## PDE実解析研究会

過去の記録 ～03/27｜次回の予定｜今後の予定 03/28～

開催情報 | 火曜日 10:30～11:30 数理科学研究科棟(駒場) 056号室 |
---|---|

担当者 | 儀我美一、石毛和弘、三竹大寿、米田剛 |

セミナーURL | http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/ |

目的 | 首都圏の偏微分方程式、実解析の研究をさらに活発にするために本研究会を東大で開催いたします。 偏微分方程式研究者と実解析研究者の討論がより日常的になることを目指しています。 そのため、講演がその分野の概観をもわかるような形になるよう配慮いたします。 また講演者との1対1の討論がしやすいように講演は火曜の午前とし、午後に討論用の場所を用意いたします。 この研究会を通して皆様に気楽に東大を訪問していただければ幸いです。 北海道大学のHPには、第1回(2004年9月29日)～第38回(2008年10月15日)の情報が掲載されております。 |

**過去の記録**

### 2020年12月01日(火)

10:30-11:30 数理科学研究科棟(駒場) Zoomによるオンライン開催 号室

Existence of the $1$-harmonic map flow (English)

**Michał Łasica 氏**(Institute of Mathematics of the Polish Academy of Sciences / University of Tokyo)Existence of the $1$-harmonic map flow (English)

[ 講演概要 ]

Similarly as in the real-valued case, the total variation of maps taking values in a Riemannian manifold extends to a lower semicontinuous functional on $L^2$. However, in general this functional is not geodesically semiconvex, so the existence of its gradient flow is not provided by general variational theory. Alternatively, one can try to apply the theory of parabolic PDE systems, mimicking the approach used for $p$-harmonic map flows, $p>1$. This poses some difficulties, because the PDE system corresponding to the flow is strongly nonlinear, singular and degenerate. However, in some cases, this approach was successful. In this talk, I will describe known results on the existence of the flow, focusing on my work with Lorenzo Giacomelli and Salvador Moll.

Similarly as in the real-valued case, the total variation of maps taking values in a Riemannian manifold extends to a lower semicontinuous functional on $L^2$. However, in general this functional is not geodesically semiconvex, so the existence of its gradient flow is not provided by general variational theory. Alternatively, one can try to apply the theory of parabolic PDE systems, mimicking the approach used for $p$-harmonic map flows, $p>1$. This poses some difficulties, because the PDE system corresponding to the flow is strongly nonlinear, singular and degenerate. However, in some cases, this approach was successful. In this talk, I will describe known results on the existence of the flow, focusing on my work with Lorenzo Giacomelli and Salvador Moll.

### 2020年10月27日(火)

10:30-11:30 数理科学研究科棟(駒場) Zoomによるオンライン開催 号室

Vanishing discount problems for Hamilton-Jacobi equations on changing domains (English)

**Son Tu 氏**(University of Wisconsin Madison)Vanishing discount problems for Hamilton-Jacobi equations on changing domains (English)

[ 講演概要 ]

We study the asymptotic behavior, as the discount factor vanishes, of the Hamilton-Jacobi equation with state-constraint on changing domains. Surprisingly, we can obtain both convergence results and non-convergence results in this convex setting. Moreover, we provide a very first result on the asymptotic expansion of the additive eigenvalue of the Hamiltonian with respect to the changing domains. The main tool we use is a duality representation of solution with viscosity Mather measures.

We study the asymptotic behavior, as the discount factor vanishes, of the Hamilton-Jacobi equation with state-constraint on changing domains. Surprisingly, we can obtain both convergence results and non-convergence results in this convex setting. Moreover, we provide a very first result on the asymptotic expansion of the additive eigenvalue of the Hamiltonian with respect to the changing domains. The main tool we use is a duality representation of solution with viscosity Mather measures.

### 2020年02月28日(金)

16:00-17:00 数理科学研究科棟(駒場) 370号室

通常の曜日・時刻・教室と異なります。

Convergence of the Allen-Cahn equation with a nonlinear Robin-boundary condition to mean curvature flow with constant contact angle (English)

通常の曜日・時刻・教室と異なります。

**Maximilian Moser 氏**(University of Regensburg)Convergence of the Allen-Cahn equation with a nonlinear Robin-boundary condition to mean curvature flow with constant contact angle (English)

[ 講演概要 ]

In this talk I will present a result for the sharp interface limit of the Allen-Cahn equation with a nonlinear Robin boundary condition in a two-dimensional domain, in the situation where an interface has developed and intersects the boundary. The boundary condition is designed in such a way that one obtains as the limit problem the mean curvature flow with constant contact angle. Convergence using strong norms is shown for contact angles close to 90° and small times, when a smooth solution to the limit problem exists. For the proof the method of de Mottoni and Schatzman is used: we construct an approximate solution for the Allen-Cahn system using asymptotic expansions based on the solution to the limit problem. Then we estimate the difference of the exact and approximate solution with a spectral estimate for the linearized (at the approximate solution) Allen-Cahn operator.

