NRIMS Symposium (Open)

Harmonic Analysis andNonlinear Partial Differential
Equations
Date:

24 June – 26 June 2024

Location:
Location:

Room 420, Research Institute for Mathematical Sciences
Kyoto University, Kyoto 606-8502

Organisers:
Organisers:

Neal Bez (Saitama University)
Yutaka Terasawa (Nagoya University)

ProgramMonday 24 June
13:15〜14:15

Takahisa Inui (Osaka University)

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 (Okinawa Institute of Science and Technology)

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

Tuesday 25 June
9:00〜10:00

Ye Zhang (Okinawa Institute of Science and Technology)

10:45〜11:45

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

Lunch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 (Saitama University)

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

Wednesday 26 June
9:30〜10:30

Mikihiro Fujii (Kyushu University)

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Abstracts
Sanghyuk Lee (Seoul National University)
Problems related to the Hermite operator I, II
Abstract: In this series of talks we are concerned with problems associated with the Hermite operator
H = −∆ + |x|2 , whose eigenfunctions are the Hermite functions. These problems include the sharp
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 theglobal space exhibit considerable differences that lead to significant difficulties compared with thelocal problems. I will present recent progress regarding those problems, including applications tostrong unique continuation for the heat equation.

Mikihiro Fujii (Kyushu University)
Ill-posedness of the stationary and time-periodic solutions to the 2D Navier–Stokes equations on thewhole plane
Abstract: 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 (Saitama University)
Optimisers for geometric Barthe–Wolff inequality
Abstract: The Brascamp–Lieb inequality is a far-reaching common generalisation of well-known
multilinear functional inequalities on euclidean spaces, such as the Hölder, Loomis–Whitney andYoung 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.

Takahisa Inui (Osaka University)
Global dynamics for threshold even solutions to the nonlinear Schrödinger equation with a repulsive
delta potential
Abstract: We consider the nonlinear Schrödinger equation with a repulsive delta potential in onedimensional space. This equation has even ground state solutions, which provide the threshold foreven 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 thethreshold.

Diogo Oliveira e Silva (Instituto Superior Técnico)
Sharp extension inequalities on finite fields
Abstract: Sharp restriction theory in euclidean space and the extension problem on finite fields haveboth 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
Abstract: The nomenclature “Landis-type results” concern the interest in the maximum vanishingrate 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
Abstract: We deal with a few recent Liouville-type theorems for the 3D stationary Navier–Stokes
equations, which were derived using iteration methods.

Ye Zhang (Okinawa Institute of Science and Technology)
Semiconcavity results of squared sub-Riemannian distance on Heisenberg group and applications toPDEs
Abstract: On Euclidean spaces, the squared distance function | · |2 is a convex function as well as asemiconcave 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 be shortest) andthe 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 group instead, it turns
out that the h-semiconcavity holds for the squared sub-Riemannian distance but the h-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.

Xiaodan Zhou (Okinawa Institute of Science and Technology)
Uniqueness of Green functions in metric spaces, yes and noAbstract: 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 have

been 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.