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Matsue Kaname Last modified date:2023.10.12

Graduate School

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Academic Degree
Dr. of Science
Country of degree conferring institution (Overseas)
Field of Specialization
Dynamical Systems, Numerical Analysis, Rigorous Numerics, Singular Perturbation Theory, Differential Equations (Blow-up Solutions, Shock Waves), Singularities, Quantum Walks, Topology Optimizations, Combustion
Total Priod of education and research career in the foreign country
Research Interests
  • Nature of premixed laminar, turbulent flames
    keyword : Premixed planar and spherical propagating flames, Darrieus-Landau instability, Michelson-Sivashinsky-type equation, hydrodynamic model, turbulent combustion, gravity effect
  • Various problems inspired by materials science
    keyword : Amorphous structured materials, crystals, minimal surface, discrete geometry, topology optimization, computational homology, structure preserving numerical method
  • Quantum walks : geometric aspects of dynamics
    keyword : quantum walks, simplicial complex, quantum search, spectral analysis
  • Finite-time singularities (blow-up solutions, extinction, quenching, canard) from the viewpoint of singularity theory with rigorous numerics, Shock waves and rigorous numerics
    keyword : blow-up, extinction, quenching, canard, shock waves, rigorous numerics, covering relations, Lyapunov functions, cone conditions
  • Geometric singular perturbation theory and rigorous numerics
    keyword : Fast-slow systems, rigorous numerics, geometric singular perturbation, isolating blocks and cones, covering relations
Academic Activities
1. Akihito Hirata, Kaname Matsue and MingWei Chen, Structural analysis of metallic glasses with computational homology, Springer, 2016.06.
1. Jean-Philippe Lessard, Kaname Matsue, Akitoshi Takayasu, Saddle-Type Blow-Up Solutions with Computer-Assisted Proofs: Validation and Extraction of Global Nature, Journal of Nonlinear Science,, 33, 2023.03, [URL], In this paper, blow-up solutions of autonomous ordinary differential equations (ODEs) which are unstable under perturbations of initial points, referred to as saddletype blow-up solutions, are studied.
Combining dynamical systems machinery (e.g., compactifications, timescale desingularizations of vector fields) with tools from
computer-assisted proofs (e.g., rigorous integrators, the parameterization method for invariant manifolds), these blow-up solutions are obtained as trajectories on local stable manifolds of hyperbolic saddle equilibria at infinity.
With the help of computer-assisted proofs, global trajectories on stable manifolds, inducing blow-up solutions,
provide a global picture organized by global-in-time solutions and blow-up solutions simultaneously.
Using the proposed methodology, intrinsic features of saddle-type blow-ups are observed: locally smooth dependence of blow-up times on initial points, level set distribution of blow-up times and decomposition of the phase space playing
a role as separatrixes among solutions, where the magnitude of initial points near those blow-ups does not matter for asymptotic behavior.
Finally, singular behavior of blow-up times on initial points belonging to different family of blow-up solutions is addressed..
2. Kaname Matsue, Moshe Matalon, Dynamics of hydrodynamically unstable premixed flames in a gravitational field - Local and global bifurcation structures, Combustion Theory and Modelling,, 27, 3, 346-374, 2023.01, [URL], The dynamics of hydrodynamically unstable premixed flames are studied using the nonlinear Michelson–Sivashinsky (MS) equation, modified appropriately to incorporate effects due to gravity.
The problem depends on two parameters: the Markstein number that characterises the combustible mixture and its diffusion properties, and the gravitational parameter that represents the ratio of buoyancy to inertial forces.
A comprehensive portrait of all possible equilibrium solutions are obtained for a wide range of parameters, using a continuation methodology adopted from bifurcation theory.
The results heighten the distinction between upward and downward propagation.
In the absence of gravity, the nonlinear development always leads to stationary solutions, namely, cellular flames propagating at a constant speed without change in shape.
When decreasing the Markstein number, a modest growth in amplitude is observed with the propagation speed reaching an upper bound. For upward propagation, the equilibrium states are also stationary solutions, but their spatial structure depends on the
initial conditions leading to their development.
The combined Darrieus–Landau and Rayleigh–Taylor instabilities create profiles of invariably larger amplitudes and sharper
crests that propagate at an increasingly faster speed when reducing the Markstein number.
For downward propagation, the equilibrium states consist in addition to stationary structures time-periodic solutions, namely, pulsating flames propagating at a constant average speed.
The stabilising influence of gravity dampens the nonlinear growth and leads to spatiotemporal changes in flame morphology, such as the formation of multicrest stationary profiles or pulsating cell splitting and merging patterns, and an overall reduction in propagation speed.
The transition between these states occurs at bifurcation and exchange of stability points, which becomes more prominent when reducing the Markstein number and/or increasing the influence of gravity.
In addition to the local bifurcation characterisation the global bifurcation structure of the equation, obtained
by tracing the continuation of the bifurcation points themselves unravels qualitative features such as the manifestation of bi-stability and hysteresis, and/or the onset and sustenance of time-periodic solutions.
