SCATTERING FOR THE TWO-DIMENSIONAL NLS WITH EXPONENTIAL NONLINEARITY

Size: px
Start display at page:

Download "SCATTERING FOR THE TWO-DIMENSIONAL NLS WITH EXPONENTIAL NONLINEARITY"

Transcription

1 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 for nonlinear Schrödinger equations with a defocusing exponential nonlinearity in two space dimensions. A certain threshold is defined based on the value of the conserved Hamiltonian, below which the exponential potential energy is dominated by the kinetic energy via a Trudinger-Moser type inequality. We prove that if the Hamiltonian is below to the critical value, then the solution approaches a free Schrödinger solution at the time infinity.. Introduction We study the scattering theory in the energy space for nonlinear Schrödinger equation (NLS): { i u + u = f(u), u : R + C, (.) u(0, x) = u 0 (x) H (R ), where the nonlinearity f : C C is defined by ) (.) f(u) = (e 4π u 4π u u. Solutions of (.) satisfy the conservation of mass and Hamiltonian (.3) M(u, t) := u dx, R ( ) (.4) H(u, t) := u + F (u) dx. Also we define (.5) E(u, t) := H(u, t) + M(u, t). R Date: August 8, Mathematics Subject Classification. 35L70, 35Q55, 35B40, 35B33, 37K05, 37L50. Key words and phrases. Nonlinear Schrödinger equation, scattering theory, Sobolev critical exponent, Trudinger-Moser inequality. S. Ibrahim is partially supported by NSERC# grant and a start up fund from the University of Victoria. M. Majdoub is grateful to the Laboratory of PDE and Applications at the Faculty of Sciences of Tunis. N. Masmoudi is partially supported by an NSF Grant DMS

2 S. IBRAHIM, M. MAJDOUB, N. MASMOUDI, AND K. NAKANISHI The exponential type nonlinearities appear in several applications, as for example the self trapped beams in plasma. (See [3]). From the mathematical point of view, Cazenave in [4] considered the Schrödinger equation with decreasing exponential and showed the global well-posedness and scattering. With increasing exponentials, the situation is much more complicated (since there is no a priori L control of the nonlinear term). The two dimensional case is particularly interesting because of its relation to the critical Sobolev (or Trudinger-Moser) embedding. On the other hand, we have subtracted the cubic part from our nonlinearity f in order to avoid another critical exponent related to the decay property of solutions. To explain these issues, we start with a brief review of the more familiar power case... The energy critical NLS. Recall the monomial defocusing semilinear Schrödinger equation in space dimension d (.6) i u + u = u p u, u : R +d C, which has the critical exponents p = d+ (only for d 3) and p d = + 4. d For the energy subcritical case (p < p ), an iteration of the local-in-time wellposedness result using the a priori upper bound on u(t) H implied by the conservation laws establishes global well-posedness for (.6) in H. Those solutions scatter when p > p [0, 5]. The energy critical case (p = p ) was actually harder than the nonlinear Klein- Gordon equation, for which the finite propagation property was crucial to exclude possible concentration of energy, whereas there is no upper bound on the propagation speed for the Schrödinger. Nevertheless, based on new ideas such as induction on the energy size and frequency split propagation estimates, Bourgain [3] proved the global well-posedness and the scattering for radially symmetric data, and it was extended to the general case by [8] using a new interaction Morawetz inequality. For the exponential nonlinearity in two spatial dimensions, small data global wellposedness together with the scattering was worked out by Nakamura-Ozawa in [4]. Later on, the size of the initial data for which one has local existence was quantified for (.) in [9], and a notion of criticality was proposed: Definition.. The Cauchy problem (.) is said to be subcritical if H(u 0 ) <, critical if H(u 0 ) = and supercritical if H(u 0 ) >. Indeed, one can construct a unique local solution if u 0 L <, and the time of existence depends only on η := u 0 L and u 0 L. Hence the maximal local solutions are indeed global in the subcritical case. The critical case is more delicate due to the possible concentration of the Hamiltonian. The following result is proved in [9]. Theorem. (Global well-posedness [9]). Assume that H(u 0 ), then the problem (.) has a unique global solution u in the class C(R, H (R )). Moreover, u L 4 loc (R, C/ (R )) and satisfies the conservation laws (.3) and (.4).

