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Nagoya Differintial Equations Seminar

Date: Mondays, 16:00 ~ (90 min. ~ 120 min.)
Place: Rm. 453, Mathematics Bldg, Nagoya University
Organizers: Mitsuru Sugimoto, Toshiaki Hishida, Kotaro Tsugawa, Jun Kato, Yutaka Terasawa

2017 / 2018

April 10
Speaker: Shinya Kinoshita (Nagoya University)
Title: Local well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov system in two and three dimensions

April 17
Speaker: Shintaro Kondo (Gifu University)
Title: Almost periodic solutions to Hasegawa-Wakatani equations

Hasegawa-Wakatani equations describing plasma phenomena of nuclear fusion are nonlinear partial differential equations with two unknown functions (plasma density and electrostatic potential). I will talk about the initial boundary value problem for Hasegawa-Wakatani equations when the initial data is almost-periodic to the uniform magnetic field direction and the problem taking the zero limit of the plasma resistivity. In this presentation I will talk about mathematical theory of the almost-periodic function mainly.

April 24
Speaker: Tokio Matsuyama (Chuo University)
Title: Local energy decay for wave equation on exterior domains

I inform the recent result on local energy decay for wave equation on exterior domains. The resolvent estimate for high frequency is assumed. This talk is based on the joint work with Vladimir S. Gueorguiev (Univ. Pisa).

May 1
Speaker: Masahiro Ikeda (RIKEN AIP Center)
Title: Global dynamics below the ground state for the nonlinear Schrödinger equations with a repulsive potential

We study global dynamics of solutions to the Cauchy problem for the focusing semi-linear Schrödinger equation with a potential on the real line. The problem is locally well-posed in the energy space. Our aim in this presentation is to study global behavior of the solution and prove a scattering result and a blow-up result for the problem with the data whose mass-energy is less than that of the ground state, where the ground state is the unique radial positive solution to the stationary Schrödinger equation without the potential. The scattering result for the defocusing version was recently studied by Lafontain.

May 8
Speaker: Kotaro Tsugawa (Nagoya University)
Title: Ill-posedness of derivative nonlinear Schrödinger equations on the torus

May 15
Speaker: Naohito Tomita (Osaka University)
Title: Bilinear pseudo-differential operators of type 1, 1 and their application to the Kato-Ponce inequality

May 22
Speaker: Yin Yin Su Win (Yangon University / Kyoto University)
Title: Local well-posedness and Gibbs measure for one dimensional periodic quantum Zakharov equation

We study one dimensional quantum Zakharov equation (qZ) on periodic boundary condition. This equation is derived from the Zakharov equation by taking account of quantum number. We find the lowest possible regularities \(s_1\) and \(s_2\) that the qZ equation is locally well-posed for Schrödinger data in \(H^{s_1}(\mathbb{T})\) and wave data in \(H^{s_2}(\mathbb{T})\times H^{s_2-2}(\mathbb{T})\) by using the Fourier restriction norm method and fixed point theorem. Furthermore, we establish the Gibbs measure corresponding to Hamiltonian of qZ equation.

May 29
Speaker: Takanobu Tokumasu (Nagoya University)
Title: Splitting method for dispersion-generalized Benjamin-Ono equations

June 5
Speaker: Takanobu Hara (Hokkaido University)
Title: The Wolff potential estimate for solutions to elliptic equations with signed data

June 12
Speaker: Mads Kyed (Technical University Darmstadt)
Title: Time-periodic solutions to parabolic boundary value problems of Agmon-Douglis-Nirenberg type

In the field of partial differential equations, a celebrated and famous result of Agmon, Douglis and Nirenberg states that if an elliptic differential operator A satisfies the so-called complementing condition with respect to a number of boundary operators, then a solution to the corresponding boundary value problem satisfies an a priori \(L^p\) estimate. This theorem is fundamental to the investigation of both linear and non-linear boundary value problems of elliptic type. In this talk, I will consider the parabolic operator related to A and present a new result, which states that a time-periodic solution to the corresponding parabolic boundary value problem satisfies a similar \(L^p\) estimate. I will present the result in such a way that it contains the theorem of Agmon, Douglis and Nirenberg as a special case. I will use an approach based solely on Fourier multipliers.

