解析学火曜セミナー
過去の記録 ~02/15|次回の予定|今後の予定 02/16~
開催情報 | 火曜日 16:00~17:30 数理科学研究科棟(駒場) 156号室 |
---|---|
担当者 | 石毛 和弘, 坂井 秀隆, 伊藤 健一 |
セミナーURL | https://www.ms.u-tokyo.ac.jp/seminar/analysis/ |
過去の記録
2025年01月14日(火)
16:00-17:30 数理科学研究科棟(駒場) 128号室
対面・オンラインハイブリッド開催,場所にご注意ください
鈴木香奈子 氏 (茨城大学)
Existence and stability of discontinuous stationary solutions to reaction-diffusion-ODE systems (Japanese)
https://forms.gle/GtA4bpBuy5cNzsyX8
対面・オンラインハイブリッド開催,場所にご注意ください
鈴木香奈子 氏 (茨城大学)
Existence and stability of discontinuous stationary solutions to reaction-diffusion-ODE systems (Japanese)
[ 講演概要 ]
We consider reaction-diffusion-ODE systems, which consists of a single reaction-diffusion equation coupled with ordinary differential equations. Such systems arise, for example, from modeling of interactions between cellular processes and diffusing growth factors.
Reaction-diffusion-ODE systems in a bounded domain with Neumann boundary condition may have two types of stationary solutions, regular and discontinuous. We can show that all regular stationary solutions are unstable. This implies that reaction-diffusion-ODE systems cannot exhibit spatial patterns, and possible stable stationary solutions must be singular or discontinuous. In this talk, we present sufficient conditions for the existence and stability of discontinuous stationary solutions.
This talk is based on joint works with A. Marciniak-Czochra (Heidelberg University), G. Karch (University of Wroclaw) and S. Cygan (University of Wroclaw).
[ 参考URL ]We consider reaction-diffusion-ODE systems, which consists of a single reaction-diffusion equation coupled with ordinary differential equations. Such systems arise, for example, from modeling of interactions between cellular processes and diffusing growth factors.
Reaction-diffusion-ODE systems in a bounded domain with Neumann boundary condition may have two types of stationary solutions, regular and discontinuous. We can show that all regular stationary solutions are unstable. This implies that reaction-diffusion-ODE systems cannot exhibit spatial patterns, and possible stable stationary solutions must be singular or discontinuous. In this talk, we present sufficient conditions for the existence and stability of discontinuous stationary solutions.
This talk is based on joint works with A. Marciniak-Czochra (Heidelberg University), G. Karch (University of Wroclaw) and S. Cygan (University of Wroclaw).
https://forms.gle/GtA4bpBuy5cNzsyX8
2024年12月24日(火)
16:00-17:30 数理科学研究科棟(駒場) 128号室
対面・オンラインハイブリッド開催,場所にご注意ください
筧知之 氏 (筑波大学)
Snapshot problems for the wave equation and for the Euler-Poisson-Darboux equation (Japanese)
https://forms.gle/2otzqXYVD6DqM11S8
対面・オンラインハイブリッド開催,場所にご注意ください
筧知之 氏 (筑波大学)
Snapshot problems for the wave equation and for the Euler-Poisson-Darboux equation (Japanese)
[ 講演概要 ]
In this talk, we deal with snapshot problems for the wave equation and for the Euler-Poisson-Darboux equation. For simplicity, let us consider the wave equation $\partial_t^2 u - \Delta u =0$ on $\mathbb{R}^n$ with the condition $u|_{t=t_1} =f_1, \cdots, u|_{t=t_m} =f_m$. It is natural to ask when the above equation has a unique solution. We call the above problem the snapshot problem for the wave equation, and the set of $m$ functions $\{ f_1, \cdots, f_m \}$ the snapshot data. Roughly speaking, one of our main results is as follows.
Theorem. We assume that $m=3$ and $(t_3-t_1)/(t_2 -t_1)$ is irrational and not a Liouville number. In addition, we assume a certain compatibility condition on the snapshot data $\{ f_1, f_2, f_3 \}$. Then the snapshot problem for the wave equation has a unique solution.
We also consider a similar snapshot problem for the Euler-Poisson-Darboux equation. This is a joint work with Jens Christensen, Fulton Gonzalez, and Jue Wang.
[ 参考URL ]In this talk, we deal with snapshot problems for the wave equation and for the Euler-Poisson-Darboux equation. For simplicity, let us consider the wave equation $\partial_t^2 u - \Delta u =0$ on $\mathbb{R}^n$ with the condition $u|_{t=t_1} =f_1, \cdots, u|_{t=t_m} =f_m$. It is natural to ask when the above equation has a unique solution. We call the above problem the snapshot problem for the wave equation, and the set of $m$ functions $\{ f_1, \cdots, f_m \}$ the snapshot data. Roughly speaking, one of our main results is as follows.
Theorem. We assume that $m=3$ and $(t_3-t_1)/(t_2 -t_1)$ is irrational and not a Liouville number. In addition, we assume a certain compatibility condition on the snapshot data $\{ f_1, f_2, f_3 \}$. Then the snapshot problem for the wave equation has a unique solution.
We also consider a similar snapshot problem for the Euler-Poisson-Darboux equation. This is a joint work with Jens Christensen, Fulton Gonzalez, and Jue Wang.
https://forms.gle/2otzqXYVD6DqM11S8
2024年10月08日(火)
16:00-17:30 数理科学研究科棟(駒場) 128号室
対面・オンラインハイブリッド開催,場所にご注意ください
Erik Skibsted 氏 (Aarhus University)
Scattering subspace for time-periodic $N$-body Schrödinger operators (English)
https://forms.gle/it1Kc4voAXK5vpcB9
対面・オンラインハイブリッド開催,場所にご注意ください
Erik Skibsted 氏 (Aarhus University)
Scattering subspace for time-periodic $N$-body Schrödinger operators (English)
[ 講演概要 ]
We propose a definition of a scattering subspace for many-body Schrödinger operators with time-periodic short-range pair-potentials. This in given in geometric terms. We then show that all channel wave operators exist, and that their ranges span the scattering subspace. This may possibly serve as an intermediate step for proving the longstanding open problem of asymptotic completeness, which may be reformulated as the assertion that the scattering subspace is the orthogonal subspace of the pure point subspace of the monodromy operator.