3 SCATTERING FOR THE TWO-DIMENSIONAL NLS... 3 Recall that C / (R ) denotes the space of /-Hölder continuous functions... Main result. The main goal in this paper is to show that every global solution of (.) with H(u) approaches solutions to the associated free equation (.7) i v + v = 0, in the energy space H as t ±. Unfortunately, we have not succeeded to handle the critical case H(u) = and we have to restrict ourselves to the subcritical one. The reason is that to trace the concentration radius, as defined in [], the finite speed of propagation of energy is essential in our argument for NLKG, which is not available for NLS. The main ingredient for the subcritical NLS is a new interaction Morawetz estimate, proved independently by Colliander et al. and Planchon-Vega [7, 6]. This estimate gives a priori global bound of u in L 4 t (L 8 x). Hence, by complex interpolation we deduce that some of the Strichartz norms used in the nonlinear estimate go to zero for large time and the scattering in the subcritical case follows. More precisely, we have Theorem.3. For any global solution u of (.) in H satisfying H(u) <, we have u L 4 (R, C / ) and there exist unique free solutions u ± of (.7) such that Moreover, the maps (u u ± )(t) H 0 u(0) u ± (0) (t ± ). are homeomorphisms between the unit balls in the nonlinear energy space and the free energy space, namely from {ϕ H ; H(ϕ) < } onto {ϕ H ; ϕ L < }. This paper is organized as follows. In Section, we recall some useful lemmas from the literature. Section is devoted to the proof of our main result (Theorem.3). In Section 4, we show the optimality of our nonlinear estimate with respect to the H norm.. Background Material In this section, we introduce some notation and recall several lemmas we use to prove the main result. First, recall the sharp Trudinger-Moser inequality on R [, 7]. It is the limit case of the Sobolev embedding. For any µ > 0 we have (.) sup ϕ Hµ (e 4π ϕ )dx <, where H µ is defined by the norm u H µ := u L +µ u L. We can change µ > 0 just by scaling ϕ(x) ϕ(x/µ). It is known that the H (R ) functions are not generally in L. The following lemma shows that we can estimate the L norm by a stronger norm but with a weaker growth (namely logarithmic).

4 4 S. IBRAHIM, M. MAJDOUB, N. MASMOUDI, AND K. NAKANISHI Lemma. (Logarithmic inequality [], Theorem.3). Let 0 < α <. For any real number λ >, a constant C πα λ exists such that for any function ϕ H0 C α ( x < ), one has ( ϕ L λ ϕ L log C λ + ϕ ) C (.) α. ϕ L We also recall the whole space version of the above inequality. Lemma. ([], Theorem.3). Let 0 < α <. For any λ > and any πα 0 < µ, a constant C λ > 0 exists such that, for any function u H (R ) C α (R ) (.3) u L λ u H µ log ( C λ + 8α µ α u C α u Hµ Finally, we recall the Strichartz estimate for the free Schrödinger equation. (See [5]). Proposition.3. Let I R be a time slab, t 0 I and (q, r), ( q, r) two admissible Strichartz couples, i.e., r, β < and q + r = α + β =. There exists a positive constant C such that if u := u(t, x) a solution in C(I, H (R )) of the linear problem then (.4) u L q (I,W,r ) C i u + u = G(t, x), u(t 0 ) H (R ), ). ( ) u(t 0 ) H + G L q (I,W, r ). In particular, note that (q, r) = (4, 4) is an admissible Strichartz couple and W,4 (R ) C / (R ). 3. Proof of Theorem.3 For a time slab I R, we define S (I) via By the Strichartz estimates we have (3.) u S (I) = u L (I,H x) + u L 4 (I,W,4 ). u S u(0) H + (i u + u) L +η η (Lx ) for any 0 < η /. The scattering result Theorem.3 is easily proved by the following two lemmas: First we have the Strichartz-type estimate on the nonlinearity Here p stands for the Lebesgue conjugate exponent of p, that is p + p =.,

