The 24th Northeastern Symposiumon Mathematical Analysis

Date: 20 - 21 February 2023
Venue: Kawai Hall, Graduate School of Science, Tohoku University

Organizers:
Goro Akagi (Tohoku University)
Masaharu Kobayashi (Hokkaido University)
Lorenzo Cavallina (Tohoku University)
Website:https://www.math.sci.hokudai.ac.jp/sympo/nema/24 en.html

This workshop is partially supported by Japan Society for the Promotion of Science, KAKENHI: Grant-in-Aid for Scientiﬁc Research (B) (No. 20H01812) “Evolution equations describing non-standard irreversible
processes – Analysis on singularities emerging in the dynamics of solutions –” Institute for Mathematics in Advanced Interdisciplinary Study

The 24th Northeastern Symposiumon Mathematical Analysis
Date： 20 - 21 February 2023
Venue： Kawai Hall, Graduate School of Science, Tohoku University(at Kita-Aobayama Campus)
6-3, Aramaki Aza-Aoba, Aoba-ku,Sendai 980-8578
https://www.sci.tohoku.ac.jp/campusmap/kita-aobayama/
URL: https://www.math.sci.hokudai.ac.jp/sympo/nema/24_en.html

Program
20 February 2023
09:45 - 09:50

Opening

09:50 - 10:40

Takayoshi Ogawa (Tohoku University)
Maximal L1 regularity for parabolic equations and a free boundary problemof the Navier–Stokes equations

10:50 - 11:40

Hiroaki Kikuchi (Tsuda University)
Threshold solutions for the 3D focusing cubic-quintic nonlinear Schrödingerequation

11:40 - 13:10

Lunch (90min)

13:10 - 13:40

Tatsu-Hiko Miura (Hirosaki University)
Error estimate for classical solutions to the heat equation in a moving thindomain and its limit equation

13:50 - 14:20

Kotaro Hisa (Tohoku University)
Initial traces and solvability for a semilinear heat equation on a half spaceof RN

14:30 - 15:00

Yusuke Ishigaki (Tokyo Institute of Technology)
Asymptotic stability of stationary solutions to outﬂow problem for compressible viscoelastic system

15:15 - 15:35

Shuntaro Tsubouchi (The University of Tokyo, D2)
Continuity of derivatives for certain very singular elliptic problems

15:35 - 15:55

Tatsuya Hosono (Tohoku University, D2)
Global existence and (8π)2 -threshold of solutions to the 4D attractionrepulsion chemotaxis system

16:00 - 16:30

Poster Preview (Zoom)

16:30 - 18:30

Poster Session (Zoom)

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21 February 2023
10:00 - 10:50

Hideo Kubo (Hokkaido University)
Global existence and blow-up for nonlinear wave equations with inversesquare potential

11:00 - 11:30

Alessandro Palmieri (University of Bari)
Blow-up results for a semilinear wave equation in the expanding de Sitter
spacetime

11:40 - 12:10

Nobuhito Miyake (The University of Tokyo)
Eventual global positivity of solutions to Cauchy problems of polyharmonicheat equations

12:10 - 13:40

Lunch (90min)

13:40 - 14:10

Takuya Sato (Tohoku University)
The initial boundary value problem for the nonlinear Schrödinger equation
with the nonlinear Neumann boundary condition in one space dimension

14:20 - 14:40

Motofumi Aoki (Tohoku University, D3)
On the relationship between the Cauchy problem and the energy conservation law for the compressible Navier–Stokes equations

14:40 - 15:00

Shun Tsuhara (Tohoku University, D2)
The boundary Strichartz estimates for the Schrödinger equation in the twodimensional half plane and its application

15:00 - 15:20

Kiichi Tashiro (Tokyo Institute of Technology, M2)
On the construction of canonical mean curvature ﬂow by elliptic regularization

15:30 - 15:50

Poster Award Ceremony

16:00 - 16:50

Philippe Souplet (Université Sorbonne Paris Nord)
Some recent Liouville type results and their applications (Zoom)