This is joint work with Helmut Abels from Regensburg.

In this talk I will present a result for the sharp interface limit of the Allen-Cahn equation with a nonlinear Robin boundary condition in a two-dimensional domain, in the situation where an interface has developed and intersects the boundary. The boundary condition is designed in such a way that one obtains as the limit problem the mean curvature flow with constant contact angle. Convergence using strong norms is shown for contact angles close to 90° and small times, when a smooth solution to the limit problem exists. For the proof the method of de Mottoni and Schatzman is used: we construct an approximate solution for the Allen-Cahn system using asymptotic expansions based on the solution to the limit problem. Then we estimate the difference of the exact and approximate solution with a spectral estimate for the linearized (at the approximate solution) Allen-Cahn operator.

This is joint work with Helmut Abels from Regensburg.

### 2019年11月26日(火)

10:30-11:30 数理科学研究科棟(駒場) 056号室

A Hamiltonian approach with penalization in shape and topology optimization (English)

**Dan Tiba 氏**(Institute of Mathematics of the Romanian Academy / Academy of Romanian Scientists)A Hamiltonian approach with penalization in shape and topology optimization (English)

[ 講演概要 ]

General geometric optimization problems involve boundary and topology variations. This research area has already almost fifty years of history and very rich applications in computer aided industrial design. Recently, a new representation of manifolds, using iterated Hamiltonian systems, has been introduced in arbitrary dimension and co-dimension. Combining this technique with a penalization procedure for the boundary conditions, a comprehensive approximation method for optimal design problems associated to elliptic equations, is obtained. It reduces shape and topology optimization problems to optimal control problems, in a general setting. It enters the category of fixed domain methods in variable/unknown domain problems and it has consistent advantages at the computational level. It allows "free" changes of the boundary and/or the topology, during the iterations. This methodology, based on iterated Hamiltonian systems and implicit parametrizations, was also applied to nonlinear programming problems in arbitrary dimension.

General geometric optimization problems involve boundary and topology variations. This research area has already almost fifty years of history and very rich applications in computer aided industrial design. Recently, a new representation of manifolds, using iterated Hamiltonian systems, has been introduced in arbitrary dimension and co-dimension. Combining this technique with a penalization procedure for the boundary conditions, a comprehensive approximation method for optimal design problems associated to elliptic equations, is obtained. It reduces shape and topology optimization problems to optimal control problems, in a general setting. It enters the category of fixed domain methods in variable/unknown domain problems and it has consistent advantages at the computational level. It allows "free" changes of the boundary and/or the topology, during the iterations. This methodology, based on iterated Hamiltonian systems and implicit parametrizations, was also applied to nonlinear programming problems in arbitrary dimension.

### 2019年11月19日(火)

10:30-11:30 数理科学研究科棟(駒場) 056号室

Starting Ricci flow with rough initial data (English)

**Peter Topping 氏**(University of Warwick)Starting Ricci flow with rough initial data (English)

[ 講演概要 ]

Ricci flow is a nonlinear PDE that is traditionally used to deform a manifold we would like to understand into a manifold we already understand. For example, Hamilton showed that a simply connected closed 3-manifold with positive Ricci curvature is deformed into a manifold of constant sectional curvature, thus allowing us to identify it as topologically a sphere.

In this talk we take a look at a different use of Ricci flow. We would like to exploit the regularising effect of parabolic PDE to turn a rough space into a smooth space by running the Ricci flow. In practice, this revolves around proving good a priori estimates on solutions, and taking unorthodox approaches to solving parabolic PDE. We will see some theory, first in 2D, then in higher dimension, and some applications.

Ricci flow is a nonlinear PDE that is traditionally used to deform a manifold we would like to understand into a manifold we already understand. For example, Hamilton showed that a simply connected closed 3-manifold with positive Ricci curvature is deformed into a manifold of constant sectional curvature, thus allowing us to identify it as topologically a sphere.

In this talk we take a look at a different use of Ricci flow. We would like to exploit the regularising effect of parabolic PDE to turn a rough space into a smooth space by running the Ricci flow. In practice, this revolves around proving good a priori estimates on solutions, and taking unorthodox approaches to solving parabolic PDE. We will see some theory, first in 2D, then in higher dimension, and some applications.

### 2019年07月23日(火)

13:00-14:00 数理科学研究科棟(駒場) 056号室

通常の開始時刻と異なります。

On the isoperimetric ratio over scalar-flat conformal classes (English)

通常の開始時刻と異なります。

**Tianling Jin 氏**(The Hong Kong University of Science and Technology)On the isoperimetric ratio over scalar-flat conformal classes (English)

[ 講演概要 ]

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n$ with smooth boundary. Suppose that $(M,g)$ admits a scalar-flat conformal metric. We prove that the supremum of the isoperimetric ratio over the scalar-flat conformal class is strictly larger than the best constant of the isoperimetric inequality on Euclidean space, and consequently is achieved, if either (i) $n \geq 12$ and the boundary has a nonumbilic point; or (ii) $n \geq 10$, the boundary is umbilic and the Weyl tensor does not vanish at some boundary point. A crucial ingredient in the proof is the expansion of solutions to the conformal Laplacian equation with blowing up Dirichlet boundary conditions.

