Nagoya Differintial Equations Seminar

Organizers: Mitsuru Sugimoto, Toshiaki Hishida, Kotaro Tsugawa, Jun Kato, Yutaka Terasawa

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.

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).

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.

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.

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.

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.

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.

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.

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.

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.

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 $H^{\frac{1}{2}}(\mathbb{R}^2)$.
Reference
[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.

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.).

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.