Overall, the results exhibit the rich and complex dynamics that occur when gravity, however small, becomes physically meaningful..
3. Kaname Matsue, Akitoshi Takayasu, Numerical validation of blow-up solutions with quasi-homogeneous compactifications, Numerische Mathematik, 10.1007/s00211-020-01125-z, 50 pages, 2020.06, [URL].
4. Kaname Matsue, Akitoshi Takayasu, Rigorous numerics of blow-up solutions for ODEs with exponential nonlinearity, Journal of Computational and Applied Mathematics, 2020.02, Our concerns here are blow-up solutions for ODEs with exponential nonlinearity from the viewpoint of dynamical systems and their numerical validations.
As an example, the finite difference discretization of $u_t = u_{xx} + e^{u^m}$ with the homogeneous Dirichlet boundary condition is considered.
Our idea is based on compactification of phase spaces and time-scale desingularization as in previous works.
In the present case, treatment of exponential nonlinearity is the main issue.
Fortunately, under a kind of exponential homogeneity of vector field, we can treat the problem in the same way as polynomial vector fields.
In particular, we can characterize and validate blow-up solutions with their blow-up times for differential equations with such exponential nonlinearity in the similar way to previous works.
A series of technical treatments of exponential nonlinearity in blow-up problems is also shown with concrete validation examples..
5. Kaname Matsue, Geometric treatments and a common mechanism in finite-time singularities for autonomous ODEs, Journal of Differential Equations, 10.1016/j.jde.2019.07.022, 267, 12, 7313-7368, 2019.12, Geometric treatments of blow-up solutions for autonomous ordinary differential equations and their blow-up rates are concerned. Our approach focuses on the type of invariant sets at infinity via compactifications of phase spaces, and dynamics on their center-stable manifolds. In particular, we show that dynamics on center-stable manifolds of invariant sets at infinity with appropriate time-scale desingularizations as well as blowing-up of singularities characterize dynamics of blow-up solutions as well as their rigorous blow-up rates. Similarities for characterizing finite-time extinction and asymptotic behavior of compacton traveling waves to blow-up solutions are also shown..
6. Kaname Matsue, Shikhar Mohan, Moshe Matalon, Effect of gravity on hydrodynamically unstable flames, The 12th Asia-Pacific Conference on Combustion, 2019.07, The hydrodynamic instability, due to the large deviation of density between fresh cold mixture and hot combusted products, was discovered by Darrieus and Landau. After seven or eight decades, many aspects of this intrinsic flame instability have been revealed, such as the effects of the flame front curvature and of flow strain rate, its influence on turbulent flames and the self-wrinkling and self-turbulization of expanding flames. In the present study we focus on the composite effects of thermal expansion, differential diffusion, and gravity on flame dynamics, based on a fully nonlinear, hydrodynamic model obtained by a multi-scale analysis that exploits the distinct length scales associated with such problems. The simulations verify the stabilization effect of gravity on planar flames propagating downwards, known from linear stability theory, and show that in the presence of gravity the nonlinear development beyond the stability threshold leads to cusp-like structures of smaller amplitude that propagate at a reduced speed. Finally, we observe that a judicious choice of the Markstein number, controlled by mixture composition and domain size, and of the Froude number creates richer morphological flame structures than in the absence of gravity..
7. Kaname Matsue, On blow-up solutions of differential equations with Poincare-type compactifications, SIAM Journal on Applied Dynamical Systems, doi:10.1137/17M1124498, 17, 2249-2288, 2018.08, [URL].
8. Kaname Matsue, Rigorous numerics for fast-slow systems with one-dimensional slow variable: topological shadowing approach, Topological Methods in Nonlinear Analysis, doi=, 50, 2, 357-468, 2017.12.
9. Kaname Matsue, Osamu Ogurisu and Etsuo Segawa, Quantum Search on Simplicial Complexes, Quantum Studies: Mathematics and Foundations,, 1-27, 2017.10.
10. Kaname Matsue, Osamu Ogurisu and Etsuo Segawa, A note on the spectral mapping theorem of quantum walk models, Interdisciplinary Information Sciences, 23, 105-114, 2017.03.
11. Kaname Matsue, Osamu Ogurisu and Etsuo Segawa, Quantum walks on simplicial complexes, Quantum Information Processing, 15, 5, 1865-1896, 2016.02.
12. Kaname Matsue, Hisashi Naito, Numerical studies of the optimization of the first eigenvalue of the heat diffusion in inhomogeneous media, Japan Journal of Industrial and Applied Mathematics, 32, 2, 489-512, 2015.10.
13. Akihiko Hirata, L.J. Kang, Takeshi Fujita, B. Klumov, Kaname Matsue, Motoko Kotani, A.R. Yavari and Mingwei Chen, Geometric frustration of icosahedron in metallic glasses, Science, 341, 6144, 376-379, 2013.07.
1. Kaname Matsue, Dynamics of hydrodynamically unstable premixed flames in a gravitational field, ANZIAM2023 (Australian and New Zealand Industrial and Applied Mathematics), 2023.02.
