Kenji Nakanishi

J-GLOBAL         Last updated: Sep 20, 2019 at 15:25
Kenji Nakanishi
Kyoto University
Research Institute for Mathematical Sciences Basic Mathematics Research Section

Academic & Professional Experience

Kyoto University 

Published Papers

Kenji Nakanishi, Takuto Yamamoto
Mathematical Research Letters   26(1) 253-279   Jun 2019   [Refereed]
We consider the final-data problem for systems of nonlinear Schrödinger
equations with Tex subcritical nonlinearity. An asymptotically free solution
is uniquely obtained for almost every randomized asymptotic profile in
Tex, ex...
Slim Ibrahim, Hiroaki Kikuchi, Kenji Nakanishi, Juncheng Wei
   May 2019
Slim Ibrahim, Nader Masmoudi, Kenji Nakanishi, Federica Sani
Journal of Functional Analysis      Aug 2019   [Refereed]
Zihua Guo, Ji Li, Kenji Nakanishi, Lixin Yan
J. Differential Equations   265(11) 5656-5675   Dec 2018   [Refereed]
We consider the Tex estimates for the solutions to the wave and
Schrödinger equations in high dimensions. For the homogeneous estimates, we
show Tex estimates fail at the critical regularity in high
dimensions by using sta...
Kenji Nakanishi
Journal of the Mathematical Society of Japan   69(4) 1353-1401   Oct 2017   [Refereed]
Kenji Nakanishi
Communications in Mathematical Physics   354(1) 161-212   May 2017   [Refereed]
Zihua Guo, Zaher Hani, Kenji Nakanishi
Communications in Mathematical Physics   359(1) 265-295   Apr 2018   [Refereed]
We study the Cauchy problem for the 3D Gross-Pitaevskii equation. The global
well-posedness in the natural energy space was proved by Gérard
\cite{Gerard}. In this paper we prove scattering for small data in the same
space with some additional a...
Slim Ibrahim, Nader Masmoudi, Kenji Nakanishi
Analysis & PDE   9(2) 503-514   Mar 2016   [Refereed]
This article resolves some errors in the paper "Scattering threshold for the
focusing nonlinear Klein-Gordon equation", Analysis & PDE 4 (2011) no. 3,
405-460. The errors are in the energy-critical cases in two and higher
Kenji Nakanishi, Tristan Roy
Communications on Pure and Applied Analysis   15(6) 2023-2058   2016   [Refereed]
Yvan Martel, Frank Merle, Kenji Nakanishi, Pierre Raphael
Communications in Mathematical Physics   342(3) 1075-1106   Nov 2015   [Refereed]
Ioan Bejenaru, Zihua Guo, Sebastian Herr, Kenji Nakanishi
Analysis & PDE 8-8 (2015), 2029-2055      Apr 2015
The Cauchy problem for the Zakharov system in four dimensions is considered.
Some new well-posedness results are obtained. For small initial data, global
well-posedness and scattering results are proved, including the case of initial
data in the e...
Dan-Andrei Geba, Kenji Nakanishi, Xiang Zhang
International Mathematics Research Notices   2015(22) 11549-11565   Feb 2015   [Refereed]
The aim of this article is to prove that for the 2+1-dimensional equivariant
Faddeev model, which is a quasilinear generalization of the corresponding
nonlinear sigma model, small initial data in critical Besov spaces evolve into
global solutions ...
Zihua Guo, Sanghyuk Lee, Kenji Nakanishi, Chengbo Wang
Communications in Mathematical Physics   331(1) 239-259   Oct 2014   [Refereed]
We obtain scattering for the 3D Zakharov system with non-radial small data in
the energy space with angular regularity of degree one. The main ingredient is
a generalized Strichartz estimate for the Schrödinger equation in the space
of Tex ang...
Zihua Guo, Kenji Nakanishi, Shuxia Wang
Communications in Partial Differential Equations   39(6) 1158-1184   May 2014   [Refereed]
We consider the global dynamics below the ground state energy for the
Klein-Gordon-Zakharov system in the 3D radial case; and obtain the dichotomy
between scattering and finite time blow up.
Joachim Krieger, Kenji Nakanishi, Wilhelm Schlag