16:50 - 16:55

Closing2

Posters
P1. Dáithı́ Ó hAodha (Tohoku University, D2)
Large-Time Behaviour of Solutions to the Linearised Compressible Navier-Stokes Equations
P2. Yudai Kanda (Tohoku University, M2)
Gradient inequality for a Sobolev gradient ﬂow and its application to Gross-Pitaevskii
eigenvalue problem
P3. Sho Katayama (The University of Tokyo, M2)
Thresholds for the existence of solutions to supercritical elliptic problems
P4. Shunsuke Kitamura (Tohoku University, D1)
The lifespan estimates of classical solutions of one dimensional semilinear wave equations of derivative type with characteristic weights
P5. Mizuki Kojima (Tokyo Institute of Technology, D1)
On a time-fractional doubly critical equation, and its quantitative approach to theclassical counterpart
P6. Kenta Kumagai (Tokyo Institute of Technology, M2)
Regularity of extremal solutions of semilinear elliptic equations with general nonlinearities
P7. Fuya Hiroi (Tohoku University, M2)
Curve diﬀusion ﬂow for planar open curves on V-shaped polygonal lines
P8. Kazuya Hirose (Hokkaido University, M2)
A dynamical approach to lower gradient estimates for viscosity solutions of HamiltonJacobi equations
P9. Shimpei Makida (Hokkaido University, D1)
Stability of viscosity solutions on expanding networks
P10. Hiroki Miyakawa (Tohoku University, M2)
Maximal regularity for degenerate elliptic and parabolic equations of p-Laplacian type
P11. Yoshihito Nakajima (Tohoku University, M2)
Time-fractional evolution equations and applications to degenerate parabolic equations
P12. Shozo Ogino (Tohoku University, M2)
Singular limit problem of the initial value problem to the compressible Navier-Stokesequations in the critical Besov space
P13. Yusuke Oka (The University of Tokyo, M2)Existence of solutions for time fractional semilinear parabolic equations in Besov–Morrey spaces
P14. Akihito Ohgane (Hokkaido University, M2)
Appropriate selection of sensory input produces good swing movement

3P15. Florian Salin (Tohoku University, D1)
Implicit Scheme for Fractional Nonlinear Diﬀusion Equation Preserving Decay of Energy
P16. Kotaro Sato (Tohoku University, D2)
On some unidirectional evolution equation arising from fracture mechanics

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Abstracts
Maximal L1 regularity for parabolic equations anda free boundary problem of the Navier-Stokes equations
Takayoshi Ogawa (Tohoku University)
In this talk, we consider the free boundary problem of the incompressible Navier-Stokes
equations near the half Euclidean space. We introduce maximal L1 -regularity estimate forthe parabolic type equation ﬁrst and apply it to the free boundary problem. The key idea forshowing end-point maximal regularity is to obtain the almost orthogonal estimates between
the Green’s functions of the derivatives of the Stokes ﬂow and the Littlewood-Paley dyadic
decomposition of unity. The sharp boundary trace estimate as well as the inner and boundary
bilinear estimates is used for the proof. This talk is based on the joint works with Prof. Senjo
Shimizu (Kyoto University).

********** ********** **********

Threshold solutions for the 3D focusing cubic-quintic
nonlinear Schrödinger equation
Hiroaki Kikuchi (Tsuda University)
In this talk, we consider the focusing cubic-quintic nonlinear Schrödinger equation inthree space dimensions. Especially, we study the global dynamics of solutions whose energy
and mass equal to those of the ground state in the sprits of Duyckaerts and Merle (’09).
When we try to obtain the corresponding results of Duyckaerts and Merle, we meet several
diﬃculties due to the cubic-quintic nonlinearity. We overcome them by using the one-passtheorem (no return theorem) developed by Nakanishi and Schlag (’12). This talk is basedon a joint work with Masaru Hamano (Waseda University) and Minami Watanabe (TsudaUniversity).

