1,720,971 research outputs found
SCATTERING FOR DEFOCUSING GENERALIZED BENJAMIN-ONO EQUATION IN THE ENERGY SPACE H-1/2 (R)
No full textWe prove the scattering for the defocusing generalized Benjamin-Ono equation in the energy space H-1/2 (R). We first establish the monotonicity formula that describes the unidirectional propagation. More precisely, it says that the center of energy moves faster than the center of mass. This type of monotonicity was first observed by Tao in the defocusing gKdV equations. We use the monotonicity in the setting of compactness-contradiction argument to prove the large data scattering in the energy space H-1/2 (R). On the way, we extend the critical local theory of Vento to the subcritical regime. Indeed, we obtain subcritical local theory and global well-posedness in the energy space.
On the fifth-order KdV equation: Local well-posedness and lack of uniform continuity of the solution map
No full textAbstractIn this paper we prove that the following fifth-order equation arising from the KdV hierarchy{∂tu+∂x5u+c1∂xu∂x2u+c2u∂x3u=0,u:Rt×Rx→R,u(0,x)=u0(x),u0∈Hs(R) is locally well-posed in Hs(R) for s>52. Also, we prove the solution map of the equation is not uniformly continuous on a bounded set
The modified scattering of two dimensional semi-relativistic Hartree equations
No full textIn this paper, we consider the asymptotic behaviors of small solutions to the semi-relativistic Hartree equations in two dimension. The nonlinear term is the cubic one convolved with the Coulomb potential |x|-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}|x|<^>{-1}\end{document}, and it produces the long-range interaction in the sense of scattering phenomenon. From this observation, one anticipates that small solutions converge to modified scattering states, although they decay as linear solutions. We show the global well-posedness and the modified scattering for small solutions in weighted Sobolev spaces. Our proof follows a road map of exploiting the space-time resonance by Germain et al. (Int Math Res Not 2009(3):414-432, 2008), and Pusateri (Commun Math Phys 332(3):1203-1234, 2014). Compared to the result in three dimensional case (Pusateri 2014), weaker time decay in two dimension is one of the main obstacles.
On finite time blow-up for the mass-critical Hartree equations
No full textWe consider the fractional Schrodinger equations with focusing Hartree-type nonlinearities. When the energy is negative, we show that the solution blows up in a finite time. For this purpose, based on Glassey's argument, we obtain a virial-type inequality
Global existence versus finite time blowup dichotomy for the system of nonlinear Schrödinger equations
No full textWe construct an extremizer for the Lieb-Thirring energy inequality (except the endpoint cases) developing the concentration-compactness technique for operator valued inequality in the formulation of the profile decomposition. Moreover, we investigate the properties of the extremizer, such as the system of Euler-Lagrange equations, regularity and summability. As an application, we study a dynamical consequence of a system of nonlinear Schrodinger equations with focusing cubic nonlinearities in three dimension when each wave function is restricted to be orthogonal. Using the critical element of the Lieb-Thirring inequality, we establish a global existence versus finite time blowup dichotomy. This result extends the single particle result of Holmer-Roudenko [35] to infinitely many particles system. (C) 2018 Elsevier Masson SAS. All rights reserved.
5계 분산방정식의 low regularity Cauchy problem에 관한 연구
No full text학위논문(박사) - 한국과학기술원 : 수리과학과, 2016.8
,[iii, 151 p. :]In this thesis, we are going to mainly discuss about the low regularity Cauchy problem for fifth order dispersive equations, in particular, the (integrable) fifth-order modified KdV equation
- + + 10u^2\partial_x^{3}u10(\partial_xu)^3 = 0 and the (integrable) fifth-order KdV equation
- + + + = 0
under the periodic boundary condition. Both equations are locally well-popsed via the standard energy method in the Sobolev space for s > 2 and , respectively, and the fifth-oreder KdV equation is, in particular, globally well-posed in the energy space thanks to the conservation law of the Hamiltonian.
