 3, ξ 5, (ξ, ξ 4, ξ 6 } is resonance, if and only if the following conditions hold; (i ξ 1 + ξ 3 + ξ 5 = ξ + ξ 4 + ξ 6, (ii two of ξ 1, ξ 3, ξ 5 are elements of A 1, that are n + 3k + j 1, n + 3k + j 3, (iii one of ξ 1, ξ 3, ξ 5 is element of A, that is n + j 5, (iv two of ξ, ξ 4, ξ 6 are elements of A 3, that are n + k + j, n + k + j 4, (v one of ξ, ξ 4, ξ 6 is element of A 4, that is n + 4k + j 6, (vi {j 1, j 3 } = {j, j 4 } and j 5 = j 6 = j 1+j 3 = j +j 4. From the relation in the above definition, we have the following lemma. emma 3.1. f {(ξ 1, ξ 3, ξ 5, (ξ, ξ 4, ξ 6 } is resonance, then ϕ(ξ 1, ξ, ξ 3, ξ 4, ξ 5, ξ 6 = 0. Proof. Since the equation (n + 3k + (n + 3k + n = (n + k + (n + k + (n + 4k, we calculate 1 (π ϕ(ξ 1, ξ.ξ 3, ξ 4, ξ 5, ξ 6 = k (3(j 1 + j 3 (j + j 4 4j ϕ(j 1, j, j 3, j 4, j 5, j 6, which is equal to zero by j 6 = j 1+j 3 = j +j 4 and ϕ(j 1, j, j 3, j 4, j 5, j 6 = 0. 4

5 3. Finite dimensional model Now suppose that u is a smooth solution and let us start with the ansatz u(t, x = e igt a ξ (te ixξ itξ (dξ, where ϕ(x = e ixξ ϕ(ξ (dξ := 1 e ixξ ϕ(ξ. ξ πz/ We choose the parameter G = 6M 0 where M 0 = u(0. By direct calculation, a ξ = a ξ (t satisfies ( ȧ ξ = 6M 0 a ξ 3 a 3 3 ξ 4 (dξ + 4 a ξ 4 a 4 ξ + a ξ1 a ξ a ξ3 a ξ4 a ξ5 e itϕ(ξ 1,ξ,ξ 3,ξ 4,ξ 5,ξ, (3.1 {ξ 1,ξ 3,ξ 5 } {ξ,ξ 4,ξ} where ϕ(ξ 1, ξ, ξ 3, ξ 4, ξ 5, ξ 6 = ξ1 + ξ ξ3 + ξ4 ξ5 + ξ6. Then plugging the observation in emma 3.1 into the equation (3.1, we have the resonant formula ( iṙ ξ = 6M 0 r ξ 3 r 3 3 ξ 4 (dξ + 4 r ξ 4 r 4 ξ + r ξ1 r ξ r ξ3 r ξ4 r ξ5, (3. res(ξ where res(ξ denotes the resonant modes with respect to ξ j, 1 j 5 such that the set {(ξ 1, ξ 3, ξ 5, (ξ, ξ 4, ξ} is resonance. 3.3 A priori estimates Next, observe the conserved quantities for (3.1. emma 3.. et {r ξ (t} be a global solution to recasted NS (3.. relations; d dt d dt Then we have the r ξ (t = 0, (3.3 ξ C ξ r ξ (t = 0. (3.4 ξ C Remark 3.1. These corresponds to the mass and the energy conservations, respectively. 5

6 Proof of emma 3.. t will be convenience to change the index ξ by ξ 6. We first prove (3.3. Multiplying r ξ6 to (3. and taking the imaginary part, we have that m(iṙ ξ6 r ξ6 = 1 4 m res(ξ 6 r ξ1 r ξ r ξ3 r ξ4 r ξ5 r ξ6. Note that the left-hand side will be 1 d r dt ξ 6. Then after the summation over the resonant set, we arrive at the following; 1 d dt ξ 6 C r ξ6 (t = 1 i 4 ξ 6 C res(ξ 6 which is zero, since by symmetry. Next we prove (3.4. n a similar way to above, we have d dt which is deduced by (r ξ1 r ξ r ξ3 r ξ4 r ξ5 r ξ6 r ξ1 r ξ r ξ3 r ξ4 r ξ5 r ξ6, ξ r ξ (t = Re ξ 6 ṙ ξ6 r ξ6 ξ C ξ 6 C = m ξ 4 6 r ξ1 r ξ r ξ3 r ξ4 r ξ5 r ξ6, ξ 6 C res(ξ 6 1 ( (ξ 3i ξ3 + ξ5 (ξ + ξ4 + ξ6 r ξ1 r ξ r ξ3 r ξ4 r ξ5 r ξ6, res since by symmetrization. Th e last term is zero, since (ξ1 + ξ3 + ξ5 (ξ + ξ4 + ξ6 = 0 for the resonance interaction modes. et us consider the ansatz r ξ (t = ξ (te iθξ(t. Once again, using this coordinate, we obtain the following lemma. emma 3.3. For j [0, ], we have that d ( dt n+ j (t + n+4k+ j (t = 0, d ( dt n+3k+ j (t + n+k+ j (t = 0, Moreover, assuming additional constrains of j (t and θ j (t, we have the following lemma. emma 3.4. Assume n+ j (t = n+ m (t, θ n+ j (t = θ n+ m (t, 6

