### 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

 2018 年度

12月10日（月）

We consider the semilinear heat equation with supercritical power nonlinearity, and show the existence of backward self-similar solutions by an ODE shooting method. As an application, we will construct peaking solutions by connecting a backward self-similar solution with a forward self-similar solution. In particular, we show the existence of incomplete blow-up solutions with blow-up profile above the singular steady state. This talk is based on a joint work with Takasi Senba (Fukuoka University).

12月17日（月）

In this talk, we develop the R-boundedness for the generalized Stokes resolvent problem in an infinite layer, with Neumann boundary condition on both upper and lower boundary. This has not been proved for such a boundary condition, while it has been proved for Neumann and Dirichlet boundary condition on upper and lower boundary, respectively. As an application, we also establish the local well-posedness for the incompressible Navier-Stokes equation in an infinite layer with a free surface for both upper and lower boundaries.

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

プログラム：TBA

1月21日（月）

 過去のセミナー

4月16日（月）

4月23日（月）

Levinson の定理は, 量子系の散乱に由来するある量がその系の束縛状態の数に等しいという主張です. Levinson (1949) が球対称ポテンシャルをもつ Schrödinger 作用素に対して証明して以来, 様々なモデルに対して調べられてきました. 一般的な証明の多くが複素解析などに基づく一方, Kellendonk-Richard (2007) は全く異なるアプローチを提案しました. 彼らは, C* 環の K 理論というトポロジーの手法を用いることで, Levinson の定理の正体が実は Atiyah-Singer の指数定理であることを明らかにしました. これにより束縛状態が有限個の場合, Levinson の定理の証明はある種の C* 環の問題に帰着されます. それでは, 束縛状態の数が無限個ある場合はどうでしょうか? 本講演では, 半直線上のある Schrödinger 作用素をモデルとして, 束縛状態が無限個の場合の Levinson の定理について考えます. このモデルに対して, (概) 周期的擬微分作用素から生成される C* 環を考えることで, 無限個の場合でも意味のある等式が得られることを紹介します. 本講演は, S. Richard 氏 (名大) との共同研究に基づきます.

5月7日（月）

We study the Cauchy problem of the linear damped wave equation and give sharp $L^p$-$L^q$ estimates of the solution. This is an improvement of the so-called Matsumura estimates. Moreover, as its application, we consider the nonlinear problem with slowly decaying initial data, and determine the critical exponent. In particular, we prove that the small data global existence holds in the critical case if the initial data does not belong to $L^1$. This talk is based on a joint work with Masahiro Ikeda (RIKEN), Mamoru Okamoto (Shinshu University), and Takahisa Inui (Osaka University).

5月14日（月）

5月21日（月）

5月28日（月）

We study the periodic traveling wave solutions of the derivative nonlinear Schrödinger equation (DNLS). It is known that (DNLS) has two types of solitons on the whole line; one has exponential decay and the other has algebraic decay. The latter corresponds to the soliton for the massless case. In the new global results recently obtained by Fukaya, Hayashi and Inui, the properties of two-parameter of the solitons are essentially used in the proof, and especially the soliton for the massless case plays an important role. To investigate further properties of the solitons, we construct exact periodic traveling wave solutions which yield the solitons on the whole line including the massless case in the long-period limit. Moreover, we study the regularity of the convergence of these exact solutions in the long-period limit.

6月4日（月）

In this talk I will focus on the asymptotic behavior of nonsmooth radial solutions of semilinear Schrödinger equations with a barely supercritical nonlinearity (i.e a nonlinearity that grows faster than the critical power but not faster than a logarithm). It is known that we have scattering of smooth radial solutions of defocusing loglog energy-supercritical Schrödinger equations. I will recall the techniques used to prove this result. Then I will explain how we can use Jensen-type inequalities to prove scattering of nonsmooth radial solutions of defocusing loglog energy-supercritical Schrödinger equations.

6月11日（月）

6月18日（月）

In this talk, I will discuss about a critical exponent for semilinear wave equations with time-dependent damping. When the damping is “effective,” it is Fujita exponent which is known to be the one for semilinear heat equations. Recently, by showing a sub-critical blow-up result, I have introduced a new conjecture that it is Strauss exponent which is known to be one for semilinear wave equations as far as the damping is “scattering.” I will also discuss about other nonlinearities and an intermediate situation, namely, the scaling invariant case. All the results in this talk are joint works with Ning-An Lai (Lishui University, China).

7月2日（月）

[集中講義 ] 7月9〜13日

1. 序 (動機)
2. 定義
3. 扱える方程式の例
4. 比較原理 (一意性)
5. 比較定理
6. 比較原理再訪
7. 最大値原理
8. ハルナック不等式

[研究集会 ] 9月18日 (火) 〜 19日 (水)
「Workshop on the Navier–Stokes flow」

プログラム：PDF file (Website)

10月1日（月）

In this talk we show boundedness of spectral multipliers for Schrödinger operators on an arbitrary open set. Furthermore, we present its application to the theory of Besov spaces and bilinear estimates. This talk is based on the joint work with T. Iwabuchi (Tohoku Univ.) and T. Matsuyama (Chuo Univ.).

