## 応用解析セミナー

開催情報 木曜日　16:00～17:30　数理科学研究科棟(駒場) 002号室 石毛 和弘

### 2023年02月06日(月)

16:00-18:10   数理科学研究科棟(駒場) 126号室

Marek Fila 氏 (Comenius University) 16:00-17:00
Solutions with moving singularities for nonlinear diffusion equations (ENGLISH)
[ 講演概要 ]
We give a survey of results on solutions with singularities moving along a prescribed curve for equations of fast diffusion or porous medium type. These results were obtained in collaboration with J.R. King, P. Mackova, J. Takahashi and E. Yanagida.
Petra Mackova 氏 (Comenius University) 17:10-18:10
Fast diffusion equation: uniqueness of solutions with a moving singularity (ENGLISH)
[ 講演概要 ]
This talk focuses on open questions in the area of the uniqueness of distributional solutions of the fast diffusion equation with a given source term. The existence of different sets of such solutions is known from previous research, and the natural next issue is to examine their uniqueness. Assuming that the source term is a measure, the existence of different classes of solutions is known, however, their uniqueness is an open problem. The existence of a class of asymptotically radially symmetric solutions with a singularity that moves along a prescribed curve was proved by M. Fila, J. Takahashi, and E. Yanagida. More recently, it has been established by M. Fila, P. M., J. Takahashi, and E. Yanagida that these solutions solve the corresponding problem with a moving Dirac source term. In this talk, we discuss the uniqueness of these solutions. This is a joint work with M. Fila.
[ 参考URL ]
https://forms.gle/nKa4XATuuGPwZWbUA

### 2022年11月24日(木)

16:00-17:30   数理科学研究科棟(駒場) 370号室

シュタルク・シュレディンガー作用素に対する放射条件評価と定常散乱理論 (Japanese)
[ 講演概要 ]

[ 参考URL ]

### 2022年06月30日(木)

16:00-17:00   オンライン開催
Xingzhi Bian 氏 (Shanghai University)
A brief introduction to a class of new phase field models (English)
[ 講演概要 ]
Existence of weak solutions for a type of new phase field models, which are the system consisting of a degenerate parabolic equation of order parameter coupled to a linear elasticity sub-system. The models are applied to describe the phase transitions in elastically deformable solids.
[ 参考URL ]
https://forms.gle/esc7Y6KGASwbFro97

### 2022年04月21日(木)

16:00-17:30   オンライン開催

Effect of decay rates of initial data on the sign of solutions to Cauchy problems of some higher order parabolic equations (Japanese)
[ 講演概要 ]

[ 参考URL ]
https://forms.gle/96bBNEAEHrsdXfH57

### 2021年12月16日(木)

16:00-17:00   オンライン開催
Zhanpeisov Erbol 氏 (東大数理)
Existence of solutions for fractional semilinear parabolic equations in Besov-Morrey spaces (Japanese)
[ 講演概要 ]

[ 参考URL ]
https://forms.gle/whpkgAwYvyQKQMzM8

### 2021年12月02日(木)

16:00-17:00   オンライン開催

[ 講演概要 ]

[ 参考URL ]
https://forms.gle/6cKyu9meCxSv72N19

### 2021年11月25日(木)

16:00-17:30   オンライン開催

Euler方程式のカレント値弱解とその応用 (日本語)
[ 講演概要 ]

[ 参考URL ]
https://forms.gle/xBAgncTERzYfauJE6

### 2021年10月28日(木)

16:00-17:00   オンライン開催
Xiaodan Zhou 氏 (OIST)
Quasiconformal and Sobolev mappings on metric measure
[ 講演概要 ]
The study of quasiconformal mappings has been an important and active topic since its introduction in the 1930s and the theory has been widely applied to different fields including differential geometry, harmonic analysis, PDEs, etc. In the Euclidean space, it is a fundamental result that three definitions (metric, geometric and analytic) of quasiconformality are equivalent. The theory of quasiconformal mappings has been extended to metric measure spaces by Heinonen and Koskela in the 1990s and their work laid the foundation of analysis on metric spaces. In general, the equivalence of the three characterizations will no longer hold without appropriate assumptions on the spaces and mappings. It is a question of general interest to find minimal assumptions on the metric spaces and on the mapping to guarantee the metric definition implies the analytic characterization or geometric characterization. In this talk, we will give an brief review of the above mentioned classical theory and present some recent results we achieved in obtaining the analytic property, in particular, the Sobolev regularity of a metric quasiconformal mapping with relaxed spaces and mapping conditions. Unexpectedly, we can apply this to prove results that are new even in the classical Euclidean setting. This is joint work with Panu Lahti (Chinese Academy of Sciences).
[ 参考URL ]
https://forms.gle/QATECqmwmWGvXoU56