5Error estimate for classical solutions to the heat equation
in a moving thin domain and its limit equation
Tatsu-Hiko Miura (Hirosaki University)
We consider the Neumann type problem of the heat equation in a moving thin domainaround a given closed moving hypersurface. Our aim is to compare a solution to the thindomain problem with a solution to a limit equation on the moving hypersurface whichappears in the thin-ﬁlm limit of the heat equation. We show that the diﬀerence of classicalsolutions to the thin domain problem and to the limit equation is of order the thicknessof the thin domain in the sup-norm. To prove such an error estimate, we derive an apriori estimate for a classical solution to the thin domain problem with a constant explicitly
depending on the thickness of the thin domain. Based on that a priori estimate, we constructa suitable approximate solution to the thin domain problem from a classical solution to thelimit equation by a formal asymptotic expansion of the thin domain problem.

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Initial traces and solvability for a semilinear heat equation
on a half space of RN
Kotaro Hisa (Tohoku University)
In this talk, we show the existence and the uniqueness of initial traces of nonnegative
solutions to a semilinear heat equation on a half space of RN under the zero Dirichletboundary condition. Furthermore, we obtain necessary conditions and suﬃcient conditions
on the initial data for the solvability of the corresponding Cauchy–Dirichlet problem. Our
necessary conditions and suﬃcient conditions are sharp and enable us to ﬁnd optimal singularities of initial data for the solvability of the Cauchy–Dirichlet problem. This talk is basedon a joint work with Prof. K. Ishige (The University of Tokyo) and Prof. J. Takahashi (TokyoInstitute of Technology).

********** ********** **********

Asymptotic stability of stationary solutions to outﬂow problem
for compressible viscoelastic system
Yusuke Ishigaki (Tokyo Institute of Technology)
We investigate the existence and stability of stationary solutions to the outﬂow problem
for compressible viscoelastic system in the one-dimensional half space. We classify the existence of stationary solutions by determining suitable conditions for several parameters, suchas the Mach number and propagation speed of elastic wave. We next establish its stability
result under the small initial perturbation. This talk is based on a joint work with Yoshihiro
Ueda (Kobe University).

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Continuity of derivatives for certain very singular elliptic problems
Shuntaro Tsubouchi (The University of Tokyo)
In this talk, we consider a one-Laplace equation or system perturbed by p-Laplacian.
These elliptic problems become no longer uniformly elliptic near a facet, the place where agradient degenerates. Recently, the speaker found it possible to prove continuous diﬀerentiability of weak solutions even across facets. The aim of this talk is to explain brieﬂy how
gradient continuity is shown.

********** ********** **********
Global existence and (8π)2 -threshold of solutions tothe 4D attraction-repulsion chemotaxis system
Tatsuya Hosono (Tohoku University)
We consider the Cauchy problem of an attraction-repulsion chemotaxis system, which
was proposed as a chemotaxis model in response to a combination of chemoattractant andchemorepellent secreted by the cells themselves. One of main topics in the study of sucha system is the presence of the L1 threshold. Indeed, critical mass phenomena called “8πproblem” are well-known in the 2-dimensional case. In this talk we establish that the solution
exists globally in time under the 4-dimensional setting, provided that the initial mass is lessthan the 4-dimensional L1 threshold value (8π)2 . To this end, we introduce the Brezis–Merle
type inequality for the 4th-order elliptic equation and rearrangement arguments. This talkis based on a joint work with Professor Takayoshi Ogawa (Tohoku University).