7 for all j, m. Then n+3k+ j (t = n+3k+ m (t, n+4k+ j (t = n+4k+ m (t, n+k+ j (t = n+k+ m (t, θ n+3k+ j (t = θ n+3k+ m (t, θ n+4k+ j (t = θ n+4k+ m (t, θ n+k+ j (t = θ n+k+ m (t d ( dt n+3k+ j (t n+ j (t = 0, d ( dt n+k+ j (t n+4k+ j (t = 0. Proof of emmas 3.3 and 3.4. The proof of emmas 3.3 and 3.4 is similar to the one of emma 3.. Then by emmas 3., 3.3 and 3.4, it is reasonable that we recast the equation (3. into the following Toy model form: A = 3! + 3( + 1 ( A A4 A 1 A3 sin (θ A + θ A4 θ A1 θ A3, A1 (t = 3!( ( A A4 A 1 A3 sin (θ A + θ A4 θ A1 θ A3, A4 = 3! + 3( + 1 ( A 1 A3 A A4 sin (θ A1 + θ A3 θ A θ A4 A (t = 3!( ( A 1 A3 A A4 sin (θ A1 + θ A3 θ A θ A4, θ A3 = 6M 0 1( A 1 + 3! + 3( + 1 ( ( 4 A3 + A1 + A4 + A A 1 3 A A4 A 1 cos (θ A + θ A4 θ A1 θ A3, A3 θ A1 = 6M 0 1( A 1 ( 4 A3 + A1 + A4 + A A 1 + 3!( ( A A4 A3 cos (θ A + θ A4 θ A1 θ A3, 7

8 θ A4 = 6M 0 1( A 4 + 3! + 3( + 1 ( ( 4 A3 + A1 + A4 + A A 4 A 1 A3 A cos (θ A + θ A4 θ A1 θ A3 A4 and θ A = 6M 0 1( A ( 4 A3 + A1 + A4 + A A + 3!( ( A 1 A3 A4 cos (θ A + θ A4 θ A1 θ A3. Remark 3.. A straightforward calculation show that this ODE system enjoys the conservation of the Hamiltonian following Hamiltonian H = 3M 0 3 ( A1 + A + A 3 + A 4 6( ( A1 + A + B1 + B ( 3 A1 + 3 A + 3 A A 4 ndeed, we see that for C = A j. + 3!( ( A1 A 1 A 3 1 A4 cos (θ A + θ A4 θ A1 θ A3. { θc = H C, θ C, C = H n virtue of d dt (k A 4 + k A 3 k A k A 1 = 0 for k = 1,, we obtain the particular dynamics of Aj, θ Aj. Proposition 3.1. There exists a solution ( Aj, θ Aj 1 j 4 to (Toy model s.t. θ A + θ A4 θ A1 θ A3 π and A1 (t = A3 (t = 1 K(t, A (t = A4 (t = 1 + K(t, K(t sin(arctan t 4 3 Proof of Proposition 3.1. The proof uses the symplectic change of variables. 8

9 3.4 Approximation lemma We show how the toy model approximates the original NS. Proposition 3.. et {a ξ (t} ξ Z/ and {r ξ (t} ξ C be a solution to the Fourier transformed NS equation and its resonant NS, respectively, with a ξ (0 = r ξ (0 for ξ C and a ξ (0 l s (ξ C 1 1/ ε. Then for s 1 and t c 1/ ε, a ξ (t r ξ (t l s 1 1/ ε, where r ξ (t = 0 if ξ C, and f l s = ξ s f(ξ l. Proof of Proposition 3.. The proof uses a priori estimates of M[u], E[u], energy argument, bootstrap argument and the perturbation argument. By Propositions 3.1 and 3., we conclude the proof of Theorem.. References [1] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, part : Schrödinger equation, part : The KdV-equation, Geom. Func. Anal., 3 (1993, , [] J. Bourgain, On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE, nternat. Math. Res. Notices, 1996, [3] T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in H s, Nonlinear Anal., 14 (1990, [4] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on R and T, J. Amer. Math. Soc., 16 (003, [5] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation, nvent. Math., 181 (010, [6] B. Dodson, Global well-posedness and scattering for the defocusing, -critical, nonlinear Schödinger equation when d = 1, Amer. J. Math., 138 (016, [7] E. Faou, P. Germain and Z. Hani, The weakly nonlinear large-box limit of the d cubic nonlinear Schrödinger equation, J. Amer. Math. Soc., 9 (015, [8] J. Ginibre and G. Velo, On the class of nonlinear Schrödinger equations, J. Funct. Anal., 3 (1979, 1 3, [9] R. T. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys., 18 (1977,

10 [10] B. Grébert and. Thomann, Resonant dynamics for the quintic nonlinear Schrödinger equation, Ann.. H. Poincaré, 9 (01, [11] H. Takaoka, Energy transfer model for the derivative nonlinear Schrödinger equations on the torus, Discrete Contin. Dyn. Syst., 37 (017, [1] Y. Tsutsumi, -solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcialaj Ekvacioj, 30 (1987, Department of Mathematics Kobe University Kobe, , Japan takaoka@math.kobe-u.ac.jp 10

DISPERSIVE EQUATIONS: A SURVEY

DISPERSIVE EQUATIONS: A SURVEY DISPERSIVE EQUATIONS: A SURVEY GIGLIOLA STAFFILANI 1. Introduction These notes were written as a guideline for a short talk; hence, the references and the statements of the theorems are often not given

More information

Nonlinear Schrödinger Equation BAOXIANG WANG. Talk at Tsinghua University 2012,3,16. School of Mathematical Sciences, Peking University.

