NRIMS Symposium (Open)

Harmonic Analysis andNonlinear Partial Differential
Equations
Date:

3 July – 5 July 2023

Location:
Location:

Maskawa Hall, Maskawa Building for Education and Research
Kyoto University, Kyoto 606-8502

Organisers:
Organisers:

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

ProgramMonday 3 July
13:15〜14:15

Masaki Kawamoto (Ehime University)

10:15〜11:15

Modified scattering for nonlinear Schrödinger equations with long-range potentials

14:30〜15:30

Naoto Shida (Nagoya University)

10:30〜11:10

Boundedness of bilinear pseudo-differential operators with symbols in the bilinear

10:30〜11:10

Hörmander class S0,0 on Besov spaces

15:45〜16:45

Konstantin Merz (Osaka University)

10:15〜11:15

Random Schrödinger operators with complex decaying potentials

Tuesday 4 July
9:00〜10:00

Kenta Oishi (Waseda University)

10:15〜11:15

On the global well-posedness and decay of a free boundary problem of the

10:30〜11:10

Navier-Stokes equation in the two-dimensional half space

10:15〜11:15

Keisuke Takasao (Kyoto University)

10:30〜11:10

Phase field method for mean curvature flow and vanishing of discrepancy measure

11:15〜13:15

Lunch13:15〜14:15

Yasunori Maekawa (Kyoto University)

10:15〜11:15

Optimal rate of convergence to nondegenerate asymptotic profiles for fast diffusion

14:30〜15:30

Po-Lam Yung (Australian National University)

10:15〜11:15

Fourier decoupling via the high-low method

15:45〜16:45

Joshua Zahl (University of British Columbia)

10:15〜11:15

Sticky Kakeya sets, and the sticky Kakeya conjecture

Wednesday 5 July
9:30〜10:30

Helmut Abels (Universität Regensburg)

10:15〜11:15

Sharp interface limits of diffuse interface models I

10:45〜11:45

Helmut Abels (Universität Regensburg)

10:30〜11:10Sharp interface limits of diffuse interface models II

Abstracts
Helmut Abels (Universität Regensburg)
Sharp interface limits of diffuse interface models I, II
Abstract: We consider the dynamics of two components, which are partly miscible on a small lengthscale proportional to some parameter ε > 0. This is described by an order parameter as e.g. a
concentration difference, which is close to two distinct values (e.g. 1 and −1) in most of the domain,
and varies smoothly, but with a large gradient, between these two values in a small interfacial regionof thickness proportional to ε. Such models are called diffuse interface models and have various
applications. We are interested in rigorous results on convergence to classical sharp interface models
as ε tends to zero, where the components fill two disjoint regions is separated by a lower dimensional
interface. We will start with the case of the scalar Allen-Cahn equation, discuss the method of formally
matched asymptotics and show how to use this method in a rigorous proof of convergence. Afterwards
we will consider results on a coupled Navier-Stokes/Allen-Cahn system, which describes a two-phaseflow of macroscopically immiscible viscous Newtonian fluids. In this case the results depend essentiallyon the choice of a mobility coefficients in Allen-Cahn equation.

Masaki Kawamoto (Ehime University)
Modified scattering for nonlinear Schrödinger equations with long-range potentials
Abstract: We consider the scattering problem for the nonlinear Schrödinger equation with linear
potentials. In particular, we mainly focus on the case where both of nonlinearities and potentials
are of long-range type in the sense of scattering. In this talk, we introduce a concrete form of theasymptotic behavior of solutions of this equation and show the modified scattering for the associated
final state problem, as well as the existence of modified wave operators. This talk is based on the joint
work with Haruya Mizutani (Osaka University).

Yasunori Maekawa (Kyoto University)
Optimal rate of convergence to nondegenerate asymptotic profiles for fast diffusion
Abstract: In this talk we consider the (possibly sign-changing) solutions to the Cauchy-Dirichletproblem for the fast diffusion equation,∂t B(u) = ∆uin Ω × (0, ∞),u = 0 on ∂Ω × (0, ∞),(1)u = u0 on Ω × {0}.
Here Ω is a bounded Lipschitz domain in RN (N ≥ 2), B(u) = |u|q−2 u, and u0 ∈ H01 (Ω) \ {0}. We
focus on the case of the fast diffusion, that is,
2 < q < 2∗ :=(2)2N.
(N − 2)+Under this condition on q, the diffusion coefficient diverges on the boundary, due to the homogeneous
Dirichlet boundary condition. As a result, it is well known that every weak solution u = u(x, t) of (1)vanishes at a finite time t∗ = t∗ (u0 ), which is uniquely determined by the initial datum u0 . The rateof the extinction is also known to be estimated as(3)