2. Kaname Matsue, Interaction between flame and physical phenomena - Asymptotic studies, results and reviews -, I2CNER-IMI Joint International Workshop - Engineering and Mathematics: Where Do We Meet?-, 2023.01.
3. Kaname Matsue, On Recent progress in Blow-Up Solutions from the viewpoint of Dynamical Systems -Theory and Rigorous Numerics-, SIAM Conference on Nonlinear Waves and Coherent Structures (NWCS22), 2022.09.
4. Kaname Matsue, Blow-up Solutions for Ordinary Differential Equations from a viewpoint of dynamical systems, La Trobe-Kyushu Joint Seminar on Mathematics for Industry, 2022.07.
5. Kaname Matsue, Recent progress in blow-up characterization of ODEs -theory and rigorous numerics-, International Workshop on Reliable Computing and Computer-Assisted Proofs (ReCAP2022), 2022.03, [URL].
6. Kaname Matsue, Jean-Philippe Lessard, Akitoshi Takayasu, Rigorous numerics of blow-up separatrix in autonomous ODEs, SCAN2020, 2021.09, This talk is concerned with special blow-up solutions for ordinary differential equations (ODEs) separating sets of initial points into two regions, one of which admits solutions global in time, and another admits blow-up solutions..
7. Kaname Matsue, Rigorous numerics of blow-up solutions for autonomous ODEs, CRM CAMP in Nonlinear Analysis, 2021.05, [URL].
8. , [URL].
9. Kaname Matsue, On numerical and Mathematical description of premixed flame dynamics, I2CNER Hydrogen Materials Compatibility Division Winter Retreat, 2020.02.
10. Kaname Matsue, On numerical and Mathematical description of premixed flame dynamics, I2CNER-IMI International Joint Workshop "Applied Math for Energy: Future Directions", 2020.01.
11. Kaname Matsue, On numerical and Mathematical description of premixed flame dynamics, I2CNER Institute Interest Seminar Series, 2020.01.
12. Kaname Matsue, Premixed Flame Dynamics: Modeling, Numerical and Mathematical Studies , Mathematical Science Workshop in Yamaguchi 2019 Presented by RITS, 2019.11.
13. , [URL].
14. Kaname Matsue, Shikhar Mohan, Moshe Matalon, Effect of gravity on hydrodynamically unstable flames , The 12th Asia-Pacific Conference on Combustion, 2019.07, [URL].
15. Kaname Matsue, Blow-up solutions for ODEs from the viewpoint of dynamical systems : theory and applications, HADES (Harmonic Analysis and Differential Equations Seminar), 2018.11.
16. Kaname Matsue, Rigorous numerics and asymptotic analysis of finite-time singularities : qualitative and quantitative natures , SCAN2018, 2018.09.
17. Walls and Doors : Dynamical Systems and Rigorous Numerics.
18. Kaname Matsue, Rigorous numerics of finite-time singularity for ODEs , EASIAM2018, 2018.06, [URL].
19. Kaname Matsue, Finite-time singularity for ODEs from the viewpoint of dynamical systems, EASIAM2018, 2018.06, [URL].
20. Kaname Matsue, Mathematical treatment of flame dynamics toward foundation of combustion, FMfI2017 (Forum of Mathematics for Industry), 2017.10.
21. Kaname Matsue, Rigorous numerics of blow-up solutions for autonomous ODEs, A3 workshop on Fluid Dynamics and Materials Science in CSIAM 2017, 2017.10.
22. , [URL].
23. Kaname Matsue, Mathematics in Combustion -Fundamentals and Trends- , I2CNER Hydrogen Materials Compatibility Research Division Summer Retreat 2017, 2017.06.
24. Kaname Matsue, Technology for New Energy Generation - Mathematics of Combustion - , I2CNER Site Visit, 2017.06.
Membership in Academic Society
  • Combustion Society of Japan
  • The Japan Society for Industrial and Applied Mathematics
  • Mathematical Society of Japan
Educational Activities
Exercise for freshmen and sophomores, and relay-style lectures for graduate students (topic : mathematics in combustion).
In FY2019, Matsue has given a teaching class "Complex Functions" for the 2nd grade undergraduate students (Engineering).
From FY2020, he also gives a class "Numerical analysis: lecture and exercises" for graduate students, as well as "Complex Functions" for undergraduates (Engineering).
From FY2021, he also gives a class in "Graduate Program for Mathematics for Innovation", as well as "Ordinary Differential Equations" for undergraduates (Engineering).
In FY2022, Seminars for graduate students (MMA course), Core-Seminar for the 1st grade undergraduate students (Mathematics), "Graduate Program for Mathematics for Innovation" for graduate students and "Complex Functions" for undergraduates (Engineering).
In FY2023, Seminars for graduate students (MMA course), Core-Seminar for the 1st grade undergraduate students (Mathematics), "Graduate Program for Mathematics for Innovation" for graduate students and "Complex Functions" for undergraduates (Engineering). "Numerical analysis: lecture and exercises" is also given in the 2nd semester.