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n$ with smooth boundary. Suppose that $(M,g)$ admits a scalar-flat conformal metric. We prove that the supremum of the isoperimetric ratio over the scalar-flat conformal class is strictly larger than the best constant of the isoperimetric inequality on Euclidean space, and consequently is achieved, if either (i) $n \geq 12$ and the boundary has a nonumbilic point; or (ii) $n \geq 10$, the boundary is umbilic and the Weyl tensor does not vanish at some boundary point. A crucial ingredient in the proof is the expansion of solutions to the conformal Laplacian equation with blowing up Dirichlet boundary conditions.

### 2019年06月18日(火)

10:30-11:30 数理科学研究科棟(駒場) 056号室

Ways to treat a diffusion problem with the fractional Caputo derivative

**Piotr Rybka 氏**(University of Warsaw)Ways to treat a diffusion problem with the fractional Caputo derivative

[ 講演概要 ]

The problem

\[

u_t = (D^\alpha u)_x + f

\]

augmented with initial and boundary data appear in model of subsurface flows. Here, $D^\alpha u$ denotes the fractional Caputo derivative of order $\alpha \in (0,1)$.

We offer three approaches:

1) from the point of view of semigroups;

2) from the point of view of the theory of viscosity solutions;

3) from the point of view of numerical simulations.

This is a joint work with T. Namba, K. Ryszewska, V. Voller.

The problem

\[

u_t = (D^\alpha u)_x + f

\]

augmented with initial and boundary data appear in model of subsurface flows. Here, $D^\alpha u$ denotes the fractional Caputo derivative of order $\alpha \in (0,1)$.

We offer three approaches:

1) from the point of view of semigroups;

2) from the point of view of the theory of viscosity solutions;

3) from the point of view of numerical simulations.

This is a joint work with T. Namba, K. Ryszewska, V. Voller.

### 2019年06月04日(火)

10:30-11:30 数理科学研究科棟(駒場) 056号室

Recent progresses in nonlinear potential theory (English)

**Giuseppe Mingione 氏**(Università di Parma)Recent progresses in nonlinear potential theory (English)

[ 講演概要 ]

Nonlinear Potential Theory aims at studying the fine properties of solutions to nonlinear, potentially degenerate nonlinear elliptic and parabolic equations in terms of the regularity of the give data. A major model example is here given by the $p$-Laplacean equation

$$ -\operatorname{div}(|Du|^{p-2}Du) = \mu \quad\quad p > 1, $$

where $\mu$ is a Borel measure with finite total mass. When $p = 2$ we find the familiar case of the Poisson equation from which classical Potential Theory stems. Although many basic tools from the classical linear theory are not at hand - most notably: representation formulae via fundamental solutions - many of the classical information can be retrieved for solutions and their pointwise behaviour. In this talk I am going to give a survey of recent results in the field. Especially, I will explain the possibility of getting linear and nonlinear potential estimates for solutions to nonlinear elliptic and parabolic equations which are totally similar to those available in the linear case. I will also draw some parallels with what is nowadays called Nonlinear Calderón-Zygmund theory.

Nonlinear Potential Theory aims at studying the fine properties of solutions to nonlinear, potentially degenerate nonlinear elliptic and parabolic equations in terms of the regularity of the give data. A major model example is here given by the $p$-Laplacean equation

$$ -\operatorname{div}(|Du|^{p-2}Du) = \mu \quad\quad p > 1, $$

where $\mu$ is a Borel measure with finite total mass. When $p = 2$ we find the familiar case of the Poisson equation from which classical Potential Theory stems. Although many basic tools from the classical linear theory are not at hand - most notably: representation formulae via fundamental solutions - many of the classical information can be retrieved for solutions and their pointwise behaviour. In this talk I am going to give a survey of recent results in the field. Especially, I will explain the possibility of getting linear and nonlinear potential estimates for solutions to nonlinear elliptic and parabolic equations which are totally similar to those available in the linear case. I will also draw some parallels with what is nowadays called Nonlinear Calderón-Zygmund theory.

### 2019年01月29日(火)

10:30-11:30 数理科学研究科棟(駒場) 056号室

The regularity of area minimizing currents modulo $p$ (English)

**Salvatore Stuvard 氏**(The University of Texas at Austin)The regularity of area minimizing currents modulo $p$ (English)

[ 講演概要 ]

The theory of integer rectifiable currents was introduced by Federer and Fleming in the early 1960s in order to provide a class of generalized surfaces where the classical Plateau problem could be solved by direct methods. Since then, a number of alternative spaces of surfaces have been developed in geometric measure theory, as required for theory and applications. In particular, Fleming introduced currents modulo $2$ to treat non-orientable surfaces, and currents modulo $p$ (where $p \geq 2$ is an integer) to study more general surfaces occurring as soap films.