Mathematische Annalen   361(1-2) 1-50   Feb 2015   [Refereed]
We construct a center-stable manifold of the ground state solitons in the
energy space for the critical wave equation without imposing any symmetry, as
the dynamical threshold between scattering and blow-up, and also as a
collection of solutions w...
Chongsheng Cao, Slim Ibrahim, Kenji Nakanishi, Edriss S. Titi
Communications in Mathematical Physics   337(2) 473-482   Jul 2015   [Refereed]
In an earlier work we have shown the global (for all initial data and all
time) well-posedness of strong solutions to the three-dimensional viscous
primitive equations of large scale oceanic and atmospheric dynamics. In this
paper we show that for...
Joachim Krieger, Kenji Nakanishi, Wilhelm Schlag
Communications in Mathematical Physics   327(1) 309-332   Apr 2014   [Refereed]
For the critical focusing wave equation Tex on Tex in the
radial case, we establish the role of the "center stable" manifold Tex
constructed in \cite{KS} near the ground state Tex as a threshold between
type I blowup and...
Zihua Guo, Kenji Nakanishi, Shuxia Wang
Mathematical Research Letters   21(4) 733-755   2014   [Refereed]
We prove small energy scattering for the 3D Klein-Gordon-Zakharov system with
radial symmetry. The idea of proof is the same as the Zakharov system studied
in \cite{GN}, namely to combine the normal form reduction and the
radial-improved Strichart...
Zihua Guo, Kenji Nakanishi, Shuxia Wang
Advances in Mathematics   238(1) 412-441   May 2013   [Refereed]
We consider the global dynamics below the ground state energy for the
Zakharov system in the 3D radial case. We obtain dichotomy between the
scattering and the growup.
Zihua Guo, Kenji Nakanishi
International Mathematics Research Notices   2014(9) 2327-2342   Jan 2013   [Refereed]
We prove small energy scattering for the 3D Zakharov system with radial
symmetry. The main ingredients are normal form reduction and the
radial-improved Strichartz estimates.
Joachim Krieger, Kenji Nakanishi, Wilhelm Schlag
Discrete and Continuous Dynamical Systems   33(6) 2423-2450   2013   [Refereed]
In this paper we establish the existence of certain classes of solutions to
the energy critical nonlinear wave equation in dimensions 3 and 5 assuming that
the energy exceeds the ground state energy only by a small amount. No radial
assumption is ...
Slim Ibrahim, Nader Masmoudi, Kenji Nakanishi
Trans. Amer. Math. Soc.   366 5653-5669   May 2014   [Refereed]
We study global dynamics for the focusing nonlinear Klein-Gordon equation
with the energy-critical nonlinearity in two or higher dimensions when the
energy equals the threshold given by the ground state of a mass-shifted
equation, and prove that t...
Slim Ibrahim, Nader Masmoudi, Kenji Nakanishi
Journal of the European Mathematical Society   17(4) 819-835   2015   [Refereed]
Trudinger-Moser inequality is a substitute to the (forbidden) critical
Sobolev embedding, namely the case where the scaling corresponds to Tex.
It is well known that the original form of the inequality with the sharp
exponent (proved by Mos...
Dan-Andrei Geba, Kenji Nakanishi, Sarada G. Rajeev
Communications on Pure & Applied Analysis   11(5) 1923-1933   Sep 2012   [Refereed]
We study equivariant maps corresponding to the classical Skyrme model and the
Adkins-Nappi model, for which we prove global existence and scattering in
critical Sobolev-Besov spaces.
Kenji Nakanishi, Wilhelm Schlag
SIAM Journal on Mathematical Analysis   44(2) 1175-1210   Apr 2012   [Refereed]
We construct center-stable and center-unstable manifolds, as well as stable
and unstable manifolds, for the nonlinear Klein-Gordon equation with a focusing
energy sub-critical nonlinearity, associated with a family of solitary waves
which is gener...