In Chapter 1, we provide general theory concerning the low regularity Cauchy problem for dispersive equations, in particular, the Fourier restriction norm method. Moreover, we introduce KdV equation as the complete integrable system and provide a short proof of the local well-posedness, which is based on the Picard iteration method in addition to the space. The fifth-order KdV and modified KdV equations are also introduced in this chapter. Reviews and main idea to show the local well-posedness of certain equations will be provided.
In Chapter 2, we are going to show that nonlinear estimates for the fifth-order KdV and modified KdV equations fail in the standard space for any regularity. Also, we construct the short time -type function space to overcome the failure of nonlinear estimates and introduce good properties of this new space. At the end of this chapter, we provide a short proof of the local well-posedness for the non-periodic fifth-order KdV equation by using the short-times space, which is contained in the author's first work [12].
In Chapters 3 and 4, we provide main ingredients in this dissertation: Nonlinear estimates and energy estimates concerning both the fifth-order modified KdV and the fifth-order KdV equations, and proof of the local well-posedness for the fifth- order modified KdV equation on T. In particular, we show the unconditional local well-posedness for the fifth-order modified KdV equation for s > 7/2 in Chapter 3. Chapters 3 and 4 are based on the author's works [27] and [28], respectively.
In Appendix, for the completeness of the proof of the local well-posedness for the KdV equation on R in Section 1.2, we introduce Tao's [kZ]-multiplier, and using this, we prove the bilinear estimate.한국과학기술원 :수리과학과
무한 차원 해밀토니안 방정식의 심플렉틱 동역학
No full text학위논문(박사) - 한국과학기술원 : 수리과학과, 2017.2,[i, 47 p. :]We consider an invariance of the symplectic capacity and the nonsqueezing theorem for infinite dimensional Hamiltonian systems. The symplectic capacity for infinite dimensional Hamiltonian systems was first introduced by Kuksin[34]. This result also contained the invariance of the symplectic capacity for Hamiltonian systems in the specific conditions. Many authors have studied the nonsqueezing theorem for the Hamiltonian systems which do not satisfy Kuksin’s condition. They only proved the nonsqueezing theorem by estimating between original and frequency truncated solution flow, without considering the symplectic capacity. For example, Bourgain [7] proved the nonsqueezing property of the 1D cubic nonlinear equation. However, we focus back to the symplectic capacity. Applying the idea of Bourgain [7] to the method of Kuksin [34], we relax the conditions of the Hamiltonian system, which was used by Kuksin [34]. Heuristically, we prove the invariance of the symplectic capacity by approximating the solutions to the original infinite dimensional Hamiltonian system by a modified Hamiltonian system which has linear flow on high frequencies and nonlinear flow on low frequencies. We also consider concrete examples such as the higher-order KdV equation and the Zakharov system. The nonsqueezing property of the higher-order Korteweg-de Vries flow was proved by Hong and Kwak [25], but we can extend this result to the symplectic capacity. Furthermore, we also prove the invariance of the symplectic capacity by the Zakharov flow on L^2_x(\BbbT) \times H^{-1/2}_x,(\BbbT) \times H^{-3/2}_x (\BbbT), the sharp space has that the local well-posedness which can be obtained by space.한국과학기술원 :수리과학과
Well-posedness and ill-posedness of the fifth-order modified KdV equation
Get PDFWe consider the initial value problem of the fifth-order modified KdV equation on the Sobolev spaces.
\displaylines{
\partial_t u - \partial_x^5u + c_1\partial_x^3(u^3)
+ c_2u\partial_x u\partial_x^2 u + c_3uu\partial_x^3 u =0\cr
u(x,0)= u_0(x)
}
where and 's are real. We show the local well-posedness in for via the contraction principle on space. Also, we show that the solution map from data to the solutions fails to be uniformly continuous below . The counter example is obtained by approximating the fifth order mKdV equation by the cubic NLS equation.Mathematic
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
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