### 2021年10月14日(木)

16:00-17:00   オンライン開催

[ 講演概要 ]

[ 参考URL ]
https://forms.gle/s4zMhkwpih3FrdhE7

### 2021年07月29日(木)

16:00-17:00   オンライン開催
Dongyuan Xiao 氏 (Univ. of Montpellier・IMAG)
Lotka-Volterra competition-diffusion system: the critical case
[ 講演概要 ]
We consider the reaction-diffusion competition system u_t=u_{xx}+u(1-u-v), v_t=dv_{xx}+rv(1-v-u), which is the so-called critical case. The associated ODE system then admits infinitely many equilibria, which makes the analysis quite intricate. We first prove the non-existence of monotone traveling waves by applying the phase plane analysis. Next, we study the long time behavior of the solution of the Cauchy problem with a compactly supported initial datum. We not only reveal that the ''faster'' species excludes the ''slower'' species (with an identified ''spreading speed''), but also provide a sharp description of the profile of the solution, thus shedding light on a new ''bump phenomenon''.
[ 参考URL ]
https://forms.gle/LHj5mVUdpQ3Jxkrd6

### 2021年06月17日(木)

16:00-17:00   オンライン開催

メトリックグラフ上の半線形楕円型方程式の正値解について (Japanese)
[ 講演概要 ]
メトリックグラフとは、辺と頂点の集合であるグラフにおいて、各辺の長さを考え、各辺と区間と同一視したものである。その上の半線形楕円型方程式は、グラフの辺の数だけ未知関数を持つ常微分方程式系に帰着される。本講演では、特異極限問題を考え、最小エネルギー解に代表される正値解の漸近挙動や解構造について考察する。そして、解が集中する位置や解の個数とメトリックグラフの幾何的な情報との関係について、得られている結果を紹介する。本研究は、倉田和浩氏(東京都立大学)との共同研究に基づく。
[ 参考URL ]
https://forms.gle/apD358V3Jn3ztKVK8

### 2021年04月22日(木)

16:30-18:00   オンライン開催

[ 講演概要 ]

すなわち線型計画問題である。線型計画問題において、最小化因子は制約を与える集合の境界に現れ、そして勾配法は

そしてCuturiの提案とは異なる緩和最小化因子を見つけるアルゴリズムを紹介する。
[ 参考URL ]
https://forms.gle/yg9XZDVdxYG6qMos8

### 2021年04月15日(木)

16:00-17:30   オンライン開催

[ 講演概要 ]

[ 参考URL ]
https://forms.gle/61xaUyw6Pk44QVZi9

### 2020年11月05日(木)

16:00-17:30   数理科学研究科棟(駒場) オンライン開催 号室
【中止】講演者の体調不良により中止となりました。

Hölder gradient estimates on L^p-viscosity solutions of fully nonlinear parabolic equations with VMO coefficients (Japanese)
[ 講演概要 ]
We discuss fully nonlinear second-order uniformly parabolic equations, including parabolic Isaacs equations. Isaacs equations arise in the theory of stochastic differential games. In 2014, N.V. Krylov proved the existence of L^p-viscosity solutions of boundary value problems for equations with VMO (vanishing mean oscillation) “coefficients” when p>n+2. Furthermore, the solutions were in the parabolic Hölder space C^{1,α} for 0<α<1. Our purpose is to show C^{1,α} estimates on L^p-viscosity solutions of fully nonlinear parabolic equations under the same conditions as in Krylov’s result.
[ 参考URL ]

### 2020年10月08日(木)

16:00-17:30   数理科学研究科棟(駒場) オンライン開催 号室

Refined construction of Type II blow-up solutions for semilinear heat equations with Joseph–Lundgren supercritical nonlinearity (Japanese)
[ 講演概要 ]

[ 参考URL ]

### 2019年12月19日(木)

16:00-17:30   数理科学研究科棟(駒場) 128 号室

[ 講演概要 ]

### 2019年10月31日(木)

16:00-17:30   数理科学研究科棟(駒場) 128 (TBD)号室
Marius Ghergu 氏 (University College Dublin)
Behaviour around the isolated singularity for solutions of some nonlinear elliptic inequalities and systems (English)
[ 講演概要 ]
We present some results on the behaviour around the isolated singularity for solutions of nonlinear elliptic inequalities driven by the Laplace operator. We derive optimal conditions that imply either a blow-up or the existence of pointwise bounds for solutions. We obtain that whenever a pointwise bound exists, then an optimal bound is given by the fundamental solution of the Laplace operator. This situation changes in case of systems of inequalities where other types of optimal bounds may occur. The approach relies on integral representation of solutions combined with various nonlinear potential estimates. Further extensions to the parabolic case will be presented. This talk is based on joint works with S. Taliaferro (Texas A&M University) and I. Verbitsky (Missouri University).