June 19
Speaker: Hirokazu Saito (Waseda University)
Title: Analytic semigroups and the maximal $L_p$-$L_q$ regularity for a compressible fluid model of Korteweg type

June 26
Speaker: Tatsuki Kawakami (Ryukoku University)
Title: Existence of mild solutions for the Hamilton-Jacobi equation with critical fractional viscosity in the Besov spaces

July 10
Speaker: Ryo Takada (Kyushu University)
Title: Time periodic initial value problem for rotating stably stratified fluids

October 2
Speaker: Jayson Cunanan (Saitama University)
Title: Smoothing estimates for the kinetic transport equation at the critical regularity

We prove smoothing estimates for velocity averages of the kinetic transport equation in hyperbolic Sobolev spaces at the critical regularity, leading to a complete characterization of the allowable regularity exponents. Such estimates will be deduced from some mixed-norm estimates for the cone multiplier operator at a certain critical index. This is a joint work with Neal Bez and Sanghyuk Lee.

October 16
Speaker: Takahiro Okabe (Hirosaki University)
Title: Remark on the strong solvability of the Navier-Stokes equations in the weak $L^n$ space

October 23
Speaker: Sachiko Ishida (Chiba University)
Title: Blow-up solution for a fully parabolic degenerate Keller-Segel system in the super critical case

In this talk we consider a fully parabolic Keller-Segel system with "degenerate" diffusion. In the case of a "non-degenerate" diffusion, Cieślak-Stinner (2012, 2014) proved there is a finite-time blow-up result under the super critical condition. Although there is a results on blow-up in "degenerate" system, we do not know whether it is finite time or infinite time (I.-Seki-Yokota (2014)). Therefore we will consider the blow-up time in this talk.

October 30
Speaker: Rossella Agliardi (University of Bologna)
Title: Fourier methods and pseudodifferential equations in Finance

We show how new pricing formulas for exotic options can be derived within a Lévy framework. To the purpose, a unifying formula is obtained by solving some nested Cauchy problem for pseudodifferential equations generalizing Black–Scholes PDE. Several examples of pricing formulas under the Lévy processes are provided to illustrate the flexibility of the method.

November 6
Speaker: Hiroshi Wakui (Tohoku University)
Title: Unboundedness and concentration domain for solutions to a degenerate drift-diffusion equation with the mass critical exponent

November 20
Speaker: Shohei Nakamura (Tokyo Metropolitan University)
Title: Unboundedness and concentration domain for solutions to a degenerate drift-diffusion equation with the mass critical exponent

Recently, the orthonormal Strichartz estimate (say ONS) which is one of the generalization of the classical Strichartz estimate for free Schrödinger propagator is studied by few mathematicians — R. Frank, M. Lewin, E. Lieb, J. Sabin, R. Seiringer motivated by the theory for many-body fermions. They proved ONS for some pairs of exponents $p,q$ but completing the picture of admissible pairs $p,q$ for ONS is still open. I will talk about recent our result concerning to this problem and provide few techniques from Harmonic and real analysis to extend the picture. This talk is based on the joint work with N. Bez, Y. Hong, S. Lee and Y. Sawano.