1/(q−2)c1 (t∗ − t)+1/(q−2)≤ ∥u(·, t)∥H01 (Ω) ≤ c2 (t∗ − t)+with c1 , c2 > 0 for all t ≥ 0, provided that u0 ̸≡ 0. Then the transformation(4)v(x, s) = (t∗ − t)−1/(q−2) u(x, t) and s = log(t∗ /(t∗ − t))is well-defined, which has been used to study the asymptotic behavior of the solution near the extinction
time. We show that, if the transformed (possibly sign-changing) solution v = v(x, s) converges stronglyin H01 (Ω) to a nondegenerate (possibly sign-changing) profile ϕ = ϕ(x) as s → +∞, then the rate ofq−1
the convergence is estimated as O(e−λ0 s ), where λ0 = µk0 − q−2. Here { µ1k }k≥1 , 0 < µ1 < µ2 < . . .,are the eigenvalues of the compact self-adjoint operator (−∆)−1 Bϕ′ with Bϕ′ = (q − 1)|ϕ|q−2 in H01 (Ω),
and k0 is the number such that µk0 − q−1
q−2 firstly excesses zero. This convergence rate has been known
for positive nondegenerate profiles in smooth bounded domains, while our result is new for possibly
sign-changing nondegenerate profiles in bounded Lipschitz domains. We also discuss the optimalityof this convergence rate. This talk is based on the joint work with Goro Akagi (Tohoku University).

Konstantin Merz (Osaka University)
Random Schrödinger operators with complex decaying potentials
Abstract: We estimate complex eigenvalues of continuum random Schrödinger operators of Anderson
type. Our analysis relies on methods of J. Bourgain (Discrete Contin. Dyn. Syst., 2002, Lecture Notes
in Math., 2003) related to almost sure scattering for random lattice Schrödinger operators, and allows
us to consider potentials which decay almost twice as slowly as in the deterministic case. The talk isbased on joint work with Jean-Claude Cuenin.

Kenta Oishi (Waseda University)
On the global well-posedness and decay of a free boundary problem of the Navier-Stokes equation in
the two-dimensional half space
Abstract: We establish the global well-posedness and some decay properties for a free boundary
problem of the incompressible Navier-Stokes equations in RN+ with N ≥ 2. For N = 2, the decay ofthe solution to the free boundary problem is too slow to control the nonlinear term on the boundary
and that is why we could not handle the case N = 2 in our preceding work. We overcome this difficulty
1−1/q
1/q
owing to the trace estimate ∥f (x′ , 0)∥Lq (RN −1 ) ≤ ∥f ∥L (RN ) ∥∇f ∥L (RN ) , where x′ = (x1 , · · · , xN −1 ).
q+q+This work was supported by JSPS KAKENHI, Grant-in-Aid for Young Scientists, Grant Number
22K13945.

Naoto Shida (Nagoya University)
Boundedness of bilinear pseudo-differential operators with symbols in the bilinear Hörmander classS0,0 on Besov spaces
Abstract: We consider bilinear pseudo-differential operators with symbols in the bilinear Hörmanderclass S0,0 . The boundedness of these operators on Lebesgue spaces has been established by MiyachiTomita (2013) and Kato-Miyachi-Tomita (2022). In this talk, we discuss the boundedness of theseoperators in the settings of Besov spaces.

Keisuke Takasao (Kyoto University)
Phase field method for mean curvature flow and vanishing of discrepancy measure
Abstract: In this talk, we consider the mean curvature flow with forcing term and the volume
preserving mean curvature flow. To construct of the weak solutions for the flows, we use the AllenCahn equations with suitable additional terms. If the L1 -norm of the difference of the Dirichlet energy
and the potential energy converges to 0 (the vanishing of the discrepancy measure), then the AllenCahn equation converges to the weak solution to the flow in the sense of varifolds. We will show thevanishing via the Cole-Hopf transformation and the monotonicity formula.

Po-Lam Yung (Australian National University)
Fourier decoupling via the high-low method
Abstract: We will explain, using examples, how Guth, Wang and Maldague proved decoupling for theparabola in R2 , and how Guo, Li and myself adapted it to study some related decoupling estimates.

Joshua Zahl (University of British Columbia)
Sticky Kakeya sets, and the sticky Kakeya conjecture
Abstract: A Kakeya set is a compact subset of Rn , that contains a unit line segment pointing inevery direction. The Kakeya conjecture asserts that such sets must have dimension n. This conjecture
is closely related to several open problems in harmonic analysis, and it sits at the base of a hierarchyof increasingly difficult questions about the behavior of the Fourier transform in Euclidean space.
There is a special class of Kakeya sets, called sticky Kakeya sets. Sticky Kakeya sets exhibit anapproximate self-similarity at many scales, and sets of this type played an important role in Katz,
Laba, and Tao’s groundbreaking 1999 work on the Kakeya problem. In this talk, I will discuss aspecial case of the Kakeya conjecture, which asserts that sticky Kakeya sets must have dimension n.
I will discuss the proof of this conjecture in dimension 3. This is joint work with Hong Wang.