## 大阪市大・大阪府大合同「南大阪応用数学セミナー」（2017年度）

 連絡先 ：高橋 太 〒558-8585 大阪府大阪市住吉区杉本３丁目３番１３８号 大阪市立大学大学院理学研究科数物系専攻・数学研究所 電話：06－6605－2508 E-mail ：futoshi@sci.osaka-cu.ac.jp（高橋） 運営委員 ：高橋 太、阿部 健、村井 実 、橋本 伊都子(大阪市立大・理学研究科/数学研究所）、 　壁谷 喜継（大阪府立大・工）

 第38回「南大阪応用数学セミナー」

タイトル A unification of theory of well-posedness for delay differential equations
アブストラクト Time-delay systems are dynamic systems for which the past affects the future behaviors. These systems appear in mathematical models of various dynamic phenomena, e.g., dynamics of the concentration of circulating blood cells, economic fluctuations, El nino-Sounthern Oscillation, etc. In this talk, we study the well-posedness of the initial value problems of delay differential equations, which are time-delay systems, by a unified perspective for general delay structures. This gives topological semi-dynamical systems for various delay differential equations including unbounded state-dependent delay.

タイトル Energy concentration phenomena of a semilinear Neumann problem in a non-smooth domain
アブストラクト In this talk we consider the energy concentration phenomena for singularly perturbed Neumann problems. For the energy concentration phenomena, various results are obtained, especially asymptotic formulae of the energy and asymptotic profiles of solutions. By those precedent studies, it is known that the mean curvature of the boundary of a domain plays an important role if the boundary is sufficiently smooth. In this talk we consider the singularly perturbed Neumann problem in a non-smooth domain, which is like a cone around some point at the boundary. In this case the solid angle at the non-regular point plays a similar role to the mean curvature as in the case that the boundary is smooth.

タイトル 高次元空間上におけるバーガーズ方程式の球対称解の漸近挙動について
アブストラクト 高次元空間上におけるバーガーズ方程式の球対称解について考察する． 一次元バーガーズ方程式の初期値境界値問題の漸近形としては現れない漸近形が， 高次元空間上におけるバーガーズ方程式の球対称問題において現れることを示す. 講演では，球対称解の漸近挙動を決定する種々の境界条件と空間次元数の相関を論じる． また，直近の成果として，空間3次元の場合は完全に漸近挙動を分類することに 成功したのでこの結果を紹介する． 本研究は，松村昭孝氏（大阪大学名誉教授）との共同研究に基づく．
 第37回「南大阪応用数学セミナー」

タイトル Convergence of the Allen-Cahn equation with non-local term to the volume preserving mean curvature flow
アブストラクト The Allen-Cahn equation with non-local term for the volume preserving mean curvature flow studied by Rubinstein and Sternberg is well known. However, whether the solution converges to the time global weak solution of the volume preserving mean curvature flow or not is an open problem, due to the difficulty of estimates of the Lagrange multipliers. In this talk, we consider the Allen-Cahn equation with non-local term studied by Golovaty. We show the convergence of the Allen-Cahn equation to the mean curvature flow with non-local term, which satisfies a generalized condition for the volume preserving property.

タイトル Existence and nonexistence of solutions for the heat equation with a superlinear source term
アブストラクト Classification theory on the existence and non-existence of local in time solutions for initial value problems of nonlinear heat equations are investigated. Without assuming a concrete growth rate on a nonlinear term, we reveal the threshold integrability of initial data which classify existence and nonexistence of solutions via a quasi-scaling and its invariant integral. Typical nonlinear terms, for instance polynomial type, exponential type and its sum, product and composition, can be treated as applications. This is a joint work with Yohei Fujishima (Shizuoka University).

Luca Martinazzi 氏 (Universität Basel)
タイトル An application of Q-curvature to local and non-local critical embeddings
アブストラクト I will briefly discuss the critical embedding of Trudinger, Moser and Adams of the Sobolev space $H^m_0(\Omega)$ into Orlicz spaces ($\Omega\subset R^{2m}$ bounded) and show a concentration-compactness result for critical points of such embedding. Quite interestingly the proof of this result relies on the properties of conformal metrics on $R^{2m}$ having constant Q-curvature, which I will also discuss. Recently in a joint work with A. Maalaoui and A. Schikorra, and based on a work of A. Hyder we extended the previous results to the non-local case, which I will also discuss.
 第36回「南大阪応用数学セミナー」

タイトル Global well-posedness of the two-dimensional exterior Navier-Stokes equations for non-decaying data
アブストラクト It is well known that the two-dimensional Navier-Stokes equations have non-trivial stationary solutions, which are asymptotically constant and with a finite Dirichlet integral. On the other hand, few results are known about the non-stationary problem on such non-decaying solutions. In this talk, we report some global well-posedness result for bounded initial data with a finite Dirichlet integral, and existence of asymptotically constant solutions for arbitrary large Reynolds numbers.

タイトル On asymptotic behavior of solutions to nonlinear Schrodinger equation with critical homogeneous nonlinearity
アブストラクト In this talk, we consider final value problem of the nonlinear Schrodinger equation with a critical homogeneous nonlinearity. When the nonlinearity is at the critical order, the asymptotic behavior of the solution depends on the shape of the nonlinearity. We first introduce a sufficient condition on the nonlinearity for that the NLS equation admits a solution behaves like a free solution (short-range case) or a free solution with a phase correction (long-range case). We also give a partial result on the second asymptotic profile of the solution. This talk is based on a joint work with Hayato Miyazaki (NIT, Tsuyama College) and Kota Uriya (Okayama university of Science).

タイトル Malliavin calculus for stochastic functional differential equations
アブストラクト The density for the probability law of the solution to a stochastic (ordinary) differential equation has been studied via partial differential equations or potential theory. P. Malliavin constructed the theory on the differential calculus over the Wiener space about almost 40 years ago, in order to give the probabilistic approach to the hypoelliptic problem on the second-order differential operator associated with the equation, which is now called the Malliavin calculus after his great contribution. In this talk, I will introduce a stochastic functional differential equation, in which the coefficients of the equation depend on the past histories. Since the equation determines the non-Markovian process, we cannot use any fruitful methods such as in the study of the Markovian case. The main goal is to study the properties of the density from the viewpoint of the Malliavin calculus.