[ 参考URL ]We propose a definition of a scattering subspace for many-body Schrödinger operators with time-periodic short-range pair-potentials. This in given in geometric terms. We then show that all channel wave operators exist, and that their ranges span the scattering subspace. This may possibly serve as an intermediate step for proving the longstanding open problem of asymptotic completeness, which may be reformulated as the assertion that the scattering subspace is the orthogonal subspace of the pure point subspace of the monodromy operator.
https://forms.gle/it1Kc4voAXK5vpcB9
2024年10月01日(火)
16:00-17:30 数理科学研究科棟(駒場) 128号室
東京確率論セミナーと合同開催,対面のみでオンライン配信は行いません,場所にご注意ください
Patrícia Gonçalves 氏 (IST Lisbon)
Hydrodynamics, fluctuations, and universality of exclusion processes (English)
東京確率論セミナーと合同開催,対面のみでオンライン配信は行いません,場所にご注意ください
Patrícia Gonçalves 氏 (IST Lisbon)
Hydrodynamics, fluctuations, and universality of exclusion processes (English)
[ 講演概要 ]
In the seventies, Frank Spitzer introduced interacting particle systems to the mathematics community. These systems consist of particles evolving randomly according to Markovian dynamics that conserve certain quantities. Interacting particle systems were already known in the physics and biophysics communities and served as toy models for a variety of interesting phenomena. One of the most classical interacting particle systems is the exclusion process, where particles evolve in a discrete space according to a transition probability, but at each site, only one particle is allowed. One of the goals of studying these models is to derive their hydrodynamic limit, i.e., to deduce the macroscopic equations governing the space-time evolution of the conserved quantities of the system from the underlying random motion of the microscopic particles.
In this talk, I will review the derivation of these limits for the exclusion process. I will also discuss their equilibrium fluctuations, i.e., the fluctuations around the typical profile when the system starts from the invariant measure. Our focus will then shift to the two-species exclusion process, a system with two conservation laws, namely particles of type A and B. We will see that for proper linear combinations of the conserved quantities, their evolution is autonomous. This advances our understanding of the universal behavior of these systems. This presentation is based on joint work with G. Cannizzaro, R. Misturini, and A. Occelli.
In the seventies, Frank Spitzer introduced interacting particle systems to the mathematics community. These systems consist of particles evolving randomly according to Markovian dynamics that conserve certain quantities. Interacting particle systems were already known in the physics and biophysics communities and served as toy models for a variety of interesting phenomena. One of the most classical interacting particle systems is the exclusion process, where particles evolve in a discrete space according to a transition probability, but at each site, only one particle is allowed. One of the goals of studying these models is to derive their hydrodynamic limit, i.e., to deduce the macroscopic equations governing the space-time evolution of the conserved quantities of the system from the underlying random motion of the microscopic particles.
In this talk, I will review the derivation of these limits for the exclusion process. I will also discuss their equilibrium fluctuations, i.e., the fluctuations around the typical profile when the system starts from the invariant measure. Our focus will then shift to the two-species exclusion process, a system with two conservation laws, namely particles of type A and B. We will see that for proper linear combinations of the conserved quantities, their evolution is autonomous. This advances our understanding of the universal behavior of these systems. This presentation is based on joint work with G. Cannizzaro, R. Misturini, and A. Occelli.
2024年07月09日(火)
16:00-17:30 数理科学研究科棟(駒場) 128号室
対面・オンラインハイブリッド開催,場所にご注意ください
Serge Richard 氏 (名古屋大学)
The topological nature of resonance(s) for 2D Schroedinger operators (English)
https://forms.gle/2fypneTA8CjYrLTX9
対面・オンラインハイブリッド開催,場所にご注意ください
Serge Richard 氏 (名古屋大学)
The topological nature of resonance(s) for 2D Schroedinger operators (English)
[ 講演概要 ]
In 1986, Gesztesy et al. revealed the surprising behavior of thresholds resonances for two-dimensional scattering systems: their contributions to Levinson's theorem are either 0 or 1, but not 1/2 as previously known for systems in dimension 1 and 3. During this seminar, we shall review this result, and explain how a C*-algebraic framework leads to a better understanding of this surprise. The main algebraic tool consists of a hexagonal algebra of Cordes, replacing a square algebra sufficient for systems in 1D and 3D. No prior C*-knowledge is expected from the audience. This presentation is based on a joint work with A. Alexander, T.D. Nguyen, and A. Rennie.
[ 参考URL ]In 1986, Gesztesy et al. revealed the surprising behavior of thresholds resonances for two-dimensional scattering systems: their contributions to Levinson's theorem are either 0 or 1, but not 1/2 as previously known for systems in dimension 1 and 3. During this seminar, we shall review this result, and explain how a C*-algebraic framework leads to a better understanding of this surprise. The main algebraic tool consists of a hexagonal algebra of Cordes, replacing a square algebra sufficient for systems in 1D and 3D. No prior C*-knowledge is expected from the audience. This presentation is based on a joint work with A. Alexander, T.D. Nguyen, and A. Rennie.
https://forms.gle/2fypneTA8CjYrLTX9
2024年06月18日(火)
16:00-17:30 数理科学研究科棟(駒場) 128号室
対面・オンラインハイブリッド開催,場所にご注意ください
森龍之介 氏 (明治大学)
Blocking and propagation in two-dimensional cylinders with spatially undulating boundary (Japanese)
https://forms.gle/TrFmSZQ1ZeqvSjfP7
対面・オンラインハイブリッド開催,場所にご注意ください
森龍之介 氏 (明治大学)
Blocking and propagation in two-dimensional cylinders with spatially undulating boundary (Japanese)
[ 講演概要 ]
We consider blocking and propagation phenomena of mean curvature flow with a driving force in two-dimensional cylinders with spatially undulating boundary. In this problem, Matano, Nakamura and Lou in 2006, 2013 characterize the effect of the shape of the boundary to blocking and propagation of the solutions under some slop condition about the boundary that implies time global existence of the classical solutions. In this talk, we consider the effect of the shape of the boundary to blocking and propagation of this problem under more general situation that the solutions may develop singularities near the boundary.
[ 参考URL ]We consider blocking and propagation phenomena of mean curvature flow with a driving force in two-dimensional cylinders with spatially undulating boundary. In this problem, Matano, Nakamura and Lou in 2006, 2013 characterize the effect of the shape of the boundary to blocking and propagation of the solutions under some slop condition about the boundary that implies time global existence of the classical solutions. In this talk, we consider the effect of the shape of the boundary to blocking and propagation of this problem under more general situation that the solutions may develop singularities near the boundary.
https://forms.gle/TrFmSZQ1ZeqvSjfP7
2024年05月14日(火)
16:00-18:15 数理科学研究科棟(駒場) 128号室
対面・オンラインハイブリッド開催(今回は講演が2件あります)
Heinz Siedentop 氏 (LMU University of Munich) 16:00-17:00
The Energy of Heavy Atoms: Density Functionals (English)
https://forms.gle/ZEyVso6wa9QpNfxH7
Robert Laister 氏 (University of the West of England) 17:15-18:15
Well-posedness for Semilinear Heat Equations in Orlicz Spaces (English)
https://forms.gle/ZEyVso6wa9QpNfxH7
対面・オンラインハイブリッド開催(今回は講演が2件あります)
Heinz Siedentop 氏 (LMU University of Munich) 16:00-17:00
The Energy of Heavy Atoms: Density Functionals (English)
[ 講演概要 ]
Since computing the energy of a system with $N$ particles requires solving a $4^N$ dimensional system of (pseudo-)differential equations in $3N$ independent variables, an analytic solution is practically impossible. Therefore density functionals, i.e., functionals that depend on the particle density (3 variables) only and yield the energy upon minimization, are of great interest.