It is easy to see that, in general, area minimizing currents modulo $p$ need not be smooth surfaces. In this talk, I will sketch the proof of the following result, which achieves the best possible estimate for the Hausdorff dimension of the singular set of an area minimizing current modulo $p$ in the most general hypotheses, thus answering a question of White from the 1980s: if $T$ is an area minimizing current modulo $p$ of dimension $m$ in $R^{m+n}$, then $T$ is smooth at all its interior points, except those belonging to a singular set of Hausdorff dimension at most $m-1$.

The theory of integer rectifiable currents was introduced by Federer and Fleming in the early 1960s in order to provide a class of generalized surfaces where the classical Plateau problem could be solved by direct methods. Since then, a number of alternative spaces of surfaces have been developed in geometric measure theory, as required for theory and applications. In particular, Fleming introduced currents modulo $2$ to treat non-orientable surfaces, and currents modulo $p$ (where $p \geq 2$ is an integer) to study more general surfaces occurring as soap films.

It is easy to see that, in general, area minimizing currents modulo $p$ need not be smooth surfaces. In this talk, I will sketch the proof of the following result, which achieves the best possible estimate for the Hausdorff dimension of the singular set of an area minimizing current modulo $p$ in the most general hypotheses, thus answering a question of White from the 1980s: if $T$ is an area minimizing current modulo $p$ of dimension $m$ in $R^{m+n}$, then $T$ is smooth at all its interior points, except those belonging to a singular set of Hausdorff dimension at most $m-1$.

### 2018年12月18日(火)

10:30-11:30 数理科学研究科棟(駒場) 056号室

Dynamics of singular vortex patches (English)

**In-Jee Jeong 氏**(Korea Institute for Advanced Study (KIAS))Dynamics of singular vortex patches (English)

[ 講演概要 ]

Vortex patches are solutions to the 2D Euler equations that are given by the characteristic function of a bounded domain that moves with time. It is well-known that if initially the boundary of the domain is smooth, the boundary remains smooth for all time. On the other hand, we consider patches with corner singularities. It turns out that, depending on whether the initial patch satisfies an appropriate rotational symmetry condition or not, the corner structure may propagate for all time or lost immediately. In the rotationally symmetric case, we are able to construct patches with interesting dynamical behavior as time goes to infinity. When the symmetry is absent, we present a simple yet formal evolution equation which describes the dynamics of the boundary. It suggests that the angle cusps instantaneously for $t > 0$.

This is joint work with Tarek Elgindi.

Vortex patches are solutions to the 2D Euler equations that are given by the characteristic function of a bounded domain that moves with time. It is well-known that if initially the boundary of the domain is smooth, the boundary remains smooth for all time. On the other hand, we consider patches with corner singularities. It turns out that, depending on whether the initial patch satisfies an appropriate rotational symmetry condition or not, the corner structure may propagate for all time or lost immediately. In the rotationally symmetric case, we are able to construct patches with interesting dynamical behavior as time goes to infinity. When the symmetry is absent, we present a simple yet formal evolution equation which describes the dynamics of the boundary. It suggests that the angle cusps instantaneously for $t > 0$.

This is joint work with Tarek Elgindi.

### 2018年12月11日(火)

10:30-11:30 数理科学研究科棟(駒場) 056号室

Solutions with moving singularities for equations of porous medium type (English)

**Marek Fila 氏**(Comenius University in Bratislava)Solutions with moving singularities for equations of porous medium type (English)

[ 講演概要 ]

We construct positive solutions of equations of porous medium type with a singularity which moves in time along a prescribed curve and keeps the spatial profile of singular stationary solutions. It turns out that there appears a critical exponent for the existence of such solutions. This is a joint work with Jin Takahashi and Eiji Yanagida.

We construct positive solutions of equations of porous medium type with a singularity which moves in time along a prescribed curve and keeps the spatial profile of singular stationary solutions. It turns out that there appears a critical exponent for the existence of such solutions. This is a joint work with Jin Takahashi and Eiji Yanagida.

### 2018年10月30日(火)

10:30-11:30 数理科学研究科棟(駒場) 056号室

The least gradient problem in the plain (English)

**Piotr Rybka 氏**(University of Warsaw)The least gradient problem in the plain (English)

[ 講演概要 ]

The least gradient problem arises in many application, e.g. in the free material design. We show existence of solutions in bounded, strictly convex planar regions, when the data are functions on bounded variation.

Our main goal is to show existence of solution in convex, but not necessarily strictly convex planar regions. In order to avoid technicalities we consider only continuous data, but BV data will do to. We formulate a set of admissibility conditions. We show that they are sufficient for existence.

This is a joint project with Wojciech Górny and Ahmad Sabra.

The least gradient problem arises in many application, e.g. in the free material design. We show existence of solutions in bounded, strictly convex planar regions, when the data are functions on bounded variation.

Our main goal is to show existence of solution in convex, but not necessarily strictly convex planar regions. In order to avoid technicalities we consider only continuous data, but BV data will do to. We formulate a set of admissibility conditions. We show that they are sufficient for existence.