10月15日（月）[名古屋確率論セミナーとの合同開催]

In this talk we give a simple introduction of Gubinelli-Imkeller-Perkowski's paracontrolled calculus. (This is basically a survey talk, but at the end we may present our own result a little bit.) This theory solves many formerly ill-defined, but physically important stochastic PDEs and is now competing with Hairer's regularity structure theory. Fortunately, paracontrolled calculus is based on existing theories and therefore not too big. It uses Besov space theory, in particular, Bony's paradifferential calculus. To make our presentation clear to non-experts, we give up generality and focus on the most important example, namely, the 3D dynamic $\Phi^4$-model (also known as the 3D stochastic quantization equation). It is a singular SPDE on $(0, \infty) \times T^3$ and looks like this: $$\partial_t u= \triangle_x u -u^3 +\xi \quad(\mbox{with u_0 given}).$$ Here, $\xi$ is a space-time white noise and $T^3$ is the 3 dimensional torus.

10月22日（月）

We prove existence of weak solutions for a diffuse interface model for the flow of two viscous incompressible Newtonian fluids with different densities in a bounded domain in two and three space dimensions. In contrast to previous works, we study a model with a singular non-local free energy, which controls the fractional Sobolev norm of the volume fraction. We show existence of weak solutions for large times with the aid of an implicit time discretization. This talk is based on a joint work with Helmut Abels (Regensburg).

10月29日（月）

The primitive equations is considered to be a fundamental model for geophysical flows, e.g. the ocean and the atmosphere. We can formally derive this equations from a scaled Navier-Stokes equations. In this talk we will give mathematically rigorous justification of its derivations in general settings. This talk is based on collaborative research with Prof. Giga (Tokyo univ.), Prof. Hieber (TU Darmstadt), Prof. Hussein (TU Darmstadt), Prof. Kashiwabara (Tokyo univ.) and Mr, Wrona (TU Darmstadt).

11月5日（月）

In Damage and Fracture Mechanics, the degree of damage in a test specimen is often represented in terms of a phase-field, whose evolution is usually described by a gradient flow of a free energy. On the other hand, due to strongly irreversible characteristics of damaging phenomena, the phase-field (i.e., the degree of damage) is supposed to be monotone in time. Accordingly, evolution laws of such phase-fields are often given in terms of gradient flows with constraints. This talk is concerned with an Allen-Cahn type equation with the positive-part function, which is a typical example of constrained gradient flows and would be a good test bed to develop techniques for dealing with gradient flows with constraints and to investigate influence of such constraints upon properties and behaviors of solutions. In this talk, we shall discuss well-posedness of the Allen-Cahn type equation in (possibly) unbounded domains by developing (re)formulations of the equation as well as energy techniques based on subdifferential calculus. Moreover, we shall also overview qualitative properties and asymptotic behaviors of solutions and some of them appear to be peculiar and different from classical Allen-Cahn equations.

[集中講義 ] 11月12〜16日

・自己共役作用素のスペクトル理論の復習
・加藤の滑らかな摂動の理論
・シュレディンガー作用素に対する一様ソボレフ評価
・2体散乱理論への応用：波動作用素の存在と漸近完全性

11月19日（月）[研究集会「Partial Differential Equation and General Relativity」との合同開催]
15:00 〜 16:00

In this lecture, based on joint work with Peter Hintz, I will discuss Kerr-de Sitter black holes, which are rotating black holes in a universe with a positive cosmological constant, i.e. they are explicit solutions (in 3+1 dimensions) of Einstein's equations of general relativity. They are parameterized by their mass and angular momentum. I will first discuss the geometry of these black holes as well as that of the underlying de Sitter space, and then talk about the stability question for these black holes in the initial value formulation. Namely, appropriately interpreted, Einstein's equations can be thought of as quasilinear wave equations, and then the question is if perturbations of the initial data produce solutions which are close to, and indeed asymptotic to, a Kerr-de Sitter black hole, typically with a different mass and angular momentum. In the last part of the talk I will discuss analytic aspects of the stability problem, in particular showing that Kerr-de Sitter black holes with small angular momentum are stable in this sense.

16:10 〜 17:40

2次元 Euclid 空間上の, ある種の零構造を持つ2次の微分型非線形項を伴う Schrödinger 方程式系に対する解の時間大域存在を考える. Li--Sunagawa（'16）によって1次元 Euclid 空間上の3次の微分型非線形項を持つ場合には結果が得られているが, 空間次元が2次元の場合は主に Sobolev 埋め込みに起因する困難により未解決であった. 本講演では, 2次元の場合にも同様の結果が得られたことを報告する. なお, 本講演は大阪大学の砂川秀明准教授との共同研究に基づく.

[研究集会 ] 11月20日 (火) 11:00〜
「Partial Differential Equation and General Relativity」

プログラム：PDF file

11月26日（月）

Explicit and partly sharp estimates are given of integrals over the square of Bessel functions with an integrable weight which can be singular at the origin. They are uniform with respect to the order of the Bessel functions and provide explicit bounds for the $L^{2}$ restrictions of Fourier transforms onto spheres in ${\bf R}^{n}$ which are independent of the radius of the sphere. This talk is based on a joint work with Hubert Kalf (Munich Univ.) and Osanobu Yamada (Ritsumeikan University).