This concept has been applied successfully in non-relativistic quantum mechanics. However, in relativistic quantum mechanics even the simple analogue of the Thomas-Fermi functional is not bounded from below for Coulomb potential. This problem was addressed eventually by Engel and Dreizler who derived a functional from QED. I will review some known mathematical properties of this functional and show that it yields basic features of physics, such as asymptotic correct energy, stability of matter, and boundedness of the excess charge.
[ 参考URL ]Since computing the energy of a system with $N$ particles requires solving a $4^N$ dimensional system of (pseudo-)differential equations in $3N$ independent variables, an analytic solution is practically impossible. Therefore density functionals, i.e., functionals that depend on the particle density (3 variables) only and yield the energy upon minimization, are of great interest.
This concept has been applied successfully in non-relativistic quantum mechanics. However, in relativistic quantum mechanics even the simple analogue of the Thomas-Fermi functional is not bounded from below for Coulomb potential. This problem was addressed eventually by Engel and Dreizler who derived a functional from QED. I will review some known mathematical properties of this functional and show that it yields basic features of physics, such as asymptotic correct energy, stability of matter, and boundedness of the excess charge.
https://forms.gle/ZEyVso6wa9QpNfxH7
Robert Laister 氏 (University of the West of England) 17:15-18:15
Well-posedness for Semilinear Heat Equations in Orlicz Spaces (English)
[ 講演概要 ]
We consider the local well-posedness of semilinear heat equations in Orlicz spaces, the latter prescribed via a Young function $\Phi$. Many existence-uniqueness results exist in the literature for power-like or exponential-like nonlinearities $f$, where the natural setting is an Orlicz space of corresponding type; i.e. if $f$ is power-like then $\Phi$ is power-like (Lebesgue space), if $f$ is exponential-like then $\Phi$ is exponential-like. However, the general problem of prescribing a suitable $\Phi$ for a given, otherwise arbitrary $f$ is open. Our goal is to provide a suitable framework to resolve this problem and I will present some recent results in this direction. The key is a new (to the best of our knowledge) smoothing estimate for the heat semigroup between two arbitrary Orlicz spaces. Existence then follows familiar lines via monotonicity or contraction mapping arguments. Global solutions are also presented under additional assumptions. This work is part of a collaborative project with Prof Kazuhiro Ishige, Dr Yohei Fujishima and Dr Kotaro Hisa.
[ 参考URL ]We consider the local well-posedness of semilinear heat equations in Orlicz spaces, the latter prescribed via a Young function $\Phi$. Many existence-uniqueness results exist in the literature for power-like or exponential-like nonlinearities $f$, where the natural setting is an Orlicz space of corresponding type; i.e. if $f$ is power-like then $\Phi$ is power-like (Lebesgue space), if $f$ is exponential-like then $\Phi$ is exponential-like. However, the general problem of prescribing a suitable $\Phi$ for a given, otherwise arbitrary $f$ is open. Our goal is to provide a suitable framework to resolve this problem and I will present some recent results in this direction. The key is a new (to the best of our knowledge) smoothing estimate for the heat semigroup between two arbitrary Orlicz spaces. Existence then follows familiar lines via monotonicity or contraction mapping arguments. Global solutions are also presented under additional assumptions. This work is part of a collaborative project with Prof Kazuhiro Ishige, Dr Yohei Fujishima and Dr Kotaro Hisa.
https://forms.gle/ZEyVso6wa9QpNfxH7
2024年03月12日(火)
16:00-17:30 数理科学研究科棟(駒場) 123号室
対面・オンラインハイブリッド開催,通常とは場所が異なります
Kobe Marshall-Stevens 氏 (University College London)
On the generic regularity of min-max CMC hypersurfaces (English)
https://forms.gle/7mqzgLqhtBuAovKB8
対面・オンラインハイブリッド開催,通常とは場所が異なります
Kobe Marshall-Stevens 氏 (University College London)
On the generic regularity of min-max CMC hypersurfaces (English)
[ 講演概要 ]
Smooth constant mean curvature (CMC) hypersurfaces serve as effective tools to study the geometry and topology of Riemannian manifolds. In high dimensions however, one in general must account for their singular behaviour. I will discuss how such hypersurfaces are constructed via min-max techniques and some recent progress on their generic regularity, allowing for certain isolated singularities to be perturbed away.
[ 参考URL ]Smooth constant mean curvature (CMC) hypersurfaces serve as effective tools to study the geometry and topology of Riemannian manifolds. In high dimensions however, one in general must account for their singular behaviour. I will discuss how such hypersurfaces are constructed via min-max techniques and some recent progress on their generic regularity, allowing for certain isolated singularities to be perturbed away.
https://forms.gle/7mqzgLqhtBuAovKB8
2023年11月14日(火)
16:15-17:15 数理科学研究科棟(駒場) 126号室
対面・オンラインハイブリッド開催,日時・場所にご注意ください
Arne Jensen 氏 (Aalborg University)
Resolvent expansions for magnetic Schrödinger operators (English)
https://forms.gle/qyEUeo4kVuPL1s289
対面・オンラインハイブリッド開催,日時・場所にご注意ください
Arne Jensen 氏 (Aalborg University)
Resolvent expansions for magnetic Schrödinger operators (English)
[ 講演概要 ]
I will present some new results resolvent expansions around threshold zero for magnetic Schrödinger operators in dimension three. The magnetic field and the electric potential are assumed to decay sufficiently fast. Analogous results for Pauli operators will also be presented.
Joint work with H. Kovarik, Brescia, Italy.
[ 参考URL ]I will present some new results resolvent expansions around threshold zero for magnetic Schrödinger operators in dimension three. The magnetic field and the electric potential are assumed to decay sufficiently fast. Analogous results for Pauli operators will also be presented.