This is a joint project with Wojciech Górny and Ahmad Sabra.

### 2018年10月23日(火)

10:30-11:30 数理科学研究科棟(駒場) 056号室

Least action principle for incompressible flow with free boundary (English)

**Jian-Guo Liu 氏**(Duke University)Least action principle for incompressible flow with free boundary (English)

[ 講演概要 ]

In this talk I will describe a connection between Arnold's least-action principle for incompressible flows with free boundary and geodesic paths for Wasserstein distance. The least-action problem for geodesic distance on the "manifold" of fluid-blob shapes exhibits instability due to microdroplet formation. Using a conformal map formulation we investigate singularity formation in water-wave dynamics neglecting gravity. A connection with fluid mixture models via a variant of Brenier's relaxed least-action principle for generalized Euler flows will also be discussed.

In this talk I will describe a connection between Arnold's least-action principle for incompressible flows with free boundary and geodesic paths for Wasserstein distance. The least-action problem for geodesic distance on the "manifold" of fluid-blob shapes exhibits instability due to microdroplet formation. Using a conformal map formulation we investigate singularity formation in water-wave dynamics neglecting gravity. A connection with fluid mixture models via a variant of Brenier's relaxed least-action principle for generalized Euler flows will also be discussed.

### 2018年07月02日(月)

10:30-11:30 数理科学研究科棟(駒場) 056号室

通常の曜日と異なります。

Convex integration in fluid dynamics (English)

通常の曜日と異なります。

**László Székelyhidi Jr. 氏**(Universität Leipzig)Convex integration in fluid dynamics (English)

[ 講演概要 ]

In the talk we present the technique of convex integration for constructing weak solutions to various equations in fluid mechanics.

We will focus on the recent resolution of Onsagers conjecture, but also discuss further directions and in particular the applicability to dissipative systems.

In the talk we present the technique of convex integration for constructing weak solutions to various equations in fluid mechanics.

We will focus on the recent resolution of Onsagers conjecture, but also discuss further directions and in particular the applicability to dissipative systems.

### 2018年05月22日(火)

10:30-11:30 数理科学研究科棟(駒場) 056号室

A discrete game interpretation for curvature flow equations with dynamic boundary conditions (日本語)

**柳 青 氏**(福岡大学)A discrete game interpretation for curvature flow equations with dynamic boundary conditions (日本語)

[ 講演概要 ]

A game-theoretic approach to motion by curvature was proposed by Kohn and Serfaty in 2006. They constructed a family of deterministic discrete games, whose value functions converge to the unique solution of the curvature flow equation. In this talk, we develop this method to provide an interpretation for the associated dynamic boundary value problems by including in the game setting a kind of nonlinear reflection near the boundary. We also discuss its applications to the fattening phenomenon. This talk is based on joint work with N. Hamamuki at Hokkaido University.

A game-theoretic approach to motion by curvature was proposed by Kohn and Serfaty in 2006. They constructed a family of deterministic discrete games, whose value functions converge to the unique solution of the curvature flow equation. In this talk, we develop this method to provide an interpretation for the associated dynamic boundary value problems by including in the game setting a kind of nonlinear reflection near the boundary. We also discuss its applications to the fattening phenomenon. This talk is based on joint work with N. Hamamuki at Hokkaido University.

### 2018年04月17日(火)

10:30-11:30 数理科学研究科棟(駒場) 056号室

Global existence of a strong solution to the Navier-Stokes equations in a curved thin domain (Japanese)

**三浦 達彦 氏**(東京大学)Global existence of a strong solution to the Navier-Stokes equations in a curved thin domain (Japanese)

[ 講演概要 ]

In this talk, we consider the Navier-Stokes equations with Navier's slip boundary conditions in a three-dimensional curved thin domain around a closed surface.

We establish the global-in-time existence of a strong solution for large data when the width of the thin domain is very small.

A key idea is to decompose a three-dimensional vector field into the average part which is almost two-dimensional and the residual part to which we can apply Poincaré type inequalities.

Such decomposition enables us to derive a good estimate for the inner product of the inertia term and the viscous term, which is essential for our arguments.

In this talk, we consider the Navier-Stokes equations with Navier's slip boundary conditions in a three-dimensional curved thin domain around a closed surface.

We establish the global-in-time existence of a strong solution for large data when the width of the thin domain is very small.

A key idea is to decompose a three-dimensional vector field into the average part which is almost two-dimensional and the residual part to which we can apply Poincaré type inequalities.

Such decomposition enables us to derive a good estimate for the inner product of the inertia term and the viscous term, which is essential for our arguments.

### 2017年12月12日(火)

10:30-11:30 数理科学研究科棟(駒場) 056号室

Stochastic Three-Dimensional Navier-Stokes Equations + Waves: Averaging, Convergence, Regularity and Nonlinear Dynamics (English)

**Alex Mahalov 氏**(Arizona State University)Stochastic Three-Dimensional Navier-Stokes Equations + Waves: Averaging, Convergence, Regularity and Nonlinear Dynamics (English)

[ 講演概要 ]

We establish multi-scale stochastic averaging, convergence and regularity theorems in a general framework by bootstrapping from global regularity of the averaged stochastic resonant equations. The averaged covariance operator couples stochastic and wave effects. We also present theoretical results for 3D nonlinear dynamics.

