Simon Bortz
University of Alabama, AL·Mathematics & Statistics
Heat flow, diffusion and transport processes are described by parabolic partial differential equations, equations that track how a quantity spreads out over time. In real settings these processes happen in regions with complicated, rough edges, and those edges may even move over time.
- Fields
Mathematics & Statistics
- Topics
parabolic partial differential equations
parabolic PDE
heat equation
- Grants
- 1 known grant, through Aug 2029
What they work on
- Characterize parabolic uniform rectifiability of time-varying boundaries through properties of caloric measure, the distribution of heat flowing in from the boundary, and through estimates for bounded solutions of the heat equation.
- Determine whether boundedness of certain parabolic singular integral operators can serve as a sharp test for parabolic uniform rectifiability of rough boundaries.
- Identify when caloric measure and surface measure on a rough time-dependent boundary are mutually absolutely continuous, meaning they agree on which boundary sets are negligible.
- Establish Green function estimates and quantitative absolute continuity of caloric measure for variable-coefficient parabolic operators in rough domains, and use them to solve Dirichlet and regularity problems under scale-invariant coefficient assumptions.
More about the research
This project asks a basic mathematical question: how does the shape and roughness of a region's boundary affect how solutions behave inside the region?
Why it matters
This is basic mathematics. It adds a clearer understanding of how rough, time-varying boundaries control the behavior of heat flow, diffusion and transport equations, and the tools it develops could eventually inform the analysis of these processes in complex or heterogeneous environments.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
William Minicozzi
Massachusetts Institute of Technology, MA·Mathematics & Statistics
Establish gradient estimates and monotonicity formulas for harmonic functions and heat equation solutions, clarifying how they depend on dimension and how elliptic and parabolic behavior relate.
Pablo Raúl Stinga
Iowa State University, IA·Mathematics & Statistics
Establish the analytical theory of surfaces of minimum mean curvature variation, smoothly bending surfaces used in data interpolation, image repair and hull design, including well-posedness and regularity of their equations.
Jiuyi Zhu
Louisiana State University, LA·Mathematics & Statistics
Estimate the size of zero-level (nodal) sets of Laplacian eigenfunctions for second-order and higher-order elliptic equations, and determine how that size depends on high frequency.
The school
University of Alabama: 22 labs on PI Pages, $221M a year on research, 235 research doctorates a year.
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