Nonlinear 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 information

Global well-posedness for semi-linear Wave and Schrödinger equations. Slim Ibrahim

Global 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 information

SCATTERING FOR THE TWO-DIMENSIONAL NLS WITH EXPONENTIAL NONLINEARITY

SCATTERING 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 information

Global well-posedness for KdV in Sobolev spaces of negative index

Global well-posedness for KdV in Sobolev spaces of negative index Electronic Journal of Differential Equations, Vol. (), No. 6, pp. 7. ISSN: 7-669. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) Global well-posedness for

More information

Recent 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 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 information

Scattering theory for nonlinear Schrödinger equation with inverse square potential

Scattering 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 information

NUMERICAL SIMULATIONS OF THE ENERGY-SUPERCRITICAL NONLINEAR SCHRÖDINGER EQUATION

NUMERICAL SIMULATIONS OF THE ENERGY-SUPERCRITICAL NONLINEAR SCHRÖDINGER EQUATION Journal of Hyperbolic Differential Equations Vol. 7, No. 2 (2010) 279 296 c World Scientific Publishing Company DOI: 10.1142/S0219891610002104 NUMERICAL SIMULATIONS OF THE ENERGY-SUPERCRITICAL NONLINEAR

More information

Growth of Sobolev norms for the cubic defocusing NLS Lecture 1

Growth of Sobolev norms for the cubic defocusing NLS Lecture 1 Growth of Sobolev norms for the cubic defocusing NLS Lecture 1 Marcel Guardia February 8, 2013 M. Guardia (UMD) Growth of Sobolev norms February 8, 2013 1 / 40 The equations We consider two equations:

More information

TADAHIRO OH 0, 3 8 (T R), (1.5) The result in [2] is in fact stated for time-periodic functions: 0, 1 3 (T 2 ). (1.4)

TADAHIRO OH 0, 3 8 (T R), (1.5) The result in [2] is in fact stated for time-periodic functions: 0, 1 3 (T 2 ). (1.4) PERIODIC L 4 -STRICHARTZ ESTIMATE FOR KDV TADAHIRO OH 1. Introduction In [], Bourgain proved global well-posedness of the periodic KdV in L T): u t + u xxx + uu x 0, x, t) T R. 1.1) The key ingredient

More information

Very Weak Turbulence for Certain Dispersive Equations

Very Weak Turbulence for Certain Dispersive Equations Very Weak Turbulence for Certain Dispersive Equations Gigliola Staffilani Massachusetts Institute of Technology December, 2010 Gigliola Staffilani (MIT) Very weak turbulence and dispersive PDE December,

More information

On the Cauchy problem of 3-D energy-critical Schrödinger equations with subcritical perturbations

On the Cauchy problem of 3-D energy-critical Schrödinger equations with subcritical perturbations J. Differential Equations 30 (006 4 445 www.elsevier.com/locate/jde On the Cauchy problem of 3-D energy-critical Schrödinger equations with subcritical perturbations Xiaoyi Zhang Academy of Mathematics

More information

The NLS on product spaces and applications

The NLS on product spaces and applications October 2014, Orsay The NLS on product spaces and applications Nikolay Tzvetkov Cergy-Pontoise University based on joint work with Zaher Hani, Benoit Pausader and Nicola Visciglia A basic result Consider

More information

ANALYTIC SMOOTHING EFFECT FOR NONLI TitleSCHRÖDINGER EQUATION IN TWO SPACE DIMENSIONS. Citation Osaka Journal of Mathematics.

ANALYTIC SMOOTHING EFFECT FOR NONLI TitleSCHRÖDINGER EQUATION IN TWO SPACE DIMENSIONS. Citation Osaka Journal of Mathematics. ANALYTIC SMOOTHING EFFECT FOR NONLI TitleSCHRÖDINGER EQUATION IN TWO SPACE DIMENSIONS Author(s) Hoshino, Gaku; Ozawa, Tohru Citation Osaka Journal of Mathematics. 51(3) Issue 014-07 Date Text Version publisher

More information

Growth of Sobolev norms for the cubic NLS

Growth of Sobolev norms for the cubic NLS Growth of Sobolev norms for the cubic NLS Benoit Pausader N. Tzvetkov. Brown U. Spectral theory and mathematical physics, Cergy, June 2016 Introduction We consider the cubic nonlinear Schrödinger equation

More information

POLYNOMIAL UPPER BOUNDS FOR THE INSTABILITY OF THE NONLINEAR SCHRÖDINGER EQUATION BELOW THE ENERGY NORM. J. Colliander. M. Keel. G.

POLYNOMIAL UPPER BOUNDS FOR THE INSTABILITY OF THE NONLINEAR SCHRÖDINGER EQUATION BELOW THE ENERGY NORM. J. Colliander. M. Keel. G. COMMUNICATIONS ON Website: http://aimsciences.org PURE AND APPLIED ANALYSIS Volume, Number, March 003 pp. 33 50 POLYNOMIAL UPPER BOUNDS FOR THE INSTABILITY OF THE NONLINEAR SCHRÖDINGER EQUATION BELOW THE

More information

ENERGY CASCADES FOR NLS ON T d

ENERGY CASCADES FOR NLS ON T d ENERGY CASCADES FOR NLS ON T d RÉMI CARLES AND ERWAN FAOU ABSTRACT. We consider the nonlinear Schrödinger equation with cubic (focusing or defocusing) nonlinearity on the multidimensional torus. For special

More information

Well-posedness for the Fourth-order Schrödinger Equations with Quadratic Nonlinearity

Well-posedness for the Fourth-order Schrödinger Equations with Quadratic Nonlinearity Well-posedness for the Fourth-order Schrödinger Equations with Quadratic Nonlinearity Jiqiang Zheng The Graduate School of China Academy of Engineering Physics P. O. Box 20, Beijing, China, 00088 (zhengjiqiang@gmail.com)

