Projects per year
Personal profile
Biography
Research interests
I am interested in the interplay between harmonic analysis and partial differential equations, as well as their interactions with other fields.
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Harmonic analysis
to study the boundedness in various function spaces of many linear/multilinear operators related to the Fourier restriction problems and PDE, space-time estimates for dispersive equations, adaptive function spaces and analysis tools associated to operators and PDE
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Nonlinear partial differential equation
to study the nonlinear evolutionary PDE arising in mathematical physics such as nonlinear dispersive equations, Navier-Stokes equation. Main concerns are the low-regularity local/global well-posedness theory, long-time/blow-up behavior of the solution, stability of the equation, playing tools from many other areas including harmonic analysis, functional analysis, ODE and probability, etc.
Supervision interests
The topics of the PhD project relate to the interplay between harmonic analysis and nonlinear dispersive equations. The study of dispersive PDE has been greatly transformed by the integration of harmonic analysis thinking and technology. On the other hand, dispersive PDE are a powerful source of harmonic analysis questions with real physical significance.
1) to develop the most recent linear and non-linear harmonic analysis method
2) to adapt the newly developed methods to problems in nonlinear dispersive equations, e.g., well-posedness, asymptotic behaviour.
Candidates are expected to have finished, or be about to finish, a Master's degree in mathematics or an equivalent qualification (candidates with degrees on different disciplines must show evidence that they completed a sufficient amount of mathematical training). A good command of measure theory, functional analysis, Fourier analysis, and PDEs is required.
Please feel free to contact me and introduce yourself with 1) CV; 2) Transcripts; 3) English results (optional); 4) Other documents (optional), e.g., publications, certificate.
Research area keywords
- Dispersive and wave equations
- Navier-Stokes and Euler equations
- Low regularity
- Well-posedness
- Asymptotic behaviour
- Harmonic analysis
- Fourier restriction
- Strichartz estimate
- Scattering
- Blow-up, formation of singularity
- Partial Differential Equations
Collaborations and top research areas from the last five years
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Partial differential equation: Schrodinger operator and long-time dynamics
Australian Research Council (ARC)
30/06/24 → 29/06/28
Project: Research
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Nonlinear harmonic analysis and dispersive partial differential equations
Sikora, A. (Primary Chief Investigator (PCI)), Guo, Z. (Chief Investigator (CI)), Hauer, D. (Chief Investigator (CI)) & Tacy, M. (Partner Investigator (PI))
8/04/20 → 31/12/22
Project: Research
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Harmonic analysis and dispersive partial differential equations
Guo, Z. (Primary Chief Investigator (PCI)), Li, J. (Chief Investigator (CI)), Kenig, C. (Partner Investigator (PI)) & Nakanishi, K. (Partner Investigator (PI))
Australian Research Council (ARC), Monash University, Macquarie University, University of Chicago, Osaka University
1/01/17 → 1/11/20
Project: Research
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Uniform sparse domination and quantitative weighted boundedness for singular integrals and application to the dissipative quasi-geostrophic equation
Chen, Y. & Guo, Z., 5 Jan 2024, In: Journal of Differential Equations. 378, p. 871-917 47 p.Research output: Contribution to journal › Article › Research › peer-review
2 Citations (Scopus) -
A uniform Besov boundedness and the well-posedness of the generalized dissipative quasi-geostrophic equation in the critical Besov space
Chen, Y., Guo, Z. & Tian, T., 27 Jun 2023, In: Forum Mathematicum. 36, 2, p. 403-416 14 p.Research output: Contribution to journal › Article › Research › peer-review
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Generalized singular integral with rough kernel and approximation of surface quasi-geostrophic equation
Chen, Y. & Guo, Z., 15 Feb 2023, In: Journal of Differential Equations. 346, p. 205-228 24 p.Research output: Contribution to journal › Article › Research › peer-review
7 Citations (Scopus) -
Pointwise decay of solutions to the energy critical nonlinear Schrödinger equations
Guo, Z., Huang, C. & Song, L., 5 Sept 2023, In: Journal of Differential Equations. 366, p. 71-84 14 p.Research output: Contribution to journal › Article › Research › peer-review
3 Citations (Scopus) -
Scattering for the Mass-Critical Nonlinear Klein–Gordon Equations in Three and Higher Dimensions
Cheng, X., Guo, Z. & Masaki, S., Oct 2023, In: Vietnam Journal of Mathematics. 51, 4, p. 869-909 41 p.Research output: Contribution to journal › Article › Research › peer-review
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