Joint work with H. Kovarik, Brescia, Italy.
https://forms.gle/qyEUeo4kVuPL1s289
2023年08月22日(火)
16:00-17:30 数理科学研究科棟(駒場) 126号室
対面・オンラインハイブリッド開催,通常とは場所が異なります
Daniel Parra 氏 (Universidad de Santiago de Chile)
Towards a Levinson's Theorem for Discrete Magnetic operators on tubes under finite rank perturbations (English)
https://forms.gle/VBp4nSnYYKVpXFhB9
対面・オンラインハイブリッド開催,通常とは場所が異なります
Daniel Parra 氏 (Universidad de Santiago de Chile)
Towards a Levinson's Theorem for Discrete Magnetic operators on tubes under finite rank perturbations (English)
[ 講演概要 ]
In this talk we study a family of magnetic Hamiltonians on discrete tubes under a finite rank perturbation supported on its border. We go into detail for the case of rank $2$ and show how the eigenvalues can be related to the scattering matrix to exhibit an index theorem in the tradition of Levison’s theorem. We then turn to the general case, discuss the different spectral scenarios that can occur and explain the C*-algebraic framework that could allow us to treat this case. This is an ongoing work with S. Richard (名大), V. Austen (名大) and A. Rennie (Wollongong).
[ 参考URL ]In this talk we study a family of magnetic Hamiltonians on discrete tubes under a finite rank perturbation supported on its border. We go into detail for the case of rank $2$ and show how the eigenvalues can be related to the scattering matrix to exhibit an index theorem in the tradition of Levison’s theorem. We then turn to the general case, discuss the different spectral scenarios that can occur and explain the C*-algebraic framework that could allow us to treat this case. This is an ongoing work with S. Richard (名大), V. Austen (名大) and A. Rennie (Wollongong).
https://forms.gle/VBp4nSnYYKVpXFhB9
2023年07月11日(火)
16:00-17:30 数理科学研究科棟(駒場) 123号室
対面・オンラインハイブリッド開催,通常とは場所が異なります
Julian López-Gómez 氏 (Complutense University of Madrid)
Nodal solutions for a class of degenerate BVP’s (English)
https://forms.gle/S3VgMSWg9wUP69cY6
対面・オンラインハイブリッド開催,通常とは場所が異なります
Julian López-Gómez 氏 (Complutense University of Madrid)
Nodal solutions for a class of degenerate BVP’s (English)
[ 講演概要 ]
In this talk we characterize the existence of nodal solutions for a generalized class of one-dimensional diffusive logistic type equations, including
\[−u''=\lambda u−a(x)u^3,\quad x∈[0,L],\]
under the boundary conditions $u(0)=u(L)=0$, where $\lambda$ is regarded as a bifurcation parameter, and the non-negative weight function $a(x)$ vanishes on some subinterval
\[ [\alpha,\beta]\subset (0,L)\]
with $\alpha<\beta$.
At a later stage, the general case when $a(x)$ vanishes on finitely many subintervals with the same length is analyzed. Finally, we construct some examples with classical non-degenerate weights, with $a(x)>0$ for all $x∈[0,L]$, where the BVP has an arbitrarily large number of solutions with one node in $(0,L)$. These are the first examples of this nature constructed in the literature.
References:
P. Cubillos, J. López-Gómez and A. Tellini, Multiplicity of nodal solutions in classical non-degenerate logistic equations, El. Res. Archive 30 (2022), 898—928.
J. López-Gómez, M. Molina-Meyer and P. H. Rabinowitz, Global bifurcation diagrams of one-node solutions on a class of degenerate boundary value problems, Disc. Cont. Dyn. Syst. B 22 (2017), 923—946.
J. López-Gómez and P. H. Rabinowitz, Nodal solutions for a class of degenerate one dimensional BVP’s, Top. Meth. Nonl. Anal. 49 (2017), 359—376.
J. López-Gómez and P. H. Rabinowitz, The estructure of the set of 1-node solutions for a class of degenerate BVP’s, J. Differential Equations 268 (2020), 4691—4732.
P. H. Rabinowitz, A note on a anonlinear eigenvalue problem for a class of differential equations, J. Differential Equations 9 (1971), 536—548.
[ 参考URL ]In this talk we characterize the existence of nodal solutions for a generalized class of one-dimensional diffusive logistic type equations, including
\[−u''=\lambda u−a(x)u^3,\quad x∈[0,L],\]
under the boundary conditions $u(0)=u(L)=0$, where $\lambda$ is regarded as a bifurcation parameter, and the non-negative weight function $a(x)$ vanishes on some subinterval
\[ [\alpha,\beta]\subset (0,L)\]
with $\alpha<\beta$.
At a later stage, the general case when $a(x)$ vanishes on finitely many subintervals with the same length is analyzed. Finally, we construct some examples with classical non-degenerate weights, with $a(x)>0$ for all $x∈[0,L]$, where the BVP has an arbitrarily large number of solutions with one node in $(0,L)$. These are the first examples of this nature constructed in the literature.
References:
P. Cubillos, J. López-Gómez and A. Tellini, Multiplicity of nodal solutions in classical non-degenerate logistic equations, El. Res. Archive 30 (2022), 898—928.
J. López-Gómez, M. Molina-Meyer and P. H. Rabinowitz, Global bifurcation diagrams of one-node solutions on a class of degenerate boundary value problems, Disc. Cont. Dyn. Syst. B 22 (2017), 923—946.
J. López-Gómez and P. H. Rabinowitz, Nodal solutions for a class of degenerate one dimensional BVP’s, Top. Meth. Nonl. Anal. 49 (2017), 359—376.
J. López-Gómez and P. H. Rabinowitz, The estructure of the set of 1-node solutions for a class of degenerate BVP’s, J. Differential Equations 268 (2020), 4691—4732.
P. H. Rabinowitz, A note on a anonlinear eigenvalue problem for a class of differential equations, J. Differential Equations 9 (1971), 536—548.
https://forms.gle/S3VgMSWg9wUP69cY6
2023年06月06日(火)
17:00-18:30 数理科学研究科棟(駒場) 128号室
対面・オンラインハイブリッド開催,通常とは時間と場所が異なります
Erik Skibsted 氏 (Aarhus University)
Stationary completeness; the many-body short-range case (English)
https://forms.gle/kWHDfb6J6kcjfSah8
対面・オンラインハイブリッド開催,通常とは時間と場所が異なります
Erik Skibsted 氏 (Aarhus University)
Stationary completeness; the many-body short-range case (English)
[ 講演概要 ]
For a general class of many-body Schr\"odinger operators with short-range pair-potentials the wave and scattering matrices as well as the restricted wave operators are all defined at any non-threshold energy. In fact this holds without imposing any a priori decay condition on channel eigenstates and even for models including long-range potentials of Derezi\'nski-Enss type. For short-range models we improve on the known \emph{weak continuity} statements in that we show that all non-threshold energies are \emph{stationary complete}, resolving in this case a recent conjecture. A consequence is that the above scattering quantities depend \emph{strongly continuously} on the energy parameter at all non-threshold energies (whence not only almost everywhere as previously demonstrated). Another consequence is that the scattering matrix is unitary at any such energy. Our procedure yields (as a side result) a new and purely stationary proof of asymptotic completeness for many-body short-range systems.