5 SCATTERING FOR THE TWO-DIMENSIONAL NLS... 5 Lemma 3.. For any H (0, ), there exists δ (0, ), such that for any time slab I, any T I and any H solution u of (.) with H(u) H, we have u S (I) u(t ) H + u 4δ L 4 (I,L 8 ) u 5 4δ S (I), Next we have a global a priori bound. It was proved independently by Planchon- Vega [6] and Colliander et al. [7] Lemma 3.. Let u be a global solution of (.) in H. Then u L 4 (R,L 8 ) u 3/4 L (R;L ) u /4 L (R;L ) M(u)3/8 H(u) /8. Actually both of them gave a priori bound on some Sobolev norm on u. The above is a consequence of it via the Sobolev embedding. By the above global bound, we can decompose R into a finite number of intervals on which the u L 4 L8 norm is sufficiently small. Then the first lemma gives a uniform bound on u S on each interval, and hence by summing it up for all intervals, we obtain a priori bound (3.) and thereby the scattering for u. u S (R R ) C(E(u)) <, Proof of Lemma 3.. It suffices to estimate the nonlinear term in some dual Strichartz norm as in (3.). Choose 0 < δ < and λ > 0 such that (3.3) K := H + <, π( + δ)λk =, λ > π( δ). We estimate only f(u), since the same estimate on f(u) is easier. Note that In the case u L x (3.4) f(u) f(u) u u (e 4π u ). K, we have by the Hölder inequality, δ Lx u L δ u L 4 δ e 4π u / δ L e 4π u /+δ L. The third term on the right is bounded by the Trudinger-Moser (.). For the last term we use the H µ version of the logarithmic inequality (.3) with µ := min(, ( H)/M) > 0. Since u H µ = u L + µ u L H + µ M H + <, that term is bounded by (3.5) e 4π(/+δ) u L ( + u C / δ// u Hµ ) π(+δ) u Hµ u C, / δ/ where we used (3.3) as well as u C / δ/ u L K. The case u L K is easy, since then f(u) u u 4. Now we integrate in time using the Hölder to obtain f(u) L +δ (L δ ) u L δ (L δ ) u L δ (L 4 δ ) u L 4 δ (C / δ/ ).

6 6 S. IBRAHIM, M. MAJDOUB, N. MASMOUDI, AND K. NAKANISHI Finally, the complex interpolation and the Sobolev embedding imply that (3.6) u u δ L δ (L δ ) L L u δ L 4 L u 4 S, u L δ (L 4 δ ) u δ L L u δ L 4 L 8, u L 4 δ (C / δ/ ) u L 4 δ (H, 4 +δ ) u δ L H u δ L 4 H,4 u S. Plugging them into the above, we deduce the result as desired. 4. Criticality of the nonlinear estimate by the Strichartz norms We see that the linear energy and the Strichartz estimate are not sufficient to control the nonlinearity in the critical case. Proposition 4.. For any δ > 0, there exists a sequence of radial free Schrödinger solutions v N (N ) such that v N + v N dx <, H(v N, 0) + δ, (4.) R f(v N ) L p t Lq x( t N, x N ) C δ (log N) /, for any (p, q) [, ] satisfying /p + /q = 3/. The above norm on f(v N ) is the dual Strichartz norm in Hx for the linear Schrödinger equation. Similar result was shown for the critical Klein-Gordon equation []. Proof. We take the same initial data as in ([], Proposition 7.): π (4.) v N (0) = ξ e iξx dξ. log N (π) By the Plancherel theorem, (4.3) v N (0) L = v N (0) L = π log N π log N (π) (π) < ξ <e a N < ξ <e a N < ξ <e a N ξ dξ = log N a log N, ξ 4 dξ < log N. By the sharp Trudinger-Moser inequality (.), there exists M > 0 such that for any µ > 0 (4.4) sup ψ L +µ ψ L F (ψ)dx M/µ, where µ can be removed or inserted by rescaling. Then (4.3) implies that F (v N (0))dx M a R (4.5), v N + v N dx <, H(v N, 0) < + M a, so that we get the desired nonlinear energy bound on v N by choosing a M/δ.

