Hung Tran
University of Wisconsin-Madison, WI·Mathematics & Statistics
How fast does a crystal grow, and what shape does it take? Questions like this, along with how flames spread through turbulent gas and how particles clump together and break apart, are governed by nonlinear partial differential equations (PDE), equations that relate how a quantity changes in space and time.
- Fields
Mathematics & Statistics
- Topics
nonlinear partial differential equations
viscosity solutions
Hamilton-Jacobi equations
- Grants
- 1 known grant, through Aug 2027
What they work on
- Establish regularity and large-time behavior results for Hamilton-Jacobi equations, a class of nonlinear PDE from optimal control, and use them to prove that mass-conserving solutions of critical coagulation-fragmentation equations exist.
- Determine how solutions of critical coagulation-fragmentation equations, which model particles merging and splitting, behave over long times, using their connection to Hamilton-Jacobi equations and viscosity solution methods.
- Analyze the regularity and large-time average of level-set mean curvature flow equations with driving and source terms, which describe moving surfaces such as growing crystal faces or flame fronts.
- Characterize the growth speed and shape of a crystal that grows sideways through adatoms and upward through dislocations or nucleation in a supersaturated medium, using level-set mean curvature flow models.
More about the research
These equations often have no smooth, classical solutions. Mathematicians handle this with viscosity solutions, a generalized notion of solution that still makes sense when solutions have kinks or corners. This project looks for new general principles and methods for understanding how such solutions behave. The equations involved connect to the calculus of variations, differential games, dynamical systems, geometry, homogenization theory and probability.
Why it matters
Practical use of crystal growth models depends on knowing how fast a crystal grows and what shape it takes. Rigorous results on these equations could put such predictions on firmer footing, and could also support models of turbulent combustion and particle coagulation used in chemistry, physics and engineering. Beyond these applications, the work adds general mathematical tools for nonlinear PDE and viscosity solutions.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
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.
Jonathan Zhu
University of Washington, WA·Mathematics & Statistics·Early-career
Analyze singularities of mean curvature flow near cylindrical and conical models, testing uniqueness of blow-up limits when curvature is present in the interior or at the boundary.
Haomin Zhou
Georgia Institute of Technology, GA·Mathematics & Statistics
Establish parameterized Wasserstein sub-manifolds, which represent evolving probability distributions through a neural network's parameters via a push-forward map and pullback metric, and develop their mathematical foundations.
The school
University of Wisconsin-Madison: 91 labs on PI Pages, $1.9B a year on research, 854 research doctorates a year.
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