### 2019年10月24日(木)

16:00-17:30   数理科学研究科棟(駒場) 128号室
オム　ジュンヨン 氏 (東京大学)

[ 講演概要 ]

### 2019年06月20日(木)

16:00-17:30   数理科学研究科棟(駒場) 118号室

[ 講演概要 ]

### 2019年04月25日(木)

16:00-18:00   数理科学研究科棟(駒場) 118号室
この日は２つ講演があります．教室と時間にご注意下さい．
Matteo Muratori 氏 (Polytechnic University of Milan) 16:00-17:00
The porous medium equation on noncompact Riemannian manifolds with initial datum a measure
(English)
[ 講演概要 ]
We investigate existence and uniqueness of weak solutions of the Cauchy problem for the porous medium equation on Cartan-Hadamard manifolds. We show existence of solutions that take a finite Radon measure as initial datum, possibly sign-changing. We then prove uniqueness in the class of nonnegative solutions, upon assuming a quadratic lower bound on the Ricci curvature. Our result is "optimal" in the sense that any weak solution necessarily solves a Cauchy problem with initial datum a finite Radon measure. Moreover, as byproducts of the techniques we employ, we obtain some new results in potential analysis on manifolds, concerning the validity of a modified version of the mean-value inequality for superharmonic functions and related properties of potentials of positive Radon measures. Finally, we briefly discuss some work in progress regarding stability of the porous medium equation with respect to the Wasserstein distance, on Riemannian manifolds with Ricci curvature bounded below.
Maurizia Rossi 氏 (University of Pisa) 17:00-18:00
On sharp large deviations for the bridge of a general diffusion
(English)
[ 講演概要 ]
In this talk we provide sharp Large Deviation estimates for the probability of exit from a domain for the bridge of a d-dimensional general diffusion process X, as the conditioning time tends to 0. This kind of results is motivated by applications to numerical simulation. In particular we investigate the influence of the drift b of X. It turns out that the sharp asymptotics for the exit time probability are independent of the drift, provided b enjoyes a simple condition that is always satisfied in dimension 1. On the other hand, we show that the drift can be influential if this assumption is not satisfied. This talk is based on a joint work with P. Baldi and L. Caramellino.

### 2018年11月15日(木)

16:00-17:30   数理科学研究科棟(駒場) 118号室

Inhomogeneous Dirichlet-boundary value problem for one dimensional nonlinear Schr\"{o}dinger equations (Japanese)
[ 講演概要 ]
We consider the inhomogeneous Dirichlet-boundary value problem for the cubic nonlinear Schr\"{o}dinger equations on the half line. We present sufficient conditions of initial and boundary data which ensure asymptotic behavior of small solutions to equations by using the classical energy method and factorization techniques.

### 2018年10月11日(木)

16:00-17:30   数理科学研究科棟(駒場) 118号室

Navier-Stokes方程式に対する摩擦型境界条件とその周辺 (Japanese)
[ 講演概要 ]

### 2018年10月04日(木)

16:00-17:30   数理科学研究科棟(駒場) 118号室

いくつかの微分方程式に対する反応拡散近似 (Japanese)
[ 講演概要 ]

### 2018年07月19日(木)

16:00-17:30   数理科学研究科棟(駒場) 118号室

Uniqueness and nondegeneracy of ground states to scalar field equation involving critical Sobolev exponent
(Japanese)
[ 講演概要 ]
This talk is devoted to studying the uniqueness and nondegeneracy of ground states to a nonlinear scalar field equation on the whole space. The nonlinearity consists of two power functions, and their growths are subcritical and critical in the Sobolev sense respectively. Under some assumptions, it is known that the equation admits a positive radial ground state and other ground states are made from the positive radial one. We show that if the dimensions are greater than or equal to 5 and the frequency is sufficiently large, then the positive radial ground state is unique and nondegenerate. This is based on joint work with Takafumi Akahori (Shizuoka Univ.), Slim Ibrahim (Univ. of Victoria), Hiroaki Kikuchi (Tsuda Univ.) and Hayato Nawa (Meiji Univ.).

### 2018年05月24日(木)

16:00-17:30   数理科学研究科棟(駒場) 128号室

Sign-changing solutions for a one-dimensional semilinear parabolic problem (Japanese)
[ 講演概要 ]
This talk is concerned with a nonlinear parabolic equation on a bounded interval with the homogeneous Dirichlet or Neumann boundary condition. Under rather general conditions on the nonlinearity, we consider the blow-up and global existence of sign-changing solutions. It is shown that there exists a nonnegative integer $k$ such that the solution blows up in finite time if the initial value changes its sign at most $k$ times, whereas there exists a stationary solution with more than $k$ zeros. The proof is based on an intersection number argument combined with a topological method.