More information

Scattering for cubic-quintic nonlinear Schrödinger equation on R 3

Scattering 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 information

A PHYSICAL SPACE PROOF OF THE BILINEAR STRICHARTZ AND LOCAL SMOOTHING ESTIMATES FOR THE SCHRÖDINGER EQUATION

A PHYSICAL SPACE PROOF OF THE BILINEAR STRICHARTZ AND LOCAL SMOOTHING ESTIMATES FOR THE SCHRÖDINGER EQUATION A PHYSICAL SPACE PROOF OF THE BILINEAR STRICHARTZ AND LOCAL SMOOTHING ESTIMATES FOR THE SCHRÖDINGER EQUATION TERENCE TAO Abstract. Let d 1, and let u, v : R R d C be Schwartz space solutions to the Schrödinger

More information

ALMOST CONSERVATION LAWS AND GLOBAL ROUGH SOLUTIONS TO A NONLINEAR SCHRÖDINGER EQUATION

ALMOST CONSERVATION LAWS AND GLOBAL ROUGH SOLUTIONS TO A NONLINEAR SCHRÖDINGER EQUATION ALMOST CONSERVATION LAWS AND GLOBAL ROUGH SOLUTIONS TO A NONLINEAR SCHRÖDINGER EQUATION J. COLLIANDER, M. KEEL, G. STAFFILANI, H. TAKAOKA, AND T. TAO Abstract. We prove an almost conservation law to obtain

More information

Gevrey regularity in time for generalized KdV type equations

Gevrey regularity in time for generalized KdV type equations Gevrey regularity in time for generalized KdV type equations Heather Hannah, A. Alexandrou Himonas and Gerson Petronilho Abstract Given s 1 we present initial data that belong to the Gevrey space G s for

More information

1D Quintic NLS with White Noise Dispersion

1D Quintic NLS with White Noise Dispersion 2011 年 1 月 7 日 1D Quintic NLS with White Noise Dispersion Yoshio TSUTSUMI, Kyoto University, Arnaud DEBUSSCHE, ENS de Cachan, Bretagne 1D quintic NLS with white noise dispersion idu + xu 2 dβ(t) =λ u 4

More information

Global well-posedness and scattering for the defocusing quintic nonlinear Schrödinger equation in two dimensions

Global well-posedness and scattering for the defocusing quintic nonlinear Schrödinger equation in two dimensions University of Massachusetts Amherst ScholarWorks@UMass Amherst Doctoral Dissertations Dissertations and Theses 08 Global well-posedness and scattering for the defocusing quintic nonlinear Schrödinger equation

More information

arxiv: v3 [math.ap] 14 Nov 2010

arxiv: v3 [math.ap] 14 Nov 2010 arxiv:1005.2832v3 [math.ap] 14 Nov 2010 GLOBAL WELL-POSEDNESS OF THE ENERGY CRITICAL NONLINEAR SCHRÖDINGER EQUATION WITH SMALL INITIAL DATA IN H 1 (T 3 ) SEBASTIAN HERR, DANIEL TATARU, AND NIKOLAY TZVETKOV

More information

Well-Posedness and Adiabatic Limit for Quantum Zakharov System

Well-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 information

BLOWUP THEORY FOR THE CRITICAL NONLINEAR SCHRÖDINGER EQUATIONS REVISITED

BLOWUP 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 information

1D Quintic NLS with White Noise Dispersion

1D Quintic NLS with White Noise Dispersion 2011 年 8 月 8 日 1D Quintic NLS with White Noise Dispersion Yoshio TSUTSUMI, Kyoto University, Arnaud DEBUSSCHE, ENS de Cachan, Bretagne 1D quintic NLS with white noise dispersion idu + xu 2 dβ(t) =λ u 4

More information

Sharp Well-posedness Results for the BBM Equation

Sharp Well-posedness Results for the BBM Equation Sharp Well-posedness Results for the BBM Equation J.L. Bona and N. zvetkov Abstract he regularized long-wave or BBM equation u t + u x + uu x u xxt = was derived as a model for the unidirectional propagation

More information

Sharp well-posedness of the Cauchy problem for a generalized Ostrovsky equation with positive dispersion

Sharp well-posedness of the Cauchy problem for a generalized Ostrovsky equation with positive dispersion Wang and Wang Boundary Value Problems (07) 07:86 https://doi.org/0.86/s366-07-099- RESEARCH Open Access Sharp well-posedness of the Cauchy problem for a generalized Ostrovsky equation with positive dispersion

More information

Some physical space heuristics for Strichartz estimates

Some physical space heuristics for Strichartz estimates Some physical space heuristics for Strichartz estimates Felipe Hernandez July 30, 2014 SPUR Final Paper, Summer 2014 Mentor Chenjie Fan Project suggested by Gigliola Staffilani Abstract This note records

More information

LOW REGULARITY GLOBAL WELL-POSEDNESS FOR THE ZAKHAROV AND KLEIN-GORDON-SCHRÖDINGER SYSTEMS

LOW REGULARITY GLOBAL WELL-POSEDNESS FOR THE ZAKHAROV AND KLEIN-GORDON-SCHRÖDINGER SYSTEMS LOW REGULARITY GLOBAL WELL-POSEDNESS FOR THE ZAKHAROV AND KLEIN-GORDON-SCHRÖDINGER SYSTEMS JAMES COLLIANDER, JUSTIN HOLMER, AND NIKOLAOS TZIRAKIS Abstract We prove low-regularity global well-posedness

More information

A GLOBAL COMPACT ATTRACTOR FOR HIGH-DIMENSIONAL DEFOCUSING NON-LINEAR SCHRÖDINGER EQUATIONS WITH POTENTIAL TERENCE TAO

A GLOBAL COMPACT ATTRACTOR FOR HIGH-DIMENSIONAL DEFOCUSING NON-LINEAR SCHRÖDINGER EQUATIONS WITH POTENTIAL TERENCE TAO A GLOBAL COMPACT ATTRACTOR FOR HIGH-DIMENSIONAL DEFOCUSING NON-LINEAR SCHRÖDINGER EQUATIONS WITH POTENTIAL TERENCE TAO arxiv:85.1544v2 [math.ap] 28 May 28 Abstract. We study the asymptotic behavior of