Joachim Krieger, Kenji Nakanishi, Wilhelm Schlag
Mathematische Zeitschrift   272(1-2) 297-316   Oct 2012   [Refereed]
In this paper we obtain a global characterization of the dynamics of even
solutions to the one-dimensional nonlinear Klein-Gordon (NLKG) equation on the
line with focusing nonlinearity |u|^{p-1}u, p>5, provided their energy exceeds
that of the gro...
Joachim Krieger, Kenji Nakanishi, Wilhelm Schlag
American Journal of Mathematics   135(4) 935-965   Aug 2013   [Refereed]
We study global behavior of radial solutions for the nonlinear wave equation
with the focusing energy critical nonlinearity in three and five space
dimensions. Assuming that the solution has energy at most slightly more than
the ground states and ...
Kenji Nakanishi, Tuoc Van Phan, Tai-Peng Tsai
Journal of Functional Analysis   263(3) 703-781   Aug 2012   [Refereed]
Consider a nonlinear Schrödinger equation in Tex whose linear part has
three or more eigenvalues satisfying some resonance conditions. Solutions which
are initially small in Tex and inside a neighborhood of the
first excited st...
Kenji Nakanishi, Wilhelm Schlag
Calculus of Variations and Partial Differential Equations   44(1-2) 1-45   May 2012   [Refereed]
We extend our previous result on the nonlinear Klein-Gordon equation to the
nonlinear Schrodinger equation with the focusing cubic nonlinearity in three
dimensions, for radial data of energy at most slightly above that of the ground
state. We prov...
Kenji Nakanishi, Wilhelm Schlag
Journal of Differential Equations   250(5) 2299-2333   Mar 2011   [Refereed]
We study the focusing, cubic, nonlinear Klein-Gordon equation in 3D with
large radial data in the energy space. This equation admits a unique positive
stationary solution, called the ground state. In 1975, Payne and Sattinger
showed that solutions...
Slim Ibrahim, Nader Masmoudi, Kenji Nakanishi
Analysis & PDE   4(3) 405-460   2011   [Refereed]
We show scattering versus blow-up dichotomy below the ground state energy for
the focusing nonlinear Klein-Gordon equation, in the spirit of Kenig-Merle for
the Tex critical wave and Schrödinger equations. Our result includes the
Tex critica...
Lassaad Aloui, Slim Ibrahim, Kenji Nakanishi
Communications in Partial Differential Equations   36(5) 797-818   Jan 2011   [Refereed]
We derive a uniform exponential decay of the total energy for the nonlinear
Klein-Gordon equation with a damping around spatial infinity in the whole space
or in the exterior of a star shaped obstacle.
Slim Ibrahim, Mohamed Majdoub, Nader Masmoudi, Kenji Nakanishi
Duke Math. J.   150(2) 287-329   Nov 2009   [Refereed]
We investigate existence and asymptotic completeness of the wave operators
for nonlinear Klein-Gordon and Schrödinger equations with a defocusing
exponential nonlinearity in two space dimensions. A certain threshold is
defined based on the value...
Shu-Ming Chang, Stephen Gustafson, Kenji Nakanishi, Tai-Peng Tsai
SIAM Journal on Mathematical Analysis   39(4) 1070-1111   Nov 2006   [Refereed]
Nonlinear Schrödinger (NLS) equations with focusing power nonlinearities
have solitary wave solutions. The spectra of the linearized operators around
these solitary waves are intimately connected to stability properties of the
solitary waves, an...
Stephen Gustafson, Kenji Nakanishi, Tai-Peng Tsai
International Mathematics Research Notices   (66) 3559-3584   Jan 2004   [Refereed]
In this paper we study a class of nonlinear Schrödinger equations which
admit families of small solitary wave solutions. We consider solutions which
are small in the energy space Tex, and decompose them into solitary wave and
dispersive wave c...

Books etc

Invariant Manifolds and Dispersive Hamiltonian Evolution Equations
Kenji Nakanishi & Wilhelm Schlag
European Mathematical Society   2011