Local Existence For A Class Of Evolutionary Equations
Faculty Mentor
Shijun Zheng
Location
Russell Union Ballroom
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Mathematical Physics
Type of Research
On-going
Session Format
Poster Presentation
College
College of Science & Mathematics
Department
Department of Mathematical Science
Abstract
We investigate a class of nonlinear evolutionary equations including magnetic Klein-Gordon and nonlinear Schrödinger equations (NLS) with fractional regularity or damping. For magnetic NLS the dynamics of the systems are governed by a critical power nonlinearity together with first-order gradient interactions. The energy-critical exponent p = 1 + 4/(d − 2) is the natural invariance scaling of the Schrödinger flow in the energy space H1. Magnetic effects introduce analytical challenges due to the presence of unbounded electromagnetic potential. Using dispersive and smoothing estimates for the magnetic propagator, we aim to show the local and global existence, uniqueness, and stability corresponding to certain initial data in H1. Moreover, we plan to establish scaling-critical a priori bounds that may lead to scattering for small initial data in the defocusing/focusing case. The analysis captures the threshold behavior separating dispersive evolution from finite time singularity formation in critical and subcritical regimes, reflecting the interplay between nonlinear focusing effects and magnetic dispersion. This framework provides a rigorous foundation for understanding long-time behaviors of Schrödinger flows in an electromagnetic field. The results have connections with quantum physics as well as complex geometries.
Program Description
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Start Date
4-23-2026 10:00 AM
End Date
4-23-2026 12:00 PM
Recommended Citation
Zaman, Tasbeha and Zheng, Shijun, "Local Existence For A Class Of Evolutionary Equations" (2026). GS4 Student Scholars Symposium. 28.
https://digitalcommons.georgiasouthern.edu/research_symposium/2026/2026/28
Local Existence For A Class Of Evolutionary Equations
Russell Union Ballroom
We investigate a class of nonlinear evolutionary equations including magnetic Klein-Gordon and nonlinear Schrödinger equations (NLS) with fractional regularity or damping. For magnetic NLS the dynamics of the systems are governed by a critical power nonlinearity together with first-order gradient interactions. The energy-critical exponent p = 1 + 4/(d − 2) is the natural invariance scaling of the Schrödinger flow in the energy space H1. Magnetic effects introduce analytical challenges due to the presence of unbounded electromagnetic potential. Using dispersive and smoothing estimates for the magnetic propagator, we aim to show the local and global existence, uniqueness, and stability corresponding to certain initial data in H1. Moreover, we plan to establish scaling-critical a priori bounds that may lead to scattering for small initial data in the defocusing/focusing case. The analysis captures the threshold behavior separating dispersive evolution from finite time singularity formation in critical and subcritical regimes, reflecting the interplay between nonlinear focusing effects and magnetic dispersion. This framework provides a rigorous foundation for understanding long-time behaviors of Schrödinger flows in an electromagnetic field. The results have connections with quantum physics as well as complex geometries.