More information

arxiv:math/ v1 [math.ap] 24 Apr 2003

arxiv:math/ v1 [math.ap] 24 Apr 2003 ICM 2002 Vol. III 1 3 arxiv:math/0304397v1 [math.ap] 24 Apr 2003 Nonlinear Wave Equations Daniel Tataru Abstract The analysis of nonlinear wave equations has experienced a dramatic growth in the last ten

More information

Almost sure global existence and scattering for the one-dimensional NLS

Almost sure global existence and scattering for the one-dimensional NLS Almost sure global existence and scattering for the one-dimensional NLS Nicolas Burq 1 Université Paris-Sud, Université Paris-Saclay, Laboratoire de Mathématiques d Orsay, UMR 8628 du CNRS EPFL oct 20th

More information

Necessary 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 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 information

ON THE PROBABILISTIC CAUCHY THEORY FOR NONLINEAR DISPERSIVE PDES

ON THE PROBABILISTIC CAUCHY THEORY FOR NONLINEAR DISPERSIVE PDES ON THE PROBABILISTIC CAUCHY THEORY FOR NONLINEAR DISPERSIVE PDES ÁRPÁD BÉNYI, TADAHIRO OH, AND OANA POCOVNICU Abstract. In this note, we review some of the recent developments in the well-posedness theory

More information

LECTURE NOTES : INTRODUCTION TO DISPERSIVE PARTIAL DIFFERENTIAL EQUATIONS

LECTURE 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 information

SCATTERING FOR THE NON-RADIAL 3D CUBIC NONLINEAR SCHRÖDINGER EQUATION

SCATTERING 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 information

Unimodular Bilinear multipliers on L p spaces

Unimodular Bilinear multipliers on L p spaces Jotsaroop Kaur (joint work with Saurabh Shrivastava) Department of Mathematics, IISER Bhopal December 18, 2017 Fourier Multiplier Let m L (R n ), we define the Fourier multiplier operator as follows :

More information

Normal form for the non linear Schrödinger equation

Normal form for the non linear Schrödinger equation Normal form for the non linear Schrödinger equation joint work with Claudio Procesi and Nguyen Bich Van Universita di Roma La Sapienza S. Etienne de Tinee 4-9 Feb. 2013 Nonlinear Schrödinger equation Consider

More information

THE THEORY OF NONLINEAR SCHRÖDINGER EQUATIONS: PART I

THE THEORY OF NONLINEAR SCHRÖDINGER EQUATIONS: PART I THE THEORY OF NONLINEAR SCHRÖDINGER EQUATIONS: PART I J. COLLIANDER, M. KEEL, G. STAFFILANI, H. TAKAOKA, AND T. TAO Contents 1. Introduction. Lecture # 1: The Linear Schrödinger Equation in R n : Dispersive

More information

TADAHIRO OH AND CATHERINE SULEM

TADAHIRO OH AND CATHERINE SULEM ON THE ONE-DIMENSIONAL CUBIC NONLINEAR SCHRÖDINGER EQUATION BELOW L 2 TADAHIRO OH AND CATHERINE SULEM Abstract. In this paper, we review several recent results concerning well-posedness of the one-dimensional,

More information

1-D cubic NLS with several Diracs as initial data and consequences

1-D cubic NLS with several Diracs as initial data and consequences 1-D cubic NLS with several Diracs as initial data and consequences Valeria Banica (Univ. Pierre et Marie Curie) joint work with Luis Vega (BCAM) Roma, September 2017 1/20 Plan of the talk The 1-D cubic

More information

ON BLOW-UP SOLUTIONS TO THE 3D CUBIC NONLINEAR SCHRÖDINGER EQUATION

ON BLOW-UP SOLUTIONS TO THE 3D CUBIC NONLINEAR SCHRÖDINGER EQUATION ON BOW-UP SOUTIONS TO THE 3D CUBIC NONINEAR SCHRÖDINGER EQUATION JUSTIN HOMER AND SVETANA ROUDENKO Abstract. For the 3d cubic nonlinear Schrödinger (NS) equation, which has critical (scaling) norms 3 and

More information

Dispersive numerical schemes for Schrödinger equations

Dispersive numerical schemes for Schrödinger equations Dispersive numerical schemes for Schrödinger equations Enrique Zuazua joint work with L. Ignat zuazua@bcamath.org Basque Center for Applied Mathematics (BCAM), Bilbao, Basque Country, Spain IMA Workshop:

More information

arxiv: v1 [math.ap] 11 Apr 2011

arxiv: v1 [math.ap] 11 Apr 2011 THE RADIAL DEFOCUSING ENERGY-SUPERCRITICAL CUBIC NONLINEAR WAVE EQUATION IN R +5 arxiv:04.2002v [math.ap] Apr 20 AYNUR BULUT Abstract. In this work, we consider the energy-supercritical defocusing cubic

More information

IMA Preprint Series # 1961

IMA Preprint Series # 1961 GLOBAL WELL-POSEDNESS AND SCATTERING FOR THE ENERGY-CRITICAL NONLINEAR SCHRÖDINGER EQUATION IN R 3 By J. Colliander M. Keel G. Staffilani H. Takaoka and T. Tao IMA Preprint Series # 1961 ( February 2004

More information

GLOBAL WELL-POSEDNESS OF NLS-KDV SYSTEMS FOR PERIODIC FUNCTIONS

GLOBAL WELL-POSEDNESS OF NLS-KDV SYSTEMS FOR PERIODIC FUNCTIONS Electronic Journal of Differential Equations, Vol. 66), o. 7, pp. 1. ISS: 17-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu login: ftp) GLOBAL WELL-POSEDESS

More information

M 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, 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 information

Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in R 3

Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in R 3 Annals of Mathematics, 67 (008), 767 865 Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in R 3 By J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T.