[ 参考URL ]For a general class of many-body Schr\"odinger operators with short-range pair-potentials the wave and scattering matrices as well as the restricted wave operators are all defined at any non-threshold energy. In fact this holds without imposing any a priori decay condition on channel eigenstates and even for models including long-range potentials of Derezi\'nski-Enss type. For short-range models we improve on the known \emph{weak continuity} statements in that we show that all non-threshold energies are \emph{stationary complete}, resolving in this case a recent conjecture. A consequence is that the above scattering quantities depend \emph{strongly continuously} on the energy parameter at all non-threshold energies (whence not only almost everywhere as previously demonstrated). Another consequence is that the scattering matrix is unitary at any such energy. Our procedure yields (as a side result) a new and purely stationary proof of asymptotic completeness for many-body short-range systems.
https://forms.gle/kWHDfb6J6kcjfSah8
2023年03月14日(火)
16:00-17:30 数理科学研究科棟(駒場) 126号室
対面・オンラインハイブリッド開催
Piermarco Cannarsa 氏 (University of Rome "Tor Vergata")
Parameter reconstruction for degenerate parabolic equations (English)
https://forms.gle/nejpQS824vFKRbMQ6
対面・オンラインハイブリッド開催
Piermarco Cannarsa 氏 (University of Rome "Tor Vergata")
Parameter reconstruction for degenerate parabolic equations (English)
[ 講演概要 ]
First, we study degenerate parabolic equations arising in climate dynamics, providing uniqueness and stability estimates for the determination of the insolation function. Then, we address several aspects of the reconstruction of the degenerate diffusion coefficient. Finally, we discuss systems of two equations including a vertical component into the model.
[ 参考URL ]First, we study degenerate parabolic equations arising in climate dynamics, providing uniqueness and stability estimates for the determination of the insolation function. Then, we address several aspects of the reconstruction of the degenerate diffusion coefficient. Finally, we discuss systems of two equations including a vertical component into the model.
https://forms.gle/nejpQS824vFKRbMQ6
2022年12月20日(火)
16:00-17:30 数理科学研究科棟(駒場) 126号室
対面・オンラインハイブリッド開催
片岡清臣 氏 (東京大学)
J.Boman氏の最近の2つの関連する結果,distributionの台と解析性,Radon変換と楕円体領域の特殊な関係性についての解説 (Japanese)
https://forms.gle/BpciRTzKh9FPUV8D7
対面・オンラインハイブリッド開催
片岡清臣 氏 (東京大学)
J.Boman氏の最近の2つの関連する結果,distributionの台と解析性,Radon変換と楕円体領域の特殊な関係性についての解説 (Japanese)
[ 講演概要 ]
Jan Boman's (Stockholm Univ.) recent two papers:
[1], Regularity of a distribution and of the boundary of its support, The Journal of Geometric Analysis vol.32, Article number: 300 (2022).
[2], A hypersurface containing the support of a Radon transform must be an ellipsoid. II: The general case; J. Inverse Ill-Posed Probl. 2021; 29(3): 351–367.
In [1] he proved "Let $f(x_1,…,x_n,y)$ be a non-zero distribution with support in a $C^1$ surface $N=\{y=F(x)\}$. If $f(x,y)$ is depending real analytically on x-variables, then $F(x)$ is analytic". As an application, he reinforced the main result of [2]. These results are obtained essentially by means of matrix algebra and a number theoretic method.
[ 参考URL ]Jan Boman's (Stockholm Univ.) recent two papers:
[1], Regularity of a distribution and of the boundary of its support, The Journal of Geometric Analysis vol.32, Article number: 300 (2022).
[2], A hypersurface containing the support of a Radon transform must be an ellipsoid. II: The general case; J. Inverse Ill-Posed Probl. 2021; 29(3): 351–367.
In [1] he proved "Let $f(x_1,…,x_n,y)$ be a non-zero distribution with support in a $C^1$ surface $N=\{y=F(x)\}$. If $f(x,y)$ is depending real analytically on x-variables, then $F(x)$ is analytic". As an application, he reinforced the main result of [2]. These results are obtained essentially by means of matrix algebra and a number theoretic method.
https://forms.gle/BpciRTzKh9FPUV8D7
2022年12月13日(火)
16:00-17:30 数理科学研究科棟(駒場) 126号室
対面・オンラインハイブリッド開催
只野之英 氏 (東京理科大学)
Continuum limit problem of discrete Schrödinger operators on square lattices (Japanese)
https://forms.gle/CRha8hydEuXzh71S7
対面・オンラインハイブリッド開催
只野之英 氏 (東京理科大学)
Continuum limit problem of discrete Schrödinger operators on square lattices (Japanese)
[ 講演概要 ]
We consider discrete Schrödinger operators on the square lattice with its mesh size very small. The aim of this talk is to introduce the rigorous setting of continuum limit problems in the view point of operator theory and then to give its proof for the above operators, the one of which is defined on the vertices and the other of which is defined on the edges. This talk is based on joint works with Shu Nakamura (Gakushuin University) and Pavel Exner (Czech Academy of Science, Czech Technical University).
[ 参考URL ]We consider discrete Schrödinger operators on the square lattice with its mesh size very small. The aim of this talk is to introduce the rigorous setting of continuum limit problems in the view point of operator theory and then to give its proof for the above operators, the one of which is defined on the vertices and the other of which is defined on the edges. This talk is based on joint works with Shu Nakamura (Gakushuin University) and Pavel Exner (Czech Academy of Science, Czech Technical University).
https://forms.gle/CRha8hydEuXzh71S7
2022年11月29日(火)
16:00-17:30 数理科学研究科棟(駒場) 126号室
対面・オンラインハイブリッド開催
滝本和広 氏 (広島大学)
Bernstein type theorem for the parabolic 2-Hessian equation under weaker assumptions (Japanese)
https://forms.gle/93YQ9C6DGYt5Vjuf7
対面・オンラインハイブリッド開催
滝本和広 氏 (広島大学)
Bernstein type theorem for the parabolic 2-Hessian equation under weaker assumptions (Japanese)
[ 講演概要 ]
In the early twentieth century, Bernstein proved that a minimal surface which can be expressed as the graph of a function defined in $\mathbb{R}^2$ must be a plane. For Monge-Ampère equation, it is known that a convex solution to $\det D^2 u=1$ in $\mathbb{R}^n$ must be a quadratic polynomial. Such kind of theorems, which we call Bernstein type theorems in this talk, have been extensively studied for various PDEs. For the parabolic $k$-Hessian equation, Bernstein type theorem has been proved by Nakamori and Takimoto (2015, 2016) under the convexity and some growth assumptions on the solution. In this talk, we shall obtain Bernstein type theorem for the parabolic 2-Hessian equation under weaker assumptions.
