NRIMS 共同研究（公開型）

調和解析と非線形偏微分方程式
副副副日時: 令和 6 年 6 月 24 日（月）13：15 – 6 月 26 日（水）11：45
副副副場所: 京都大学数理解析研究所 420号室
副副副場所: 京都市左京区北白川追分町
副副副場所: 市バス 京大農学部前 または 北白川 下車
研究代表者: ベズ ニール (埼玉大学大学院理工学研究科)
副副代表者: 寺澤 祐高 (名古屋大学大学院多元数理科学研究科)

プログラム
6 月 24 日（月）13:15〜14:15

戍亥 隆恭 (大阪大)10:15〜11:15

Global dynamics for threshold even solutions to the nonlinear Schrödinger equation

10:30〜11:10

with a repulsive delta potential

14:30〜15:30

Xiaodan Zhou (沖縄科学技術大学院大学)

10:30〜11:10

Uniqueness of Green functions in metric spaces, yes and no

15:45〜16:45

Luz Roncal (Basque Center for Applied Mathematics)

10:15〜11:15

Landis-type results for discrete equations

6 月 25 日（火）9:00〜10:00

Ye Zhang (沖縄科学技術大学院大学)

10:15〜11:15

Semiconcavity results of squared sub-Riemannian distance on Heisenberg group and10:30〜11:10

applications to PDEs

10:15〜11:15

Sanghyuk Lee (Seoul National University)

10:30〜11:10

Problems related to the Hermite operator I

11:15〜13:15

お昼休み13:15〜14:15

Diogo Oliveira e Silva (Instituto Superior Técnico)

10:15〜11:15

Sharp extension inequalities on finite fields

14:30〜15:30

Anthony Gauvan (埼玉大)

10:15〜11:15

Optimisers for geometric Barthe–Wolff inequality

15:45〜16:45

Sanghyuk Lee (Seoul National University)

10:15〜11:15

Problems related to the Hermite operator II

6 月 26 日（水）9:30〜10:30

藤井 幹大 (九州大)

10:15〜11:15

Ill-posedness of the stationary and time-periodic solutions to the 2D Navier–Stokes

10:30〜11:10

equations on the whole plane

10:45〜11:45

Minsuk Yang (Yonsei University)

10:30〜11:10

Liouville-type theorems for the 3D stationary Navier–Stokes equations

アブストラクト
Sanghyuk Lee (Seoul National University)
Problems related to the Hermite operator I, II
アブストラクト: In this series of talks we are concerned with problems associated with the Hermiteoperator H = −∆ + |x|2 , whose eigenfunctions are the Hermite functions. These problems include thesharp Lp bound on spectral projection and the Lp convergence of Bochner–Riesz means. In a local
setting, the problems are similar to their counterparts associated with −∆. However, the problems
in the global space exhibit considerable differences that lead to significant difficulties compared withthe local problems. I will present recent progress regarding those problems, including applications tostrong unique continuation for the heat equation.
藤井 幹大 (九州大)Ill-posedness of the stationary and time-periodic solutions to the 2D Navier–Stokes equations on thewhole plane
アブストラクト: We consider the incompressible Navier–Stokes equations on the whole plane. In
contrast to the initial value problem, the solvability of the stationary and time-periodic solutions to
the 2D Navier–Stokes equations has long been open. In this talk, we solve this problem negatively.
Our method is based on the contradiction argument via the large time behavior of the nonstationary
Navier–Stokes flow with the stationary external forces.

Anthony Gauvan (埼玉大)Optimisers for geometric Barthe–Wolff inequality
アブストラクト: The Brascamp–Lieb inequality is a far-reaching common generalisation of wellknown multilinear functional inequalities on euclidean spaces, such as the Hölder, Loomis–Whitney
and Young convolution inequalities. Recently, Barthe and Wolff made a systematic study of “inverse”
Brascamp–Lieb inequalities. In this talk, we will describe the optimisers for geometric instances of
those inverse inequalities.
戍亥 隆恭 (大阪大)Global dynamics for threshold even solutions to the nonlinear Schrödinger equation with a repulsive
delta potential
アブストラクト: We consider the nonlinear Schrödinger equation with a repulsive delta potential in
one-dimensional space. This equation has even ground state solutions, which provide the thresholdfor even solutions from the viewpoint of global dynamics. The even solutions below the threshold
either scatter or blow up. In this study, we investigate the global dynamics of the even solutions at
the threshold.