More information

arxiv:math/ v7 [math.ap] 8 Jan 2006

arxiv:math/ v7 [math.ap] 8 Jan 2006 arxiv:math/0402129v7 [math.ap] 8 Jan 2006 GLOBAL WELL-POSEDNESS AND SCATTERING FOR THE ENERGY-CRITICAL NONLINEAR SCHRÖDINGER EQUATION IN R 3 J. COLLIANDER, M. KEEL, G. STAFFILANI, H. TAKAOKA, AND T. TAO

More information

A COUNTEREXAMPLE TO AN ENDPOINT BILINEAR STRICHARTZ INEQUALITY TERENCE TAO. t L x (R R2 ) f L 2 x (R2 )

A COUNTEREXAMPLE TO AN ENDPOINT BILINEAR STRICHARTZ INEQUALITY TERENCE TAO. t L x (R R2 ) f L 2 x (R2 ) Electronic Journal of Differential Equations, Vol. 2006(2006), No. 5, pp. 6. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) A COUNTEREXAMPLE

More information

MODIFIED SCATTERING FOR THE BOSON STAR EQUATION 1. INTRODUCTION

MODIFIED SCATTERING FOR THE BOSON STAR EQUATION 1. INTRODUCTION MODIFIED SCATTERING FOR THE BOSON STAR EQUATION FABIO PUSATERI ABSTRACT We consider the question of scattering for the boson star equation in three space dimensions This is a semi-relativistic Klein-Gordon

More information

Properties of solutions to the semi-linear Schrödinger equation.

Properties of solutions to the semi-linear Schrödinger equation. Properties of solutions to the semi-linear Schrödinger equation. Nikolaos Tzirakis University of Illinois Urbana-Champaign Abstract Lecture notes concerning basic properties of the solutions to the semi-linear

More information

Piecewise Smooth Solutions to the Burgers-Hilbert Equation

Piecewise Smooth Solutions to the Burgers-Hilbert Equation Piecewise Smooth Solutions to the Burgers-Hilbert Equation Alberto Bressan and Tianyou Zhang Department of Mathematics, Penn State University, University Park, Pa 68, USA e-mails: bressan@mathpsuedu, zhang

More information

ON THE CAUCHY-PROBLEM FOR GENERALIZED KADOMTSEV-PETVIASHVILI-II EQUATIONS

ON THE CAUCHY-PROBLEM FOR GENERALIZED KADOMTSEV-PETVIASHVILI-II EQUATIONS Electronic Journal of Differential Equations, Vol. 009(009), No. 8, pp. 1 9. ISSN: 107-6691. URL: http://ejde.math.tstate.edu or http://ejde.math.unt.edu ftp ejde.math.tstate.edu ON THE CAUCHY-PROBLEM

More information

TRANSFER OF ENERGY TO HIGH FREQUENCIES IN THE CUBIC DEFOCUSING NONLINEAR SCHRÖDINGER EQUATION

TRANSFER OF ENERGY TO HIGH FREQUENCIES IN THE CUBIC DEFOCUSING NONLINEAR SCHRÖDINGER EQUATION TRANSFER OF ENERGY TO HIGH FREQUENCIES IN THE CUBIC DEFOCUSING NONLINEAR SCHRÖDINGER EQUATION J. COLLIANDER, M. KEEL, G. STAFFILANI, H. TAKAOKA, AND T. TAO Abstract. We consider the cubic defocusing nonlinear

More information

NONLINEAR PROPAGATION OF WAVE PACKETS. Ritsumeikan University, and 22

NONLINEAR PROPAGATION OF WAVE PACKETS. Ritsumeikan University, and 22 NONLINEAR PROPAGATION OF WAVE PACKETS CLOTILDE FERMANIAN KAMMERER Ritsumeikan University, 21-1 - 21 and 22 Our aim in this lecture is to explain the proof of a recent Theorem obtained in collaboration

More information

Infinite-time Exponential Growth of the Euler Equation on Two-dimensional Torus

Infinite-time Exponential Growth of the Euler Equation on Two-dimensional Torus Infinite-time Exponential Growth of the Euler Equation on Two-dimensional Torus arxiv:1608.07010v1 [math.ap] 5 Aug 016 Zhen Lei Jia Shi August 6, 016 Abstract For any A >, we construct solutions to the

More information

arxiv:math/ v2 [math.ap] 8 Jun 2006

arxiv:math/ v2 [math.ap] 8 Jun 2006 LOW REGULARITY GLOBAL WELL-POSEDNESS FOR THE KLEIN-GORDON-SCHRÖDINGER SYSTEM WITH THE HIGHER ORDER YUKAWA COUPLING arxiv:math/0606079v [math.ap] 8 Jun 006 Changxing Miao Institute of Applied Physics and

More information

arxiv: v1 [math.ap] 20 Nov 2007

arxiv: v1 [math.ap] 20 Nov 2007 Long range scattering for the Maxwell-Schrödinger system with arbitrarily large asymptotic data arxiv:0711.3100v1 [math.ap] 20 Nov 2007 J. Ginibre Laboratoire de Physique Théorique Université de Paris

More information

arxiv: v1 [math.ap] 21 Mar 2011

arxiv: v1 [math.ap] 21 Mar 2011 HIGH FREQUENCY PERTURBATION OF CNOIDAL WAVES IN KDV arxiv:113.4135v1 [math.ap] 21 Mar 211 M. B. ERDOĞAN, N. TZIRAKIS, AND V. ZHARNITSKY Abstract. The Korteweg-de Vries (KdV) equation with periodic boundary

More information

On the Asymptotic Behavior of Large Radial Data for a Focusing Non-Linear Schrödinger Equation

On the Asymptotic Behavior of Large Radial Data for a Focusing Non-Linear Schrödinger Equation Dynamics of PDE, Vol.1, No.1, 1-47, 2004 On the Asymptotic Behavior of Large adial Data for a Focusing Non-Linear Schrödinger Equation Terence Tao Communicated by Charles Li, received December 15, 2003.