We establish multi-scale stochastic averaging, convergence and regularity theorems in a general framework by bootstrapping from global regularity of the averaged stochastic resonant equations. The averaged covariance operator couples stochastic and wave effects. We also present theoretical results for 3D nonlinear dynamics.

### 2017年11月21日(火)

10:30-11:30 数理科学研究科棟(駒場) 056号室

Optimal isoperimetric inequalities for surfaces in any codimension

in Cartan-Hadamard manifolds (English)

**Felix Schulze 氏**(University College London)Optimal isoperimetric inequalities for surfaces in any codimension

in Cartan-Hadamard manifolds (English)

[ 講演概要 ]

Let $(M^n,g)$ be simply connected, complete, with non-positive sectional

curvatures, and $\Sigma$ a 2-dimensional surface in $M^n$. Let $S$ be an area

minimising 3-current such that $\partial S = \Sigma$. We use a weak mean

curvature flow, obtained via elliptic regularisation, starting from

$\Sigma$, to show that $S$ satisfies the optimal Euclidean isoperimetric

inequality: $|S| \leq 1/(6\sqrt{\pi}) |\Sigma|^{3/2}$. We also obtain the

optimal estimate in case the sectional curvatures of $M$ are bounded from

above by $\kappa < 0$ and characterise the case of equality. The proof

follows from an almost monotonicity of a suitable isoperimetric

difference along the approximating flows in one dimension higher.

Let $(M^n,g)$ be simply connected, complete, with non-positive sectional

curvatures, and $\Sigma$ a 2-dimensional surface in $M^n$. Let $S$ be an area

minimising 3-current such that $\partial S = \Sigma$. We use a weak mean

curvature flow, obtained via elliptic regularisation, starting from

$\Sigma$, to show that $S$ satisfies the optimal Euclidean isoperimetric

inequality: $|S| \leq 1/(6\sqrt{\pi}) |\Sigma|^{3/2}$. We also obtain the

optimal estimate in case the sectional curvatures of $M$ are bounded from

above by $\kappa < 0$ and characterise the case of equality. The proof

follows from an almost monotonicity of a suitable isoperimetric

difference along the approximating flows in one dimension higher.

### 2017年11月15日(水)

10:30-11:30 数理科学研究科棟(駒場) 056号室

※ 通常と曜日が異なります。

Boundary value problems for parabolic equations with measurable coefficients (English)

※ 通常と曜日が異なります。

**Kaj Nyström 氏**(Uppsala University)Boundary value problems for parabolic equations with measurable coefficients (English)

[ 講演概要 ]

In recent joint works with P. Auscher and M. Egert we establish new results concerning boundary value problems in the upper half-space for second order parabolic equations (and systems) assuming only measurability and some transversal regularity in the coefficients of the elliptic part. To establish our results we introduce and develop a first order strategy by means of a parabolic Dirac operator at the boundary to obtain, in particular, Green's representation for solutions in natural classes involving square functions and non-tangential maximal functions, well-posedness results with data in $L^2$-Sobolev spaces together with invertibility of layer potentials, and perturbation results. In addition we solve the Kato square root problem for parabolic operators with coefficients of the elliptic part depending measurably on all variables. Using these results we are also able to solve the $L^p$-Dirichlet problem for parabolic equations with real, time-dependent, elliptic but non-symmetric coefficients. In this talk I will briefly describe some of these developments.

In recent joint works with P. Auscher and M. Egert we establish new results concerning boundary value problems in the upper half-space for second order parabolic equations (and systems) assuming only measurability and some transversal regularity in the coefficients of the elliptic part. To establish our results we introduce and develop a first order strategy by means of a parabolic Dirac operator at the boundary to obtain, in particular, Green's representation for solutions in natural classes involving square functions and non-tangential maximal functions, well-posedness results with data in $L^2$-Sobolev spaces together with invertibility of layer potentials, and perturbation results. In addition we solve the Kato square root problem for parabolic operators with coefficients of the elliptic part depending measurably on all variables. Using these results we are also able to solve the $L^p$-Dirichlet problem for parabolic equations with real, time-dependent, elliptic but non-symmetric coefficients. In this talk I will briefly describe some of these developments.