7 SCATTERING FOR THE TWO-DIMENSIONAL NLS... 7 The free solution is given by π (4.6) v N (t, x) = ξ e it ξ +iξx dξ, log N (π) < ξ <e a N we have in the region where t ε N and x εn for some small fixed ε > 0 and large N N, log N (4.7) Rv N (t, x) π a + ε, e 4π v v N log N. π log N We have to estimate v. By the radial symmetry, it suffices to consider the case x = (x, 0) and v N = ( v N, 0). Then ξ v N = r v N +iξx log N ξ e it ξ dξ (4.8) = i N log N < ξ <e a N π e itρ cos θ sin(rρ cos θ)dθdρ, since 0 < tρ < ε and 0 < rρ cos θ < ε, we get (4.9) r v N N log N π rρdρ rn log N εn log N. Thus we conclude inf e4π v N v N v N L q t ε N x ( x εn ) N log N εn (εn ) /q, (4.0) log N f(v N ) L p t Lq x(t ε N, x εn ) ε /p+/q N 3 /q /p log N ε 3 log N. References [] S. Adachi and K. Tanaka, Trudinger type inequalities in R N and their best exponents, Proc. Amer. Math. Soc. 8 (000), no. 7, [] J. Bourgain, Scattering in the energy space and below for 3D NLS, J. Anal. Math. 75 (998) [3] J. Bourgain, Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Amer. Math. Soc. (999), no., [4] T. Cazenave, Equations de Schrödinger non linéaires en dimension deux. Proc. Roy. Soc. Edinburgh Sect. A 84 (979), no. 3-4, [5] T. Cazenave: Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, 0. New York University, Courant Institute of Mathematical Sciences, AMS, 003. [6] T. Cazenave and F.B. Weissler, Critical nonlinear Schrödinger Equation, Non. Anal. TMA, 4 (990), [7] J. Colliander, M. Grillakis and N. Tzirakis, Tensor products and correlation estimates with applications to nonlinear Schrödinger equations, Communications on pure and applied mathematics, 6 (009), [8] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness and scattering in the energy space for the critical nonlinear Schrödinger equation in R 3, Annals of Math., 67 (008),

8 8 S. IBRAHIM, M. MAJDOUB, N. MASMOUDI, AND K. NAKANISHI [9] J. Colliander, S. Ibrahim, M. Majdoub and N. Masmoudi, Energy critical NLS in two space dimension, Journal of Hyperbolic Differential Equations, 6 (009), [0] J. Ginibre and G. Velo, Scattering theory in the energy space for a class of nonlinear Schrödinger equations, J. Math. Pures Appl. (9) 64 (985), no. 4, [] S. Ibrahim, M. Majdoub and N. Masmoudi, Double logarithmic inequality with a sharp constant, Proc. Amer. Math. Soc. 35 (007), [] S. Ibrahim, M. Majdoub, N. Masmoudi, K. Nakanishi, Scattering for the two-dimensional energy-critical wave equation, Duke Mathematical Journal, 50 (009), [3] J. F. Lam, B. Lippman, and F. Tappert, Self trapped laser beams in plasma, Phys. Fluid 0 (977), [4] M. Nakamura and T. Ozawa, Nonlinear Schrödinger equations in the Sobolev space of critical order, J. Funct. Anal. 55 (998), [5] K. Nakanishi, Energy scattering for nonlinear Klein-Gordon and Schrödinger equations in spatial dimensions and, J. Funct. Anal. 69 (999), 0 5. [6] F. Planchon and L. Vega, Bilinear virial identities and applications, Ann. Sci. Ec. Norm. Super., 4 (009), [7] B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in R. J. Funct. Anal., 9 (005), Department of Mathematics and Statistics,, University of Victoria, PO Box 3060 STN CSC, Victoria, BC, V8P 5C3, Canada address: ibrahim@math.uvic.ca URL: Faculty of Sciences of Tunis, Department of Mathematics address: Mohamed.Majdoub@fst.rnu.tn New York University, The Courant Institute for Mathematical Sciences, address: masmoudi@courant.nyu.edu Department of Mathematics, Kyoto University address: n-kenji@math.kyoto-u.ac.jp

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

ENERGY SCATTERING FOR THE 2D CRITICAL WAVE EQUATION

ENERGY SCATTERING FOR THE 2D CRITICAL WAVE EQUATION ENERGY SCATTERING FOR THE 2D CRITICAL WAVE EQUATION S. IBRAHIM, M. MAJDOUB, N. MASMOUDI, AND K. NAKANISHI Abstract. We investigate existence and asymptotic completeness of the wave operators for nonlinear

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

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

Energy 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 information

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

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

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

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

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

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

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

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

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

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

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

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 SOLUTIONS FOR A SEMILINEAR 2D KLEIN-GORDON EQUATION WITH EXPONENTIAL TYPE NONLINEARITY

GLOBAL SOLUTIONS FOR A SEMILINEAR 2D KLEIN-GORDON EQUATION WITH EXPONENTIAL TYPE NONLINEARITY GLOBAL SOLUTIONS FOR A SEMILINEAR D KLEIN-GORDON EQUATION WITH EXPONENTIAL TYPE NONLINEARITY S. Ibrahim Department of Mathematics & Statistics, McMaster University, 180 main Street West, Hamilton, Ontario,

