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Fourier Analysis and Partial Differential Equations Lab
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This lab uses tools of Fourier analysis to study the size and smoothness of solutions to partial differential equations.
AboutAbout the lab
Deriving properties of solutions from the structure of equations
Prof. Sugimoto studies partial differential equations and Fourier analysis.
Expressing various natural phenomena as partial differential equations makes it possible to treat them mathematically.
Through the analysis of these equations, the lab aims to extract new principles that can explain concrete phenomena in a unified way.
In particular, the lab aims to apply methods that derive properties of solutions from the structure of the equations to the analysis of quantitative properties, such as the size and smoothness of solutions.
Key terms
- partial differential equation
- An equation describing how a quantity determined by several variables, such as time and position, changes
- Fourier analysis
- A method of studying a function by decomposing it into a superposition of waves of various frequencies
- Fourier integral operator
- A tool based on Fourier analysis, used to represent solutions of partial differential equations and to transform equations into forms that are easier to handle
ResearchResearch themes
The lab analyzes quantitative properties of solutions to partial differential equations, such as their size and smoothness, using Fourier integral operators. To develop the necessary tools, it also pursues research on Fourier analysis at the same time.
*Illustrative imageSmoothing Effects of Dispersive Equations
The lab studies the phenomenon in initial value problems for equations such as the Schrödinger equation, where averaging the solution over time makes it smoother than the initial data. In recent years, it has achieved results by applying to this problem a method that first reduces estimates for solutions to a normal form and then derives them.
*Illustrative imageFourier Integral Operators and Function Spaces
The information carried by the solutions of an equation is contained in Fourier integral operators as algebraic and geometric structure. To extract the size and smoothness of solutions from it, the lab develops the theory of Fourier integral operators and the theory of function spaces.
MembersFaculty
- ProfessorMitsuru Sugimoto
VoicesStudent voices
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Report an errorAbout the images
The top and research theme images are illustrative and were not taken at the lab. They are cropped from Wikimedia Commons images. Cropped CC BY-SA images are provided under the same CC BY-SA license as the original (version shown on each line).
- Top image: “Ashbridge's Bay ripples, July 2019” Jason Zhang / CC BY-SA 3.0 (cropped)
- Smoothing Effects of Dispersive Equations: “Black alder twig and water ripples at Wolfsberger Mühle” Radomianin / CC BY-SA 4.0 (cropped)
- Fourier Integral Operators and Function Spaces: “Water Interference” Daniela_deGol / CC BY 4.0 (cropped)
- Related lab (Fourier Analysis and Partial Differential Equations Lab): “Ocean waves water” Jon Sullivan / Public domain (cropped)
- Related lab (Fourier Analysis and Partial Differential Equations Lab): “Unusual wave at the Gap” Avenue / CC BY-SA 3.0 (cropped)
- Related lab (Fourier Analysis, Partial Differential Equations, and Fluid Dynamics Lab): “Flow separation” Jaganath at English Wikipedia / CC BY-SA 3.0 (cropped)
- Related lab (Nakanishi Lab): “Root system A2 with labels” Mark L MacDonald; derivative of work of User:Smithers888 and User:Maksim / CC BY-SA 3.0 (cropped)
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