## 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41回「南大阪応用数学セミナー」

Chun-Hsiung Hsia氏（National Taiwan University）
タイトル On the mathematical analysis of synchronization of the Kuramoto oscillators with inertia effect.
アブストラクト In this talk, we shall give a complete account of the theory on the synchronization issues for the Kuramoto oscillators with inertia effect. Both the frequency synchronization and phase synchronization are in view. The crux of analysis on this problem is due to the collisions and the lack of the uniform boundedness of the initial velocity. We shall do some nonlocal estimates on the speed functions to give good estimates on the diameter function to overcome the aforementioned crux. This is joint work with Bongsuk Kwon and Chang-Yeol Jung.

タイトル On a maximizing problem of the Sobolev embedding related to the space of bounded variation
アブストラクト In this talk, we consider the maximizing problem associated with Sobolev embedding related to the space of bounded variation of BV-functions, which is a substitute of the Sobolev space of the marginal case. In our setting of the maximizing problem, we suffer from the non-compactness due to the vanishing phenomenon and the non-reflexivity of the space of BV-functions. In order to overcome these difficulties, we use the fact that the family of maximizers of the Sobolev embedding with BV-functions is the set of characteristic functions on balls. Simultaneously, we give a characterization of maximizers of our problem to prove that the maximizers must form characteristic functions on balls and specify their radii and heights exactly. This is a joint work with Prof. Michinori Ishiwata in Osaka University.

Marta Calanchi氏（Universita degli Studi di Milano）
タイトル Some applications of weighted Trudinger-Moser inequalities
アブストラクト We discuss some Trudinger--Moser inequalities with weighted Sobolev norms. Suitable logarithmic weights in these norms allow an improvement in the maximal growth for integrability, when one restricts to radial functions.
The main results concern the application of these inequalities to the existence of solutions for certain mean-field equations of Liouville-type. Sharp critical thresholds are found such that for parameters below these thresholds the corresponding functionals are coercive and hence solutions are obtained as global minima of these functionals. In the critical cases the functionals are no longer coercive and solutions may not exist.
We also discuss a limiting case, in which the allowed growth is of double exponential type.
Joint work with Eugenio Massa, (Universidade de S ̃ao Paulo) and Bernhard Ruf (Universita degli Studi di Milano).
 第40回「南大阪応用数学セミナー」

タイトル Global-in-time existence and asymptotic behavior of solutions to a chemotaxis model for the nest building of termites
アブストラクト 白アリの造巣過程に対する数理モデルであるDeneubourg系(Insectes Sociaux 24 (1977)) の解の存在とその挙動を考える. Deneubourg系は，時間スケールを表す係数を零にすると， 線形減衰項付きのKeller-Segel系に帰着する． Keller-Segel系は解の爆発（走化性崩壊）を起こすことが知られているが， 線形減衰でこれが抑えられるのかという文脈からも, Deneubourg系の解の時間大域存在は興味深い． 本講演では，走化性係数ないし生物密度の初期総量に対するスモールネスの下で， Deneubourg系の時間大域存在が示されることを述べる． 解の挙動については，蟻塚形成などの観点から，パターン形成が興味深いが， 現在までに，時間大域存在に関する上述スモールネスの下で， 空間一様解が不安定化するようなパラメータ領域を見つけられていない． ある予想の下で数値計算を行っても，Keller-Segel系で見られるような１点集中の解しか， 今のところ捉えられていない．そういった数値計算例についても紹介したい． 本講演内容は，中口悦史氏（東京医科歯科大学）との共同研究に基づく．

タイトル ウェーブレット解析に基づいた信号源分離問題の解法について
アブストラクト ウェーブレット解析に基づいた信号源分離問題の解法について その概略を講演する．

タイトル 四元数値関数の時間周波数解析
アブストラクト ハミルトンは，二次元ユークリッド空間における平行移動と回転を表す複素数を拡張して， 三次元ユークリッド空間における平行移動と回転を表すことができる四元数を定義した． 四元数の積は非可換であるため，四元数値関数の時間周波数解析には特別な注意が必要である． この講演では，四元数値関数の時間周波数解析の初歩を概観し，その応用について述べる．
 第39回「南大阪応用数学セミナー」

タイトル Analytic smoothing effect for global solutions to a quadratic system of nonlinear Schr\"odinger equations
アブストラクト 二次の非線形相互作用項をもつシュレディンガー方程式系の初期値問題をL^2劣臨界の 空間次元n=1,2,3の設定で考える。特に初期値が空間無限遠方で指数関数的に減衰していて 大きなノルムをもつ初期値に対して時間大域解が実解析的となることを示したい。 これまで時間大域解に対するNLSの解析的平滑化効果の結果はスケール臨界の設定で対応する ノルムの小さな初期値に対するものが多い。L^2劣臨界において大きな初期値に対する結果は ないものと思われる。L^2劣臨界ではガリレイ生成作用素による巾級数展開のノルムで特徴付けた 関数空間において時間局所解を構成することができる。一方この解はL^2保存則が成り立つ 状況では通常のL^2の解として時間大域的に延長されている。本講演では保存則が成り立つ時に 解析的な解を延長する手続きにおいて収束半径に対応するパラメータを適当に小さくすることで ガリレイ生成作用素による巾級数展開のノルムをL^2ノルムで制御し実解析性を保ったまま 解を時間大域的に延長していく方法を紹介する。

タイトル Finite energy of generalized suitable weak solutions to the Navier-Stokes equations and the Liouville-type theorem
アブストラクト Introducing a new notion of generalized suitable weak solutions, we prove validity of the energy inequality for such a class of weak solutions to the Navier-Stokes equations in the whole space. Although we need certain growth condition on the pressure, we may treat the class even with infinite energy quantity except for the initial velocity. This is a joint work with Hideo Kozono (Waseda University) and Yutaka Terasawa (Nagoya University).

タイトル Existence of a symmetry-breaking bifurcation point for the one-dimensional Liouville type equation
アブストラクト The two-point boundary value problem for the one-dimensional Liouville type equation is considered. In 2002, Jacobsen and Schmitt presented the exact multiplicity result of radial solutions for the multi-dimensional problem, which is a generalization of the well-known results by Liouville, Gel'fand, Joseph and Lundgren. Recently, Korman gave an alternative proof and introduced interesting self-similar solutions. In this talk, Morse indices of positive even solutions of the one-dimensional problem are calculated, by using Korman's solutions. Then the Leray-Schauder degrees of the operator associated with the problem can be known, and a symmetry-breaking bifurcation point is found.
 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.