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

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

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

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

VANISHING-CONCENTRATION-COMPACTNESS ALTERNATIVE FOR THE TRUDINGER-MOSER INEQUALITY IN R N

VANISHING-CONCENTRATION-COMPACTNESS ALTERNATIVE FOR THE TRUDINGER-MOSER INEQUALITY IN R N VAISHIG-COCETRATIO-COMPACTESS ALTERATIVE FOR THE TRUDIGER-MOSER IEQUALITY I R Abstract. Let 2, a > 0 0 < b. Our aim is to clarify the influence of the constraint S a,b = { u W 1, (R ) u a + u b = 1 } on

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

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

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

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

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

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

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 NLS with a potential on the line

Scattering for NLS with a potential on the line Asymptotic Analysis 1 (16) 1 39 1 DOI 1.333/ASY-161384 IOS Press Scattering for NLS with a potential on the line David Lafontaine Laboratoire de Mathématiques J.A. Dieudonné, UMR CNRS 7351, Université

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

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

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

Global unbounded solutions of the Fujita equation in the intermediate range

Global 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 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

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

Global Well-Posedness for Energy Critical Fourth-Order Schrödinger Equations in the Radial Case

Global 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 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

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

arxiv: v1 [math.ap] 28 Mar 2014

arxiv: v1 [math.ap] 28 Mar 2014 GROUNDSTATES OF NONLINEAR CHOQUARD EQUATIONS: HARDY-LITTLEWOOD-SOBOLEV CRITICAL EXPONENT VITALY MOROZ AND JEAN VAN SCHAFTINGEN arxiv:1403.7414v1 [math.ap] 28 Mar 2014 Abstract. We consider nonlinear Choquard

More information

EXISTENCE OF SOLUTIONS FOR CROSS CRITICAL EXPONENTIAL N-LAPLACIAN SYSTEMS

EXISTENCE OF SOLUTIONS FOR CROSS CRITICAL EXPONENTIAL N-LAPLACIAN SYSTEMS Electronic Journal of Differential Equations, Vol. 2014 (2014), o. 28, pp. 1 10. ISS: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTECE OF SOLUTIOS

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

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

NONLINEAR SCHRÖDINGER ELLIPTIC SYSTEMS INVOLVING EXPONENTIAL CRITICAL GROWTH IN R Introduction

NONLINEAR SCHRÖDINGER ELLIPTIC SYSTEMS INVOLVING EXPONENTIAL CRITICAL GROWTH IN R Introduction Electronic Journal of Differential Equations, Vol. 014 (014), No. 59, pp. 1 1. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu NONLINEAR SCHRÖDINGER

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

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

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/ 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

Long-term dynamics of nonlinear wave equations

Long-term dynamics of nonlinear wave equations Long-term dynamics of nonlinear wave equations W. Schlag (University of Chicago) Recent Developments & Future Directions, September 2014 Wave maps Let (M, g) be a Riemannian manifold, and u : R 1+d t,x

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

EXISTENCE OF NONTRIVIAL SOLUTIONS FOR A QUASILINEAR SCHRÖDINGER EQUATIONS WITH SIGN-CHANGING POTENTIAL

EXISTENCE 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 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

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

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

Higher 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 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 information

Strichartz Estimates for the Schrödinger Equation in Exterior Domains

Strichartz 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 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

arxiv: v3 [math.ap] 1 Sep 2017

arxiv: v3 [math.ap] 1 Sep 2017 arxiv:1603.0685v3 [math.ap] 1 Sep 017 UNIQUE CONTINUATION FOR THE SCHRÖDINGER EQUATION WITH GRADIENT TERM YOUNGWOO KOH AND IHYEOK SEO Abstract. We obtain a unique continuation result for the differential

More information

GAUSSIAN MEASURE OF SECTIONS OF DILATES AND TRANSLATIONS OF CONVEX BODIES. 2π) n

GAUSSIAN MEASURE OF SECTIONS OF DILATES AND TRANSLATIONS OF CONVEX BODIES. 2π) n GAUSSIAN MEASURE OF SECTIONS OF DILATES AND TRANSLATIONS OF CONVEX BODIES. A. ZVAVITCH Abstract. In this paper we give a solution for the Gaussian version of the Busemann-Petty problem with additional