More information

Presenter: Noriyoshi Fukaya

Presenter: 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 information

CUTOFF RESOLVENT ESTIMATES AND THE SEMILINEAR SCHRÖDINGER EQUATION

CUTOFF 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 information

Invariant measures and the soliton resolution conjecture

Invariant 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 information

Sharp Sobolev Strichartz estimates for the free Schrödinger propagator

Sharp Sobolev Strichartz estimates for the free Schrödinger propagator Sharp Sobolev Strichartz estimates for the free Schrödinger propagator Neal Bez, Chris Jeavons and Nikolaos Pattakos Abstract. We consider gaussian extremisability of sharp linear Sobolev Strichartz estimates

More information

ISABELLE GALLAGHER AND MARIUS PAICU

ISABELLE GALLAGHER AND MARIUS PAICU REMARKS ON THE BLOW-UP OF SOLUTIONS TO A TOY MODEL FOR THE NAVIER-STOKES EQUATIONS ISABELLE GALLAGHER AND MARIUS PAICU Abstract. In [14], S. Montgomery-Smith provides a one dimensional model for the three

More information

Four-Fermion Interaction Approximation of the Intermediate Vector Boson Model

Four-Fermion Interaction Approximation of the Intermediate Vector Boson Model Four-Fermion Interaction Approximation of the Intermediate Vector Boson odel Yoshio Tsutsumi Department of athematics, Kyoto University, Kyoto 66-852, JAPAN 1 Introduction In this note, we consider the

More information

Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrodinger equation

Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrodinger equation Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrodinger equation The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story

More information

Exponential Energy Decay for the Kadomtsev-Petviashvili (KP-II) equation

Exponential 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 information

Almost Conservation Laws for KdV and Cubic NLS. UROP Final Paper, Summer 2018

Almost Conservation Laws for KdV and Cubic NLS. UROP Final Paper, Summer 2018 Almost Conservation Laws for KdV and Cubic NLS UROP Final Paper, Summer 2018 Zixuan Xu Mentor: Ricardo Grande Izquierdo Project suggested by: Gigliola Staffilani Abstract In this paper, we explore the

More information

Solutions to the Nonlinear Schrödinger Equation in Hyperbolic Space

Solutions 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 information

Global solutions for the cubic non linear wave equation

Global 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 information

ON GLOBAL SOLUTIONS OF A ZAKHAROV-SCHULMAN TYPE SYSTEM

ON GLOBAL SOLUTIONS OF A ZAKHAROV-SCHULMAN TYPE SYSTEM ON GLOBAL SOLUTIONS OF A ZAKHAROV-SCHULMAN TYPE SYSTEM THOMAS BECK, FABIO PUSATERI, PHIL SOSOE, AND PERCY WONG Abstract. We consider a class of wave-schrödinger systems in three dimensions with a Zakharov-

More information

From quantum many body systems to nonlinear dispersive PDE, and back

From quantum many body systems to nonlinear dispersive PDE, and back From quantum many body systems to nonlinear dispersive PDE, and back Nataša Pavlović (University of Texas at Austin) Based on joint works with T. Chen, C. Hainzl, R. Seiringer, N. Tzirakis May 28, 2015

More information

Sharp blow-up criteria for the Davey-Stewartson system in R 3

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 information

arxiv: v2 [math.ap] 4 Dec 2013

arxiv: v2 [math.ap] 4 Dec 2013 ON D NLS ON NON-TRAPPING EXTERIOR DOMAINS FARAH ABOU SHAKRA arxiv:04.768v [math.ap] 4 Dec 0 Abstract. Global existence and scattering for the nonlinear defocusing Schrödinger equation in dimensions are

More information

RESONANT DYNAMICS FOR THE QUINTIC NON LINEAR SCHRÖDINGER EQUATION

RESONANT DYNAMICS FOR THE QUINTIC NON LINEAR SCHRÖDINGER EQUATION RESONANT DYNAMICS FOR THE QUINTIC NON LINEAR SCHRÖDINGER EQUATION by Benoît Grébert & Laurent Thomann Abstract. We consider the quintic nonlinear Schrödinger equation (NLS) on the circle i@ tu + @ 2 xu

More information

Bielefeld Course on Nonlinear Waves - June 29, Department of Mathematics University of North Carolina, Chapel Hill. Solitons on Manifolds

Bielefeld 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 information

Blow-up on manifolds with symmetry for the nonlinear Schröding

Blow-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 information

Regularity results for the Dirac-Klein-Gordon equations

Regularity results for the Dirac-Klein-Gordon equations Achenef Tesfahun Temesgen Regularity results for the Dirac-Klein-Gordon equations Thesis for the degree of Philosophiae Doctor Trondheim, January 2009 Norwegian University of Science and Technology Faculty

More information

DETERMINATION OF THE BLOW-UP RATE FOR THE SEMILINEAR WAVE EQUATION

DETERMINATION OF THE BLOW-UP RATE FOR THE SEMILINEAR WAVE EQUATION DETERMINATION OF THE LOW-UP RATE FOR THE SEMILINEAR WAVE EQUATION y FRANK MERLE and HATEM ZAAG Abstract. In this paper, we find the optimal blow-up rate for the semilinear wave equation with a power nonlinearity.