[ 参考URL ]In the early twentieth century, Bernstein proved that a minimal surface which can be expressed as the graph of a function defined in $\mathbb{R}^2$ must be a plane. For Monge-Ampère equation, it is known that a convex solution to $\det D^2 u=1$ in $\mathbb{R}^n$ must be a quadratic polynomial. Such kind of theorems, which we call Bernstein type theorems in this talk, have been extensively studied for various PDEs. For the parabolic $k$-Hessian equation, Bernstein type theorem has been proved by Nakamori and Takimoto (2015, 2016) under the convexity and some growth assumptions on the solution. In this talk, we shall obtain Bernstein type theorem for the parabolic 2-Hessian equation under weaker assumptions.
https://forms.gle/93YQ9C6DGYt5Vjuf7
2022年10月04日(火)
16:00-17:30 数理科学研究科棟(駒場) 126号室
対面・オンラインハイブリッド開催
深尾武史 氏 (京都教育大学)
前方後方拡散分方程式を動的境界条件に持つCahn-Hilliard方程式への粘性消滅法による接近 (Japanese)
https://forms.gle/nPfEgKUX2tfUrg5LA
対面・オンラインハイブリッド開催
深尾武史 氏 (京都教育大学)
前方後方拡散分方程式を動的境界条件に持つCahn-Hilliard方程式への粘性消滅法による接近 (Japanese)
[ 講演概要 ]
4階の偏微分方程式であるCahn-Hilliard方程式は相分離現象を記述する方程式としてよく知られている. J. W. Cahn, "Science during Paradigm Creation", (2011)によると時間後方問題となる難点が4階の項によって解決される点は現象解明の副産物であったようである. 近年, 前方後方問題への接近としてこのCahn-Hilliard方程式における粘性消滅法の考察がBui-Smarrazzo-Tesei, J. Math. Anal. Appl, (2014)やKagawa-Otani, J. Math. Anal. Appl, (2022)などで行われている. 本講演ではこれらの粘性消滅法の考えを時間微分を境界条件に含む, いわゆる動的境界条件で考察する. 講演の前半では研究動機と動的境界条件下でのCahn-Hilliard方程式についての先行研究を紹介しつつ, 抽象発展方程式の枠組みで適切性を論じる流れを解説する. 後半では動的境界条件下でのCahn-Hilliard方程式の1つとしてよく知られるGMSモデルを元に証明の大枠, すなわち一様評価と極限操作を解説し, 最後にLWモデルの場合との違いについて述べる. なお, 本講演はPavia大学のP. Colli氏とMilano工科大学のL. Scarpa氏との共同研究に基づく.
[ 参考URL ]4階の偏微分方程式であるCahn-Hilliard方程式は相分離現象を記述する方程式としてよく知られている. J. W. Cahn, "Science during Paradigm Creation", (2011)によると時間後方問題となる難点が4階の項によって解決される点は現象解明の副産物であったようである. 近年, 前方後方問題への接近としてこのCahn-Hilliard方程式における粘性消滅法の考察がBui-Smarrazzo-Tesei, J. Math. Anal. Appl, (2014)やKagawa-Otani, J. Math. Anal. Appl, (2022)などで行われている. 本講演ではこれらの粘性消滅法の考えを時間微分を境界条件に含む, いわゆる動的境界条件で考察する. 講演の前半では研究動機と動的境界条件下でのCahn-Hilliard方程式についての先行研究を紹介しつつ, 抽象発展方程式の枠組みで適切性を論じる流れを解説する. 後半では動的境界条件下でのCahn-Hilliard方程式の1つとしてよく知られるGMSモデルを元に証明の大枠, すなわち一様評価と極限操作を解説し, 最後にLWモデルの場合との違いについて述べる. なお, 本講演はPavia大学のP. Colli氏とMilano工科大学のL. Scarpa氏との共同研究に基づく.
https://forms.gle/nPfEgKUX2tfUrg5LA
2022年08月23日(火)
16:00-17:30 数理科学研究科棟(駒場) 126号室
対面・オンラインハイブリッド開催(対面は本学関係者のみに限定します)
Stefan Neukamm 氏 (Dresden University/RIMS)
Quantitative homogenization for monotone, uniformly elliptic systems with random coefficients (English)
https://forms.gle/V1wxbYhT4mkPF4gY9
対面・オンラインハイブリッド開催(対面は本学関係者のみに限定します)
Stefan Neukamm 氏 (Dresden University/RIMS)
Quantitative homogenization for monotone, uniformly elliptic systems with random coefficients (English)
[ 講演概要 ]
Motivated by homogenization of nonlinearly elastic composite materials, we study homogenization rates for elliptic PDEs with monotone nonlinearity in the uniformly elliptic case. Under the assumption of a fast decay of correlations on scales larger than the microscale $\varepsilon$, we establish estimates of optimal order for the approximation of the homogenized operator by the method of representative volumes. Moreover, we discuss applications to nonlinear elasticity random laminates.
[ 参考URL ]Motivated by homogenization of nonlinearly elastic composite materials, we study homogenization rates for elliptic PDEs with monotone nonlinearity in the uniformly elliptic case. Under the assumption of a fast decay of correlations on scales larger than the microscale $\varepsilon$, we establish estimates of optimal order for the approximation of the homogenized operator by the method of representative volumes. Moreover, we discuss applications to nonlinear elasticity random laminates.
https://forms.gle/V1wxbYhT4mkPF4gY9
2022年07月26日(火)
16:00-17:30 数理科学研究科棟(駒場) 126号室
対面・オンラインハイブリッド開催(対面は本学関係者のみに限定します)
熊谷隆 氏 (早稲田大学)
Periodic homogenization of non-symmetric jump-type processes with drifts (Japanese)
https://forms.gle/ewZEy1jAXrAhWx1Q8
対面・オンラインハイブリッド開催(対面は本学関係者のみに限定します)
熊谷隆 氏 (早稲田大学)
Periodic homogenization of non-symmetric jump-type processes with drifts (Japanese)
[ 講演概要 ]
Homogenization problem is one of the classical problems in analysis and probability which is very actively studied recently. In this talk, we consider homogenization problem for non-symmetric Lévy-type processes with drifts in periodic media. Under a proper scaling, we show the scaled processes converge weakly to Lévy processes on ${\mathds R}^d$. In particular, we completely characterize the limiting processes when the coefficient function of the drift part is bounded continuous, and the decay rate of the jumping measure is comparable to $r^{-1-\alpha}$ for $r>1$ in the spherical coordinate with $\alpha \in (0,\infty)$. Different scaling limits appear depending on the values of $\alpha$.