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

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

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

EXISTENCE OF SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS WITH UNBOUNDED POTENTIAL. 1. Introduction In this article, we consider the Kirchhoff type problem

EXISTENCE 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 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

A survey on l 2 decoupling

A survey on l 2 decoupling A survey on l 2 decoupling Po-Lam Yung 1 The Chinese University of Hong Kong January 31, 2018 1 Research partially supported by HKRGC grant 14313716, and by CUHK direct grants 4053220, 4441563 Introduction

More information

On the role of geometry in scattering theory for nonlinear Schrödinger equations

On the role of geometry in scattering theory for nonlinear Schrödinger equations On the role of geometry in scattering theory for nonlinear Schrödinger equations Rémi Carles (CNRS & Université Montpellier 2) Orléans, April 9, 2008 Free Schrödinger equation on R n : i t u + 1 2 u =

More information

SYMMETRY IN REARRANGEMENT OPTIMIZATION PROBLEMS

SYMMETRY IN REARRANGEMENT OPTIMIZATION PROBLEMS Electronic Journal of Differential Equations, Vol. 2009(2009), No. 149, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SYMMETRY IN REARRANGEMENT

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

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

LIFE SPAN OF BLOW-UP SOLUTIONS FOR HIGHER-ORDER SEMILINEAR PARABOLIC EQUATIONS

LIFE SPAN OF BLOW-UP SOLUTIONS FOR HIGHER-ORDER SEMILINEAR PARABOLIC EQUATIONS Electronic Journal of Differential Equations, Vol. 21(21), No. 17, pp. 1 9. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu LIFE SPAN OF BLOW-UP

More information

Nonlinear elliptic systems with exponential nonlinearities

Nonlinear elliptic systems with exponential nonlinearities 22-Fez conference on Partial Differential Equations, Electronic Journal of Differential Equations, Conference 9, 22, pp 139 147. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu

More information

The Semilinear Klein-Gordon equation in two and three space dimensions

The Semilinear Klein-Gordon equation in two and three space dimensions Università degli Studi Roma Tre Facoltà di Scienze Matematiche, Fisiche e Naturali Corso di Laurea Magistrale in Matematica Tesi di Laurea Magistrale in Matematica Sintesi The Semilinear Klein-Gordon equation

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

Endpoint Strichartz estimates for the magnetic Schrödinger equation

Endpoint Strichartz estimates for the magnetic Schrödinger equation Journal of Functional Analysis 58 (010) 37 340 www.elsevier.com/locate/jfa Endpoint Strichartz estimates for the magnetic Schrödinger equation Piero D Ancona a, Luca Fanelli b,,luisvega b, Nicola Visciglia

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

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

STRICHARTZ ESTIMATES ON EXTERIOR POLYGONAL DOMAINS

STRICHARTZ ESTIMATES ON EXTERIOR POLYGONAL DOMAINS STRICHARTZ ESTIMATES ON EXTERIOR POLYGONAL DOMAINS DEAN BASKIN, JEREMY L. MARZUOLA, AND JARED WUNSCH Abstract. Using a new local smoothing estimate of the first and third authors, we prove local-in-time

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

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

Takens embedding theorem for infinite-dimensional dynamical systems

Takens embedding theorem for infinite-dimensional dynamical systems Takens embedding theorem for infinite-dimensional dynamical systems James C. Robinson Mathematics Institute, University of Warwick, Coventry, CV4 7AL, U.K. E-mail: jcr@maths.warwick.ac.uk Abstract. Takens

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

Global well-posedness of the primitive equations of oceanic and atmospheric dynamics

Global well-posedness of the primitive equations of oceanic and atmospheric dynamics Global well-posedness of the primitive equations of oceanic and atmospheric dynamics Jinkai Li Department of Mathematics The Chinese University of Hong Kong Dynamics of Small Scales in Fluids ICERM, Feb

More 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

Inégalités de dispersion via le semi-groupe de la chaleur

Inégalités 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 information

Nonlinear Schrödinger equation with combined power-type nonlinearities and harmonic potential

Nonlinear 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 information

Two dimensional exterior mixed problem for semilinear damped wave equations

Two dimensional exterior mixed problem for semilinear damped wave equations J. Math. Anal. Appl. 31 (25) 366 377 www.elsevier.com/locate/jmaa Two dimensional exterior mixed problem for semilinear damped wave equations Ryo Ikehata 1 Department of Mathematics, Graduate School of