More information

The Concentration-compactness/ Rigidity Method for Critical Dispersive and Wave Equations

The Concentration-compactness/ Rigidity Method for Critical Dispersive and Wave Equations The Concentration-compactness/ Rigidity Method for Critical Dispersive and Wave Equations Carlos E. Kenig Supported in part by NSF 3 In these lectures I will describe a program (which I will call the

More information

The initial-boundary value problem for the 1D nonlinear Schrödinger equation on the half-line

The initial-boundary value problem for the 1D nonlinear Schrödinger equation on the half-line The initial-boundary value problem for the 1D nonlinear Schrödinger equation on the half-line Justin Holmer Department of Mathematics, University of California Berkeley, CA 9720 1 2 Abstract We prove,

More information

Dispersive numerical schemes for linear and nonlinear Schrödinger equations

Dispersive numerical schemes for linear and nonlinear Schrödinger equations Collège de France, December 2006 Dispersive numerical schemes for linear and nonlinear Schrödinger equations Enrique Zuazua Universidad Autónoma, 28049 Madrid, Spain enrique.zuazua@uam.es http://www.uam.es/enrique.zuazua

More information

Existence 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 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 information

The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation

The 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 information

A REMARK ON AN EQUATION OF WAVE MAPS TYPE WITH VARIABLE COEFFICIENTS

A 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 information

KAM for quasi-linear KdV. Pietro Baldi, Massimiliano Berti, Riccardo Montalto

KAM for quasi-linear KdV. Pietro Baldi, Massimiliano Berti, Riccardo Montalto KAM for quasi-linear KdV Pietro Baldi, Massimiliano Berti, Riccardo Montalto Abstract. We prove the existence and stability of Cantor families of quasi-periodic, small amplitude solutions of quasi-linear

More information

SYMMETRY RESULTS FOR PERTURBED PROBLEMS AND RELATED QUESTIONS. Massimo Grosi Filomena Pacella S. L. Yadava. 1. Introduction

SYMMETRY RESULTS FOR PERTURBED PROBLEMS AND RELATED QUESTIONS. Massimo Grosi Filomena Pacella S. L. Yadava. 1. Introduction Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 21, 2003, 211 226 SYMMETRY RESULTS FOR PERTURBED PROBLEMS AND RELATED QUESTIONS Massimo Grosi Filomena Pacella S.

More information

Smoothing Effects for Linear Partial Differential Equations

Smoothing Effects for Linear Partial Differential Equations Smoothing Effects for Linear Partial Differential Equations Derek L. Smith SIAM Seminar - Winter 2015 University of California, Santa Barbara January 21, 2015 Table of Contents Preliminaries Smoothing

More information

ANNALES DE L I. H. P., SECTION A

ANNALES DE L I. H. P., SECTION A ANNALES DE L I. H. P., SECTION A NAKAO HAYASHI PAVEL I. NAUMKIN Asymptotic behavior in time of solutions to the derivative nonlinear Schrödinger equation Annales de l I. H. P., section A, tome 68, n o

More information

THE UNIVERSITY OF CHICAGO UNIFORM ESTIMATES FOR THE ZAKHAROV SYSTEM AND THE INITIAL-BOUNDARY VALUE PROBLEM FOR THE KORTEWEG-DE VRIES

THE UNIVERSITY OF CHICAGO UNIFORM ESTIMATES FOR THE ZAKHAROV SYSTEM AND THE INITIAL-BOUNDARY VALUE PROBLEM FOR THE KORTEWEG-DE VRIES THE UNIVERSITY OF CHICAGO UNIFORM ESTIMATES FOR THE ZAKHAROV SYSTEM AND THE INITIAL-BOUNDARY VALUE PROBLEM FOR THE KORTEWEG-DE VRIES AND NONLINEAR SCHRÖDINGER EQUATIONS ADISSERTATIONSUBMITTEDTO THE FACULTY

More information

The work of Terence Tao

The work of Terence Tao The work of Terence Tao Charles Fefferman Mathematics at the highest level has several flavors. On seeing it, one might say: (A) What amazing technical power! (B) What a grand synthesis! (C) How could

More information

DRIFT OF SPECTRALLY STABLE SHIFTED STATES ON STAR GRAPHS

DRIFT OF SPECTRALLY STABLE SHIFTED STATES ON STAR GRAPHS DRIFT OF SPECTRALLY STABLE SHIFTED STATES ON STAR GRAPHS ADILBEK KAIRZHAN, DMITRY E. PELINOVSKY, AND ROY H. GOODMAN Abstract. When the coefficients of the cubic terms match the coefficients in the boundary

More information

UNIQUE CONTINUATION PROPERTY FOR THE KADOMTSEV-PETVIASHVILI (KP-II) EQUATION

UNIQUE CONTINUATION PROPERTY FOR THE KADOMTSEV-PETVIASHVILI (KP-II) EQUATION Electronic Journal of Differential Equations, Vol. 2525), No. 59, pp. 1 12. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu login: ftp) UNIQUE CONTINUATION

More information

The Dirichlet s P rinciple. In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation:

The Dirichlet s P rinciple. In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation: Oct. 1 The Dirichlet s P rinciple In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation: 1. Dirichlet s Principle. u = in, u = g on. ( 1 ) If we multiply

More information

Stability and Instability of Standing Waves for the Nonlinear Fractional Schrödinger Equation. Shihui Zhu (joint with J. Zhang)

Stability and Instability of Standing Waves for the Nonlinear Fractional Schrödinger Equation. Shihui Zhu (joint with J. Zhang) 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 P1 iu t ( + k 2 )

More information