This talk is based on joint work with Xin Chen, Zhen-Qing Chen and Jian Wang (Ann. Probab. 2021).
[ 参考URL ]Homogenization problem is one of the classical problems in analysis and probability which is very actively studied recently. In this talk, we consider homogenization problem for non-symmetric Lévy-type processes with drifts in periodic media. Under a proper scaling, we show the scaled processes converge weakly to Lévy processes on ${\mathds R}^d$. In particular, we completely characterize the limiting processes when the coefficient function of the drift part is bounded continuous, and the decay rate of the jumping measure is comparable to $r^{-1-\alpha}$ for $r>1$ in the spherical coordinate with $\alpha \in (0,\infty)$. Different scaling limits appear depending on the values of $\alpha$.
This talk is based on joint work with Xin Chen, Zhen-Qing Chen and Jian Wang (Ann. Probab. 2021).
https://forms.gle/ewZEy1jAXrAhWx1Q8
2022年06月28日(火)
16:00-17:30 数理科学研究科棟(駒場) 126号室
対面・オンラインハイブリッド開催(対面は本学関係者のみに限定します)
石田敦英 氏 (東京理科大学)
Mourre inequality for non-local Schödinger operators (Japanese)
https://forms.gle/sBSeNH9AYFNypNBk9
対面・オンラインハイブリッド開催(対面は本学関係者のみに限定します)
石田敦英 氏 (東京理科大学)
Mourre inequality for non-local Schödinger operators (Japanese)
[ 講演概要 ]
We consider the Mourre inequality for the following self-adjoint operator $H=\Psi(-\Delta/2)+V$ acting on $L^2(\mathbb{R}^d)$, where $\Psi: [0,\infty)\rightarrow\mathbb{R}$ is an increasing function, $\Delta$ is Laplacian and $V: \mathbb{R}^d\rightarrow\mathbb{R}$ is an interaction potential. Mourre inequality immediately yields the discreteness and finite multiplicity of the eigenvalues. Moreover, Mourre inequality has the application to the absence of the singular continuous spectrum by combining the limiting absorption principle and, in addition, Mourre inequality is also used for proof of the minimal velocity estimate that plays an important role in the scattering theory. In this talk, we report that Mourre inequality holds under the general $\Psi$ and $V$ by choosing the conjugate operator $A=(p\cdot x+x\cdot p)/2$ with $p=-\sqrt{-1}\nabla$, and that the discreteness and finite multiplicity of the eigenvalues hold. This talk is a joint work with J. Lőrinczi (Hungarian Academy of Sciences) and I. Sasaki (Shinshu University).
[ 参考URL ]We consider the Mourre inequality for the following self-adjoint operator $H=\Psi(-\Delta/2)+V$ acting on $L^2(\mathbb{R}^d)$, where $\Psi: [0,\infty)\rightarrow\mathbb{R}$ is an increasing function, $\Delta$ is Laplacian and $V: \mathbb{R}^d\rightarrow\mathbb{R}$ is an interaction potential. Mourre inequality immediately yields the discreteness and finite multiplicity of the eigenvalues. Moreover, Mourre inequality has the application to the absence of the singular continuous spectrum by combining the limiting absorption principle and, in addition, Mourre inequality is also used for proof of the minimal velocity estimate that plays an important role in the scattering theory. In this talk, we report that Mourre inequality holds under the general $\Psi$ and $V$ by choosing the conjugate operator $A=(p\cdot x+x\cdot p)/2$ with $p=-\sqrt{-1}\nabla$, and that the discreteness and finite multiplicity of the eigenvalues hold. This talk is a joint work with J. Lőrinczi (Hungarian Academy of Sciences) and I. Sasaki (Shinshu University).
https://forms.gle/sBSeNH9AYFNypNBk9
2022年05月31日(火)
16:00-17:30 数理科学研究科棟(駒場) 126号室
対面・オンラインハイブリッド開催(対面は本学関係者のみに限定します)
岡部真也 氏 (東北大学)
Convergence of Sobolev gradient trajectories to elastica (Japanese)
https://forms.gle/wkCbqdmNuz9zr3vA8
対面・オンラインハイブリッド開催(対面は本学関係者のみに限定します)
岡部真也 氏 (東北大学)
Convergence of Sobolev gradient trajectories to elastica (Japanese)
[ 講演概要 ]
In this talk we consider a higher order Sobolev gradient flow for the modified elastic energy defined on closed space curves. The $L^2$-gradient flow for the modified elastic energy has been well studied, and standard results are solvability of the flow for smooth initial curve and subconvergence of solutions to elastica. Moreover, stronger convergence results, so called full limit convergence, are generally up to reparametrisation and sometimes translation. In this talk, we consider $H^2$-gradient flow for the modified elastic energy and prove (i) the solvability of the flow for initial curve in the energy class, (ii) full limit convergence to elastica by way of a Lojasiewicz—Simon gradient inequality. This talk is based on a joint work with Philip Schrader (Murdoch University).
[ 参考URL ]In this talk we consider a higher order Sobolev gradient flow for the modified elastic energy defined on closed space curves. The $L^2$-gradient flow for the modified elastic energy has been well studied, and standard results are solvability of the flow for smooth initial curve and subconvergence of solutions to elastica. Moreover, stronger convergence results, so called full limit convergence, are generally up to reparametrisation and sometimes translation. In this talk, we consider $H^2$-gradient flow for the modified elastic energy and prove (i) the solvability of the flow for initial curve in the energy class, (ii) full limit convergence to elastica by way of a Lojasiewicz—Simon gradient inequality. This talk is based on a joint work with Philip Schrader (Murdoch University).
https://forms.gle/wkCbqdmNuz9zr3vA8
2022年05月24日(火)
16:00-17:30 オンライン開催
Michael Goesswein 氏 (東京大学/University of Regensburg)
Stability analysis for the surface diffusion flow on double bubbles using the Lojasiewicz-Simon (English)
https://forms.gle/Cam3mpSSEKKVppZr9
Michael Goesswein 氏 (東京大学/University of Regensburg)
Stability analysis for the surface diffusion flow on double bubbles using the Lojasiewicz-Simon (English)
[ 講演概要 ]
Many strategies for stability analysis use precise knowledge of the set of equilibria. For example, Escher, Mayer, and Simonett used center manifold analysis to study the surface diffusion flow on closed manifolds. Especially in higher dimensional situations with boundaries, this can cause problems as the set of equilibria will have a lot of degrees of freedom. In such situations approaches with a Lojasiewicz-Simon inequality gives an elegant way to avoid this problem. In this talk, we will both explain the general tools and ideas for this strategy and use them to prove the stability of standard double bubbles with respect to the surface diffusion flow. The talk is based on joint work with H. Garcke.