More information

EXISTENCE OF SOLUTIONS TO ASYMPTOTICALLY PERIODIC SCHRÖDINGER EQUATIONS

EXISTENCE 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 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

RANDOM PROPERTIES BENOIT PAUSADER

RANDOM PROPERTIES BENOIT PAUSADER RANDOM PROPERTIES BENOIT PAUSADER. Quasilinear problems In general, one consider the following trichotomy for nonlinear PDEs: A semilinear problem is a problem where the highest-order terms appears linearly

More information

DECAY ESTIMATES FOR THE KLEIN-GORDON EQUATION IN CURVED SPACETIME

DECAY ESTIMATES FOR THE KLEIN-GORDON EQUATION IN CURVED SPACETIME Electronic Journal of Differential Equations, Vol. 218 218), No. 17, pp. 1 9. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu DECAY ESTIMATES FOR THE KLEIN-GORDON EQUATION

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

Non-radial solutions to a bi-harmonic equation with negative exponent

Non-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 information

MEAN CURVATURE FLOW OF ENTIRE GRAPHS EVOLVING AWAY FROM THE HEAT FLOW

MEAN CURVATURE FLOW OF ENTIRE GRAPHS EVOLVING AWAY FROM THE HEAT FLOW MEAN CURVATURE FLOW OF ENTIRE GRAPHS EVOLVING AWAY FROM THE HEAT FLOW GREGORY DRUGAN AND XUAN HIEN NGUYEN Abstract. We present two initial graphs over the entire R n, n 2 for which the mean curvature flow

More information

Proceedings of the 5th International Conference on Inverse Problems in Engineering: Theory and Practice, Cambridge, UK, 11-15th July 2005

Proceedings of the 5th International Conference on Inverse Problems in Engineering: Theory and Practice, Cambridge, UK, 11-15th July 2005 Proceedings of the 5th International Conference on Inverse Problems in Engineering: Theory and Practice, Cambridge, UK, 11-15th July 2005 SOME INVERSE SCATTERING PROBLEMS FOR TWO-DIMENSIONAL SCHRÖDINGER

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

SOLUTIONS TO SINGULAR QUASILINEAR ELLIPTIC EQUATIONS ON BOUNDED DOMAINS

SOLUTIONS TO SINGULAR QUASILINEAR ELLIPTIC EQUATIONS ON BOUNDED DOMAINS Electronic Journal of Differential Equations, Vol. 018 (018), No. 11, pp. 1 1. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu SOLUTIONS TO SINGULAR QUASILINEAR ELLIPTIC EQUATIONS

More information

HARNACK S INEQUALITY FOR GENERAL SOLUTIONS WITH NONSTANDARD GROWTH

HARNACK S INEQUALITY FOR GENERAL SOLUTIONS WITH NONSTANDARD GROWTH Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 37, 2012, 571 577 HARNACK S INEQUALITY FOR GENERAL SOLUTIONS WITH NONSTANDARD GROWTH Olli Toivanen University of Eastern Finland, Department of

More information

Singular Limits of the Klein-Gordon Equation

Singular Limits of the Klein-Gordon Equation Singular Limits of the Klein-Gordon Equation Abstract Chi-Kun Lin 1 and Kung-Chien Wu 2 Department of Applied Mathematics National Chiao Tung University Hsinchu 31, TAIWAN We establish the singular limits

More information

PARTIAL REGULARITY OF BRENIER SOLUTIONS OF THE MONGE-AMPÈRE EQUATION

PARTIAL REGULARITY OF BRENIER SOLUTIONS OF THE MONGE-AMPÈRE EQUATION PARTIAL REGULARITY OF BRENIER SOLUTIONS OF THE MONGE-AMPÈRE EQUATION ALESSIO FIGALLI AND YOUNG-HEON KIM Abstract. Given Ω, Λ R n two bounded open sets, and f and g two probability densities concentrated

More information

Xiaoyi Zhang. Educational and Professional History 2003 Ph.D. in Mathematics, Graduate school of China Academy of Engineering Physics at Beijing.

Xiaoyi Zhang. Educational and Professional History 2003 Ph.D. in Mathematics, Graduate school of China Academy of Engineering Physics at Beijing. Xiaoyi Zhang Personal Information: Current Work Address: Department of Mathematics 14 Maclean Hall University of Iowa Iowa city, IA, 52242 Office Phone: 319-335-0785 E-mail: xiaozhang@math.uiowa.edu Educational

More information