### 2017年10月17日(火)

10:30-11:30 数理科学研究科棟(駒場) 056号室

Some perspectives on negative index materials (English)

**Hoài-Minh Nguyên 氏**(École Polytechnique Fédérale de Lausanne)Some perspectives on negative index materials (English)

[ 講演概要 ]

Negative index materials (NIMs) are artificial structures whose refractive index has negative value over some frequency range. These materials were first investigated theoretically by Veselago in 1964. The existence of NIMs was confirmed experimentally by Shelby, Smith, and Schultz in 2001. New fabrication techniques now allow the construction of NIMs at scales that are interesting for applications. NIMs have attracted a lot of attention from the scientific community, not only because of potentially interesting applications, but also because of challenges in understanding their peculiar properties. Mathematically, the study of NIMs faces two difficulties. First, the equations describing the phenomenon have sign changing coefficients, hence the ellipticity and the compactness are lost in general. Second, the localized resonance, i.e., the field explodes in some regions and remains bounded in some others as the loss goes to 0, might appear. In this talk I will discuss various mathematics techniques used to understand various applications of NIMs such as cloaking and superlensing and to develop new designs for them.

Negative index materials (NIMs) are artificial structures whose refractive index has negative value over some frequency range. These materials were first investigated theoretically by Veselago in 1964. The existence of NIMs was confirmed experimentally by Shelby, Smith, and Schultz in 2001. New fabrication techniques now allow the construction of NIMs at scales that are interesting for applications. NIMs have attracted a lot of attention from the scientific community, not only because of potentially interesting applications, but also because of challenges in understanding their peculiar properties. Mathematically, the study of NIMs faces two difficulties. First, the equations describing the phenomenon have sign changing coefficients, hence the ellipticity and the compactness are lost in general. Second, the localized resonance, i.e., the field explodes in some regions and remains bounded in some others as the loss goes to 0, might appear. In this talk I will discuss various mathematics techniques used to understand various applications of NIMs such as cloaking and superlensing and to develop new designs for them.

### 2017年07月18日(火)

10:30-11:30 数理科学研究科棟(駒場) 056号室

軸対称非圧縮Euler方程式の或る瞬間爆発について (日本語)

**米田 剛 氏**(東京大学)軸対称非圧縮Euler方程式の或る瞬間爆発について (日本語)

[ 講演概要 ]

本講演では、軸対称非圧縮Euler方程式の瞬間爆発についての結果を報告する。より具体的には、$C^{2,\alpha}$ ($0<\alpha<1$)に入る初期速度場に対応する解が、任意の$T$における$C^1([0,T):C^2)$には入らないという定理を紹介する。定理の証明には、特異積分作用素の$L^\infty$-非有界性は一切使わず、代わりにFrenet-Serret formulasやorthonormal moving frameといった幾何学的概念を本質的に使う。時間があれば、この洞察の物理的背景も紹介したい。

本講演では、軸対称非圧縮Euler方程式の瞬間爆発についての結果を報告する。より具体的には、$C^{2,\alpha}$ ($0<\alpha<1$)に入る初期速度場に対応する解が、任意の$T$における$C^1([0,T):C^2)$には入らないという定理を紹介する。定理の証明には、特異積分作用素の$L^\infty$-非有界性は一切使わず、代わりにFrenet-Serret formulasやorthonormal moving frameといった幾何学的概念を本質的に使う。時間があれば、この洞察の物理的背景も紹介したい。

### 2016年12月20日(火)

10:30-11:30 数理科学研究科棟(駒場) 056号室

On the stability of the 3D Couette Flow (English)

**Nader Masmoudi 氏**(Courant Institute, NYU)On the stability of the 3D Couette Flow (English)

[ 講演概要 ]

We will discuss the dynamics of small perturbations of the plane, periodic Couette flow in the 3D incompressible Navier-Stokes system at high Reynolds number. For sufficiently regular initial data, we determine the stability threshold for small perturbations and characterize the long time dynamics of solutions near this threshold. For rougher data, we obtain an estimate of the stability threshold which agrees closely with numerical experiments. The primary linear stability mechanism is an anisotropic enhanced dissipation resulting from the mixing caused by the large mean shear. There is also a linear inviscid damping similar to the one observed in 2D. The main linear instability is a non-normal instability known as the lift-up effect. There is clearly a competition between these linear effects. Understanding the variety of nonlinear resonances and devising the correct norms to estimate them form the core of the analysis we undertake. This is based on joint works with Jacob Bedrossian and Pierre Germain.

We will discuss the dynamics of small perturbations of the plane, periodic Couette flow in the 3D incompressible Navier-Stokes system at high Reynolds number. For sufficiently regular initial data, we determine the stability threshold for small perturbations and characterize the long time dynamics of solutions near this threshold. For rougher data, we obtain an estimate of the stability threshold which agrees closely with numerical experiments. The primary linear stability mechanism is an anisotropic enhanced dissipation resulting from the mixing caused by the large mean shear. There is also a linear inviscid damping similar to the one observed in 2D. The main linear instability is a non-normal instability known as the lift-up effect. There is clearly a competition between these linear effects. Understanding the variety of nonlinear resonances and devising the correct norms to estimate them form the core of the analysis we undertake. This is based on joint works with Jacob Bedrossian and Pierre Germain.

