### 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

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）

11月5日（月）

[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）

11月26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.