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2024 | Book

Mathematical Physics and Its Interactions

In Honor of the 60th Birthday of Tohru Ozawa, Tokyo, Japan, August 2021

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About this book

This publication comprises research papers contributed by the speakers, primarily based on their planned talks at the meeting titled 'Mathematical Physics and Its Interactions,' initially scheduled for the summer of 2021 in Tokyo, Japan. It celebrates Tohru Ozawa's 60th birthday and his extensive contributions in many fields.

The works gathered in this volume explore interactions between mathematical physics, various types of partial differential equations (PDEs), harmonic analysis, and applied mathematics. They are authored by research leaders in these fields, and this selection honors the spirit of the workshop by showcasing cutting-edge results and providing a forward-looking perspective through discussions of problems, with the goal of shaping future research directions.

Originally planned as an in-person gathering, this conference had to change its format due to limitations imposed by COVID, more precisely to avoid inducing people into unnecessary vaccinations.

Table of Contents

Frontmatter
Positive Solutions of Superlinear Elliptic Equations with Respect to the Schrödinger Operator
Abstract
We study positive solutions of semilinear elliptic equations with respect to the Schrödinger operator in the superlinear case. In a bounded smooth domain, a priori estimates for solutions and their gradients, the Harnack inequality and the boundary Harnack principle are presented. As applications, we also investigate the asymptotic behavior of positive solutions with isolated boundary singularities and the removability of isolated boundary singularities.
Kentaro Hirata
Convexity Phenomena Arising in an Area-Preserving Crystalline Curvature Flow
Abstract
An area-preserving crystalline curvature flow is regarded as a simple model of the deformation process of a negative crystal. In the present paper, behavior of polygonal curves by area-preserving crystalline curvature flow is discussed. We show “convexity phenomena”, that is, the solution polygon from a non-convex initial polygon becomes convex in a finite time. In order to show this assertion, we classify edge-disappearing patterns completely and prove that all zero-curvature edges disappear in a finite time, and we also show that evolution process of the flow can be continued beyond such edge-disappearing singularities.
Tetsuya Ishiwata, Shigetoshi Yazaki
Rate of Convergence for Approximate Solutions in Obstacle Problems for Nonlinear Operators
Abstract
The rate of convergence of approximate solutions via penalization and regularization for bilateral obstacle problems of parabolic partial differential equations is shown by the nonlinear adjoint method introduced by Evans [6]. In particular, quasilinear operators arising in the level set mean curvature flow and the p-Laplace operator for \(p>2\) with obstacles can be applied.
Shigeaki Koike, Takahiro Kosugi
On a Compatibility Condition for the Navier-Stokes Solutions in Maximal -regularity Class
Abstract
We consider a compatibility condition on the initial data a and the external force f for the initial-boundary value problem of the Navier-Stokes equations with no-slip condition on \(\partial \Omega \) in a bounded domain \(\Omega \subset \mathbb R^n\). Our class of solutions is based on that of maximal \(L^s\)-regularity as \(W^{1,s}(0, T; \mathcal {D}(A_r^\varepsilon )) \cap L^s(0, T; \mathcal {D}(A^{1+\varepsilon }_r))\), where \(A_r\) denotes the Stokes operator in \(L^r_\sigma (\Omega )\). We show that if the solution belongs to such a class for \(\varepsilon > 1/s + 1/2r\), then a and f necessarily satisfy
$$\begin{aligned} A_ra + P_r(a\cdot \nabla a) = P_rf(0) \quad \text {on }\partial \Omega , \end{aligned}$$
where \(P_r\) denotes the \(L^r\)-Helmholtz projection in \(\Omega \). Simultaneously, we construct the solution in such a class with \(0\le \varepsilon < 1/2r\) for a and f without such a compatibility condition as above provided \(2/s + n/r =3 + 2\varepsilon \).
Hideo Kozono, Senjo Shimizu
Asymptotic Behavior in Time of Solution to System of Cubic Nonlinear Schrödinger Equations in One Space Dimension
Abstract
In this paper, we consider the large time asymptotic behavior of solutions to systems of two cubic nonlinear Schrödinger equations in one space dimension. It turns out that for a system there exists a small solution of which asymptotic profile is a sum of two parts oscillating in a different way. This kind of behavior seems new. Further, several examples of systems which admit solution with several types of behavior such as modified scattering, nonlinear amplification, and nonlinear dissipation, are given. We also extend our previous classification result of nonlinear cubic systems.
Satoshi Masaki, Jun-Ichi Segata, Kota Uriya
Remarks on Blow up of Solutions of Nonlinear Wave Equations in Friedmann-Lemaître-Robertson-Walker Spacetime
Abstract
Consider nonlinear wave equations in the spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) spacetimes. We improve some upper bounds of the lifespan of blow-up solutions for the power nonlinearity. We also show upper bounds of the lifespan in the critical cases for the time derivative nonlinearity.
Kimitoshi Tsutaya, Yuta Wakasugi
The Frisch–Parisi Formalism for Fluctuations of the Schrödinger Equation
Abstract
We consider the solution of the Schrödinger equation u in \(\mathbb {R}\) when the initial datum tends to the Dirac comb. Let \(h_{\text {p}, \delta }(t)\) be the fluctuations in time of \(\int \left|x\right|^{2\delta }\left|u(x,t)\right|^2\,dx\), for \(0 < \delta < 1\), after removing a smooth background. We prove that the Frisch–Parisi formalism holds for \(H_\delta (t) = \int _{[0,t]}h_{\text {p}, \delta }(2s)\,ds\), which is morally a simplification of Riemann’s non-differentiable curve R. Our motivation is to understand the evolution of the vortex filament equation of polygonal filaments, which are related to R.
Sandeep Kumar, Felipe Ponce-Vanegas, Luz Roncal, Luis Vega
Recent Developments in Spectral Theory for Non-self-adjoint Hamiltonians
Abstract
The objective of this survey is to collect and elaborate on different tools, both well-established and more recent ones, which have been developed in the last decades to investigate spectral properties of non-self-adjoint operators of the form \(H=H_0+V\). More specifically, we will show how Hardy-type and Sobolev inequalities, together with Virial theorems and Birman-Schwinger principles enter into play in the analysis of the spectrum of these Hamiltonians.
Lucrezia Cossetti, Luca Fanelli, Nico M. Schiavone
Boussinesq, Schrödinger and Euler-Korteweg
Abstract
The aim of this paper of survey nature is to emphasize the links between two specific strongly dispersive “abcd” Boussinesq systems and nonlinear Schrödinger equations, one of those systems being a particular case of the Euler-Korteweg (EK) systems. We will in particular translate results for the later the known results for the EK system. We also comment on a one-dimensional integrable version of those systems, known as the Kaup-Broer-Kupperschmidt system. Finally we discuss another Boussinesq system that can be viewed as a (weakly) dispersive perturbation of the Saint-Venant (shallow-water) system.
Jean-Claude Saut, Li Xu
Representations of Pauli–Fierz Type Models by Path Measures

Functional integral representations of the semigroups generated by Pauli–Fierz type Hamiltonians in quantum field theory are reviewed. Firstly we introduce functional integral representations for Schrödinger type operators. Secondly those for Pauli–Fierz type Hamiltonians are shown. Finally inequalities derived from functional integral representations are shown.

Fumio Hiroshima
Metadata
Title
Mathematical Physics and Its Interactions
Editor
Shuji Machihara
Copyright Year
2024
Publisher
Springer Nature Singapore
Electronic ISBN
978-981-9703-64-7
Print ISBN
978-981-9703-63-0
DOI
https://doi.org/10.1007/978-981-97-0364-7

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