### 2016年11月22日(火)

10:30-11:30 数理科学研究科棟(駒場) 056号室

De Giorgi conjecture and minimal surfaces for integro-differential operators (English)

**Yannick Sire (Johns Hopkins University) 氏**De Giorgi conjecture and minimal surfaces for integro-differential operators (English)

[ 講演概要 ]

I will review the classical De Giorgi conjecture and its link with minimal surfaces. Then I will move on recent results for flatness of level sets of solutions of semi linear equations involving anomalous diffusion. First I will deal with the fractional laplacian; second with quite general integral operators in 2 dimensions.

I will review the classical De Giorgi conjecture and its link with minimal surfaces. Then I will move on recent results for flatness of level sets of solutions of semi linear equations involving anomalous diffusion. First I will deal with the fractional laplacian; second with quite general integral operators in 2 dimensions.

### 2016年10月11日(火)

10:30-11:30 数理科学研究科棟(駒場) 056号室

Global solutions to the second boundary value problem of the prescribed affine mean curvature and Abreu's equations (English)

**Nam Quang Le 氏**(Indiana University)Global solutions to the second boundary value problem of the prescribed affine mean curvature and Abreu's equations (English)

[ 講演概要 ]

The second boundary value problem of the prescribed affine mean curvature equation is a nonlinear, fourth order, geometric partial differential equation. It was introduced by Trudinger and Wang in 2005 in their investigation of the affine Plateau problem in affine geometry. The previous works of Trudinger-Wang, Chau-Weinkove and the author solved this global problem under some restrictions on the sign or integrability of the affine mean curvature. In this talk, we explain how to remove these restrictions and obtain global solutions under optimal integrability conditions on the affine mean curvature. Our analysis also covers the case of Abreu's equation arising in complex geometry.

The second boundary value problem of the prescribed affine mean curvature equation is a nonlinear, fourth order, geometric partial differential equation. It was introduced by Trudinger and Wang in 2005 in their investigation of the affine Plateau problem in affine geometry. The previous works of Trudinger-Wang, Chau-Weinkove and the author solved this global problem under some restrictions on the sign or integrability of the affine mean curvature. In this talk, we explain how to remove these restrictions and obtain global solutions under optimal integrability conditions on the affine mean curvature. Our analysis also covers the case of Abreu's equation arising in complex geometry.

### 2016年08月29日(月)

10:30-11:30 数理科学研究科棟(駒場) 268号室

通常の開催曜日、会場と異なります。

The Navier-Stokes equations: stationary existence, conditional regularity, and self-similar singularities (English)

https://www.math.lsu.edu/~pcnguyen/

通常の開催曜日、会場と異なります。

**Nguyen Cong Phuc 氏**(Louisiana State University)The Navier-Stokes equations: stationary existence, conditional regularity, and self-similar singularities (English)

[ 講演概要 ]

In this talk, both stationary and time-dependent Navier-Stokes equations are discussed. The common theme is that the quadratic nonlinearity and the pressure are both treated as weights generally belonging to a Sobolev space of negative order. We obtain the unique existence of solutions to stationary Navier-Stokes equations with small singular external forces that belong to a critical space. This result can be viewed as the stationary counterpart of an existence result obtained by H. Koch and D. Tataru for the free non-stationary Navier-Stokes equations with small initial data in $BMO^{-1}$. In another direction, some new local energy bounds are obtained for the time-dependent Navier-Stokes equations which imply the regularity condition $L_{t}^{\infty}(X)$, where $X$ is a non-endpoint borderline Lorentz space $X=L_{x}^{3, q}, q\not=\infty$. The analysis also allows us to rule out the existence of Leray's backward self-similar solutions to the Navier–Stokes equations with profiles in $L^{12/5}(\mathbb{R}^3)$ or in the Marcinkiewicz space $L^{q, \infty}(\mathbb{R}^{3})$ for any $q \in (12/5, 6)$.

This talk is based on joint work with Tuoc Van Phan and Cristi Guevara.

[ 参考URL ]In this talk, both stationary and time-dependent Navier-Stokes equations are discussed. The common theme is that the quadratic nonlinearity and the pressure are both treated as weights generally belonging to a Sobolev space of negative order. We obtain the unique existence of solutions to stationary Navier-Stokes equations with small singular external forces that belong to a critical space. This result can be viewed as the stationary counterpart of an existence result obtained by H. Koch and D. Tataru for the free non-stationary Navier-Stokes equations with small initial data in $BMO^{-1}$. In another direction, some new local energy bounds are obtained for the time-dependent Navier-Stokes equations which imply the regularity condition $L_{t}^{\infty}(X)$, where $X$ is a non-endpoint borderline Lorentz space $X=L_{x}^{3, q}, q\not=\infty$. The analysis also allows us to rule out the existence of Leray's backward self-similar solutions to the Navier–Stokes equations with profiles in $L^{12/5}(\mathbb{R}^3)$ or in the Marcinkiewicz space $L^{q, \infty}(\mathbb{R}^{3})$ for any $q \in (12/5, 6)$.

This talk is based on joint work with Tuoc Van Phan and Cristi Guevara.

https://www.math.lsu.edu/~pcnguyen/