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Global existence and blow-up for nonlinear wave equations
with inverse-square potential
Hideo Kubo (Hokkaido University)
In this talk I’d like to present blow-up and global existence results for the Cauchy problem to the nonlinear wave equation with the inverse-square potential. Because of the strong
singularity of the inverse-square potential, one may expect the feature of the problem becomes quite diﬀerent from that of the corresponding problem without potential. However,as long as small amplitude solutions are concerned, we will see that the critical exponentof the nonlinearity which separates the blow-up and global existence is unchanged, though
we would meet technical diﬃculties. The key point of our analysis is the study of Rellich
type inequality for the Laplacian with the inverse-square potential. In fact, it provides usnice alternative to Klainerman’s inequality which is the one of the most important tool
for the study of the nonlinear wave equation. This talk is based on the joint works withProf. V. Georgiev, Prof. M. Sobajima, and Prof. W. Dai.
********** ********** **********
Blow-up results for a semilinear wave equation in the expandingde Sitter spacetime
Alessandro Palmieri (University of Bari)
In this talk, we consider a semilinear wave equation in the expanding de Sitter spacetime.
We will focus on blow-up results and upper-bound estimates for the lifespan of local solutions
to the corresponding Cauchy problem. The technique in the proof of these results is basedon the classical approach for the wave equation in the ﬂat case, which consists in studying
the growth of the time-dependent function given by the spatial average of a local solution.
A special emphasis will be put on a threshold case for the parameters appearing in thenonlinear term: we will combine an integral representation formula by Yagdjian-Galstianwith a technique borrowed from the critical case for the classical wave equation. Based ona joint work with Professor Hiroyuki Takamura (Tohoku University).
********** ********** **********
Eventual global positivity of solutions to Cauchy problems ofpolyharmonic heat equations
Nobuhito Miyake (The University of Tokyo)
In this talk, we consider Cauchy problems of polyharmonic heat equations. Our aim of
this talk is to show the existence of the threshold for the decay rate of initial data which
determines whether the corresponding solution to the Cauchy problem of the linear polyharmonic heat equation is eventually globally positive or not. As the application of thisresult, we construct eventually globally positive solutions to a semilinear polyharmonic heatequation.
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The initial boundary value problem for the nonlinear Schrödinger equation
with the nonlinear Neumann boundary condition in one space dimension
Takuya Sato (Tohoku University)
We consider the initial boundary value problem of the nonlinear Schrödinger equation on
the half line with a nonlinear Neumann boundary condition in one space dimension. Afterestablishing the boundary Srtichartz estimate in L2 , we consider the time local well-posednessof the problem in the lower regularity spaces as L2 (R+ ).

********** ********** **********On the relationship between the Cauchy problem and the energy conservation
law for the compressible Navier–Stokes equations
Motofumi Aoki (Tohoku University)
In this talk, we consider the energy conservation law of the full system of compressible
Navier–Stokes equations. The existence of weak solutions is known. We study suﬃcientconditions that the weak solutions satisfy energy conservation law. We also study the illposedness of the Cauchy problem for the equations of the ideal gas in the critical Besov spaces
and compare the ill-posedness condition with the suﬃcient conditions for energy conservation
law. This talk is based on a joint work with Professor Tsukasa Iwabuchi (Tohoku University).

********** ********** **********The boundary Strichartz estimates for the Schrödinger equation in thetwo-dimensional half plane and its application
Shun Tsuhara (Tohoku University)
We consider the initial boundary value problem for the nonlinear Schrödinger equationin the half plane with the nonlinear Neumann boundary condition. In the previous works,Batal–Özsari and Hayashi–Ogawa–Sato considered such a problem for the one-dimensionalcase. In this talk, we show the boundary Strichartz estimates for the two-dimensional casewith the Neumann boundary term and local well-posedness for the nonlinear Schrödinger
equation in L2 (R2+ ). This talk is based on a joint work with Prof. Takayoshi Ogawa andProf. Takuya Sato (Tohoku University).

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On the construction of canonical mean curvature ﬂow by elliptic regularization
Kiichi Tashiro (Tokyo Institute of Technology)
In this talk, we discuss the generalized mean curvature ﬂow constructed by elliptic-intime regularization. Generally, the generalized ﬂows have a non-uniqueness issue, such assuddenly disappearing. To this problem, we show the validity of an explicit identity about
the area change of the evolving grain. Such additional property resolves the non-uniquenessof the generalized ﬂows under suitable assumptions on the initial datum.

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Some recent Liouville type results and their applications
Philippe Souplet (LAGA, Université Sorbonne Paris Nord)
We present some recent Liouville type theorems for nonlinear elliptic equations, especiallyin the case of a half-space. The problems under consideration include the Lane-Emdenequation and the diﬀusive Hamilton-Jacobi (DHJ) equation. In the latter case, we will
discuss applications of the elliptic Liouville type theorem to the asymptotics of gradient
blow-up and to the loss of boundary conditions for viscosity solutions of the parabolic DHJ
equation, which arises in stochastic optimal control.
Joint works with L. Dupaigne, R. Filippucci, P. Pucci and B. Sirakov.

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