Diogo Oliveira e Silva (Instituto Superior Técnico)
Sharp extension inequalities on finite fields
アブストラクト: Sharp restriction theory in euclidean space and the extension problem on finite fields
have both received much attention in the last two decades, but so far they have not intersected. In thistalk, we discuss our first results on sharp restriction theory on finite fields. Even though our methods
for dealing with paraboloids and cones borrow some inspiration from their euclidean counterparts,
new phenomena arise which are related to the underlying arithmetic and discrete structures. This isbased on recent joint work with Cristian González-Riquelme.

Luz Roncal (Basque Center for Applied Mathematics)
Landis-type results for discrete equations
アブストラクト: The nomenclature “Landis-type results” concern the interest in the maximum vanishing rate of solutions to equations with potentials. We prove Landis-type uniqueness results for boththe semidiscrete heat and the stationary discrete Schrödinger equations. The results are obtained
through quantitative estimates within a spatial lattice which manifest an interpolation phenomenon
between continuum and discrete scales. In the case of the elliptic equation, these quantitative estimates exhibit a rate decay which, in the range close to continuum, coincides with the same exponent
as in the classical results of the Landis conjecture in the Euclidean setting. The strategy is inspired by
the classical continuum approach, namely logarithmic convexity and Carleman-type estimates. Joint
work with Aingeru Fernández-Bertolin and Diana Stan.

Minsuk Yang (Yonsei University)
Liouville-type theorems for the 3D stationary Navier–Stokes equations
アブストラクト: We deal with a few recent Liouville-type theorems for the 3D stationary Navier–Stokes equations, which were derived using iteration methods.

Ye Zhang (沖縄科学技術大学院大学)
Semiconcavity results of squared sub-Riemannian distance on Heisenberg group and applications toPDEs
アブストラクト: On Euclidean spaces, the squared distance function | · |2 is a convex function as
well as a semiconcave function. When it comes to the simplest sub-Riemannian manifold, the Heisenberg group, due to the appearance of the cut locus (the set of points where geodesics cease to beshortest) and the abnormal set (the set of endpoints of abnormal length minimizers), the squared
sub-Riemannian distance cannot be locally semiconcave nor locally semiconvex. However, if we consider the corresponding weaker horizontal convexity (or h-convexity in short) on Heisenberg groupinstead, it turns out that the h-semiconcavity holds for the squared sub-Riemannian distance but theh-semiconvexity still fails. We have two applications to PDEs:
1. the viscosity solution of some non-coercive evolutive Hamilton–Jacobi equations given by theassociated Hopf–Lax formula is h-semiconcave but it will break the h-semiconvexity;
2. h-logconcavity is not preserved by the Dirichlet heat flow by a further study of the heat kernel.
This talk is based on joint works with Federica Dragoni, Qing Liu and Xiaodan Zhou.

Xiaodan Zhou (沖縄科学技術大学院大学)
Uniqueness of Green functions in metric spaces, yes and no
アブストラクト: In Euclidean space Rn , a result of Kichenassamy and Verón shows that the n-Laplace
operator
Ln u := div(|∇u|n−2 ∇u)admits a unique global Green function, i.e., there is a unique, properly normalized singular solution
which blows up to +∞ at the origin and converges to −∞ at infinity. The argument was later simplified
and extended to the Carnot group by Balogh, Holopainen and Tyson, thus establishing uniqueness ofglobal Green functions in the conformal case p = n in this geometry.

The purpose of this talk is to discuss the uniqueness of Q-harmonic Green functions for Q > 1 inthe setting of complete metric spaces (X, d, µ) equipped with an Ahlfors Q-regular Borel measureµ, and a Poincaré inequality. Over the past 20 years, many aspects of first-order calculus havebeen systematically developed in such spaces and spaces satisfying these conditions include Euclidean
spaces, Riemannian manifolds, and many nonEuclidean spaces like sub-Riemannian manifolds such as
the Heisenberg group, Gromov-Hausdorff limits of manifolds with lower Ricci curvature bounds, visualboundaries of certain hyperbolic buildings, etc. For our particular question about the uniqueness ofGreen functions, it turns out that the answer depends on the availability of an inner product on thegeneralized cotangent space of the metric space.
The talk is based on two joint works, one with Mario Bonk and Luca Capogna, and the other withAnders Björn, Jana Björn and Sylvester Eriksson-Bique.