November 27
Speaker: Hitoshi Tanaka (Tokyo Metropolitan University)
Title: The $n$-linear embedding theorem for dyadic rectangles

Let $\sigma_i$, $i=1,\ldots,n$, denote reverse doubling weights on $\mathbb{R}^d$, let $\mathcal{DR}(\mathbb{R}^d)$ denote the set of all dyadic rectangles on $\mathbb{R}^d$ (Cartesian products of usual dyadic intervals) and let $K:\,\mathcal{DR}(\mathbb{R}^d)\to[0,\infty)$ be a map. In this talk we give the $n$-linear embedding theorem for dyadic rectangles. That is, we prove the $n$-linear embedding inequality for dyadic rectangles \[ \sum_{R\in\mathcal{DR}(\mathbb{R}^d)} K(R)\prod_{i=1}^n\left|\int_{R}f_i\,{\rm d}\sigma_i\right| \le C \prod_{i=1}^n \|f_i\|_{L^{p_i}(\sigma_i)} \] can be characterized by simple testing condition \[ K(R)\prod_{i=1}^n\sigma_i(R) \le C \prod_{i=1}^n\sigma_i(R)^{\frac{1}{p_i}} \quad R\in\mathcal{DR}(\mathbb{R}^d), \] in the range 1 $< p_{i}<\infty$ and $\sum_{i=1}^n\frac{1}{p_i} >$ 1. As a corollary to this theorem, for reverse doubling weights, we verify a necessary and sufficient condition for which the weighted norm inequality for the multilinear strong positive dyadic operator and for multilinear strong fractional integral operator to hold. This is joint work with Professor Kôzô Yabuta.

December 4
Speaker: Shan Minjie (Peking University / Kyoto University)
Title: Local well-posedness for the two-dimensional Zakharov-Kuznetsov equation

The initial value problem for the two-dimensional Zakharov-Kuznetsov equation is locally well-posed in $H^{s}(\mathbb{R}^2)$ when $\frac{1}{2}<{s}$. Local well-posedness for the 2D ZK equation in $H^{\frac{1}{2}}(\mathbb{R}^2)$ corresponds to the non-admissible endpoint Strichartz estimate, however we combine one kind of sharp Strichartz estimate with modulation decompose technique to obtain local well-posedness in $B^{\frac{1}{2}}_{2,1}(\mathbb{R}^2)$ which is a subspace of $H^{\frac{1}{2}}(\mathbb{R}^2)$.
[1] Axel Grünrock and Sebastian Herr. The Fourier restriction norm method for the Zakharov-Kuznetsov equation. Discrete Contin. Dyn. Syst., 34(5) (2014), 2061-2068.
[2] M. Hadac, S. Herr and H. Koch. Well-posedness and scattering for the KP-II equation in a critical space. Ann. Inst. H. Poincare Anal. NonLineaire 26 (2009), 917-941.

December 18
Speaker: Mamoru Okamoto (Shinshu University)
Title: Long-time behavior of solutions to the fifth-order mKdV-type equation

January 15
Speaker: Ken Abe (Osaka City University)
Title: Axisymmetric flows in an exterior domain

We consider the three-dimensional Navier-Stokes equations for axisymmetric initial data. It is known that the Cauchy problem is globally well-posed for large axisymmetric initial data in $L_3$ with finite energy, if the swirl component of initial velocity is identically zero (with no swirl). However, unique solvability is unknown in general for the case with swirl. In this talk, we study axisymmetric flows with swirl in an exterior domain subject to the slip boundary condition. We report unique existence of global solutions for large axisymmetric data in $L_3$ with finite energy, satisfying a decay condition of the swirl component. This talk is based on a joint work with G. Seregin (St. Petersburg/ Oxford U.).

January 22
Speaker: Kota Saito (Nagoya University)
Title: Relations between existence of arithmetic progressions and Fractal dimensions

In this talk, we outline the results about relations between existence of arithmetic progressions (especially 'weak' arithmetic progressions) and fractal dimensions. We provide that sets of the real numbers must contain 'weak' arithmetic progressions of given length if the dimensions of the sets are near enough to 1. We also consider higher dimensional analogues of these problems. As a consequence, we obtain a discretised version of a 'reverse Kakeya problem.' This is a joint work with Jonathan M. Fraser and Han Yu. In the later of this talk, we discuss applications to number theory. Especially we provide the weak solution to the higher dimensional expansion of Erdös–Turán conjecture.