### 名古屋微分方程式セミナー

セミナー世話人：杉本充　菱田俊明　津川光太郎　加藤淳　寺澤祐高

 2017 年度

[研究集会 ] 3月15日 (木) 〜 16日 (金)
「第10回 名古屋微分方程式研究集会」

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 過去のセミナー

4月10日（月）

4月17日（月）

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.

4月24日（月）

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).

5月1日（月）

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.

5月8日（月）

5月15日（月）

5月22日（月）

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.

5月29日（月）

splitting method は数値計算で用いられる手法の一つで, 偏微分方程式 $u_t = (A+B)u$ の近似として, $u_t = Au$, $u_t = Bu$ の二式から作られるシステムを解くというものである. 今回, 分散項を一般化した Benjamin-Ono 方程式に対し, 二種類の近似方程式系を考え, 各々の近似解と真の解との誤差について適切な評価を得ることができたので, それを紹介する.

6月5日（月）

$p$-優調和関数に対する Wolff ポテンシャルによる各点評価について考える. この評価は, $p$-調和関数の Wiener の判定条件の必要性のため Kilpeläinen-Malý (1994) によって導入された. Trudinger-Wang　(2002) はこの評価に Poisson 変形を利用した新証明を与えた. 本講演では, Poisson 変形と Kilpeläinen-Malý の手法を組み合わせることで, この評価に新証明を与える.

6月12日（月）

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.

6月19日（月）

6月26日（月）

7月10日（月）

[談話会 ] 7月12日（水） 15:00 〜 16:00

ユークリッド空間上の連続関数は, 定義域を超曲面に制限することにより, その超曲面上の連続関数と見なすことができる. この主張において, 「連続」を「可積分」に置き換えることは可能だろうか? 超曲面の測度は 0 であるので, この場合はそこへの制限を自然な方法で定義できることすら必ずしも自明ではない. このような制限の存在を保証する一連の主張は「制限定理」と総称され, 掛谷問題などの調和解析の有名な未解決問題とも関連していることが知られている. 一方, 制限定理と偏微分方程式論との密接な関連性も認識されており, 例えば Strichartz 評価式や平滑化評価式といった Schrödinger 方程式の Cauchy 問題に関する基本的な評価式は, 制限定理から導出可能であることが知られている. この講演ではこれらについて概説するとともに, 近年取り組んでいる平滑化評価式の最良定数の問題, さらにはその Schrödinger 型方程式の Cauchy 問題の適切性に関する溝畑・竹内予想との関連性などについて述べたい.

10月2日（月）

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.

10月16日（月）

10月23日（月）17:00 〜

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.

10月30日（月）

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.

11月6日（月）

[集中講義 ] 11月13〜17日

・初期値問題の適切性に関する一般論
・局所解の構成
・解の長時間挙動 その1：ライフスパンの評価
・解の長時間挙動 その2：零構造と、弱い零構造
・まとめと展望

11月20日（月）

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.

11月27日（月）15:00 〜

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.

12月4日（月）

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.

[集中講義 ] 12月11〜15日

この講義の目標は, 幾何解析, 調和解析や分散偏微分方程式に現れる幾つかの重要な不等式を熱流単調性の手法を用いて証明する事である. 熱流単調性の手法は, Hölder の不等式, Loomis-Whitney の不等式, 畳み込みに関する Young の不等式などの古曲的な不等式に関する現代的なアプローチである. この手法に基づいて本講義の内容は次の通り:

1. イントロ
2. Brascamp-Lieb の不等式
3. 畳み込みに関する Young の不等式
4. Schrödinger 方程式に対する Strichartz 評価
5. Hausdorff-Young の不等式

12月18日（月）

5 階修正 KdV 型方程式の解の漸近挙動について考える． Sobolev 空間における時間局所的適切性は, Kwon ('08) による, 逐次近似法を用いる限り最良の結果が知られている. 本講演では, 解の時間減衰を得るため, 重み付き Sobolev 空間を用いる. Fourier 制限ノルム空間において Kwon ('08) が示した 3 重線形評価式を一般化し, 正則性が低い重み付き Sobolev 空間での適切性を示す. また, Ifrim and Tataru ('15) による「波束テスト法」(method of testing by wave packets) を用いて, 5 階修正 KdV 型方程式の解は自己相似解に漸近することを示す.

[研究集会 ] 1月5日 (金) 〜 6日 (土)
「若手による流体力学の基礎方程式研究集会」

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1月15日（月）

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.).

1月22日（月）

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.