[ 参考URL ]Many strategies for stability analysis use precise knowledge of the set of equilibria. For example, Escher, Mayer, and Simonett used center manifold analysis to study the surface diffusion flow on closed manifolds. Especially in higher dimensional situations with boundaries, this can cause problems as the set of equilibria will have a lot of degrees of freedom. In such situations approaches with a Lojasiewicz-Simon inequality gives an elegant way to avoid this problem. In this talk, we will both explain the general tools and ideas for this strategy and use them to prove the stability of standard double bubbles with respect to the surface diffusion flow. The talk is based on joint work with H. Garcke.
https://forms.gle/Cam3mpSSEKKVppZr9
2022年04月26日(火)
16:00-17:30 数理科学研究科棟(駒場) 126号室
対面・オンラインハイブリッド開催
和久井洋司 氏 (東京理科大学)
Existence of a bounded forward self-similar solution to a minimal Keller-Segel model (Japanese)
https://forms.gle/mrXnjsgctSJJ1WSF6
対面・オンラインハイブリッド開催
和久井洋司 氏 (東京理科大学)
Existence of a bounded forward self-similar solution to a minimal Keller-Segel model (Japanese)
[ 講演概要 ]
In this talk, we consider existence of a bounded forward self-similar solution to the initial value problem of a minimal Keller-Segel model. It is well known that the mass conservation law plays an important role to classify its large time behavior of solutions to Keller-Segel models. On the other hand, we could not expect existence of self-similar solutions to our problem with the mass conservation law except for the two dimensional case due to the scaling invariance of our problem. We will show existence of a forward self-similar solution to our problem. The key idea to guarantee boundedness of its self-similar solution is to choose a concrete upper barrier function using the hypergeometric function.
[ 参考URL ]In this talk, we consider existence of a bounded forward self-similar solution to the initial value problem of a minimal Keller-Segel model. It is well known that the mass conservation law plays an important role to classify its large time behavior of solutions to Keller-Segel models. On the other hand, we could not expect existence of self-similar solutions to our problem with the mass conservation law except for the two dimensional case due to the scaling invariance of our problem. We will show existence of a forward self-similar solution to our problem. The key idea to guarantee boundedness of its self-similar solution is to choose a concrete upper barrier function using the hypergeometric function.
https://forms.gle/mrXnjsgctSJJ1WSF6
2022年04月12日(火)
16:00-17:30 オンライン開催
Amru Hussein 氏 (Technische Universität Kaiserslautern)
Maximal $L^p$-regularity and $H^{\infty}$-calculus for block operator matrices and applications (English)
https://forms.gle/QbQKex12dbQrt2Lw6
Amru Hussein 氏 (Technische Universität Kaiserslautern)
Maximal $L^p$-regularity and $H^{\infty}$-calculus for block operator matrices and applications (English)
[ 講演概要 ]
Many coupled evolution equations can be described via $2\times2$-block operator matrices of the form $\mathcal{A}=\begin{bmatrix}A & B \\ C & D \end{bmatrix}$ in a product space $X=X_1\times X_2$ with possibly unbounded entries. Here, the case of diagonally dominant block operator matrices is considered, that is, the case where the full operator $\mathcal{A}$ can be seen as a relatively bounded perturbation of its diagonal part though with possibly large relative bound. For such operators, the properties of sectoriality, $\mathcal{R}$-sectoriality and the boundedness of the $H^\infty$-calculus are studied, and for these properties perturbation results for possibly large but structured perturbations are derived. Thereby, the time-dependent parabolic problem associated with $\mathcal{A}$ can be analyzed in maximal $L^p_t$-regularity spaces, and this is applied to a wide range of problems such as different theories for liquid crystals, an artificial Stokes system, strongly damped wave and plate equations, and a Keller-Segel model.
This talk is based on a joint work with Antonio Agresti, see https://arxiv.org/abs/2108.01962
[ 参考URL ]Many coupled evolution equations can be described via $2\times2$-block operator matrices of the form $\mathcal{A}=\begin{bmatrix}A & B \\ C & D \end{bmatrix}$ in a product space $X=X_1\times X_2$ with possibly unbounded entries. Here, the case of diagonally dominant block operator matrices is considered, that is, the case where the full operator $\mathcal{A}$ can be seen as a relatively bounded perturbation of its diagonal part though with possibly large relative bound. For such operators, the properties of sectoriality, $\mathcal{R}$-sectoriality and the boundedness of the $H^\infty$-calculus are studied, and for these properties perturbation results for possibly large but structured perturbations are derived. Thereby, the time-dependent parabolic problem associated with $\mathcal{A}$ can be analyzed in maximal $L^p_t$-regularity spaces, and this is applied to a wide range of problems such as different theories for liquid crystals, an artificial Stokes system, strongly damped wave and plate equations, and a Keller-Segel model.
This talk is based on a joint work with Antonio Agresti, see https://arxiv.org/abs/2108.01962
https://forms.gle/QbQKex12dbQrt2Lw6
2021年11月16日(火)
16:00-17:30 オンライン開催
昨年度までと開始時間が異なるのでご注意ください
久保英夫 氏 (北海道大学)
低階項を伴う非線型波動方程式の初期値問題について (Japanese)
https://forms.gle/6ZCp8hQxKA3vq3DB9
昨年度までと開始時間が異なるのでご注意ください
久保英夫 氏 (北海道大学)
低階項を伴う非線型波動方程式の初期値問題について (Japanese)
[ 講演概要 ]
本講演では線型部分の主要部と同じオーダーを持ち空間変数に依存する低階項を伴う非線型波動方程式について考える。特に、低階項に特別な構造を課すと、低階項の効果を次元のシフトという目に見える形で示すことができることが知られている。実際、V. Georgiev氏、若狭恭平氏と行った先行研究では初期値の球対称性を仮定した上で、小振幅解の大域可解性と有限時間爆発を分ける非線型項の臨界指数が決定されていた。今回は重みつきL^2評価を利用することで、球対称性を仮定することなく、超臨界指数を持つ非線型波動方程式の小振幅解が存在し、それが漸近自由となることを報告したい。
[ 参考URL ]本講演では線型部分の主要部と同じオーダーを持ち空間変数に依存する低階項を伴う非線型波動方程式について考える。特に、低階項に特別な構造を課すと、低階項の効果を次元のシフトという目に見える形で示すことができることが知られている。実際、V. Georgiev氏、若狭恭平氏と行った先行研究では初期値の球対称性を仮定した上で、小振幅解の大域可解性と有限時間爆発を分ける非線型項の臨界指数が決定されていた。今回は重みつきL^2評価を利用することで、球対称性を仮定することなく、超臨界指数を持つ非線型波動方程式の小振幅解が存在し、それが漸近自由となることを報告したい。
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