Zachary Bradshaw
University of Arkansas, AR·Mathematics & Statistics
Weather forecasts, aircraft design and models of Earth's interior all rely on the Navier-Stokes equations, the mathematical rules for how viscous fluids such as air and water move. These equations are nonlinear, which means small effects can feed on themselves.
- Fields
Mathematics & Statistics
Physics
Engineering
- Topics
Navier-Stokes equations
Euler equations
partial differential equations
- Grants
- 2 known grants, through Aug 2029
What they work on
- Analyze the dynamic stability of multi-celled vortex solutions of the Navier-Stokes equations, such as tornado-like models with an updraft surrounding a central downdraft.
- Establish the existence and stability of singular stationary Navier-Stokes solutions near physically motivated boundaries, and determine how common viscous eddies are in corner flows.
- Find conditions under which the inviscid Euler equations set the leading-order structure of super-critically singular viscous flows, connecting the result to forecasting predictability.
- Derive estimates for how fast two Navier-Stokes solutions with identical initial data separate, using new approaches such as higher-order local time-regularity results.
- Adapt the separation-rate results to other models, including the surface quasi-geostrophic, semilinear heat and complex Ginzburg-Landau equations, to test how robust non-uniqueness is across structures.
More about the research
As a result, they can produce very complicated behavior that is hard to predict. Two projects here use rigorous mathematical analysis to ask how predictable fluid models really are. One asks whether simple steady flows, such as tornado models, capture the large-scale behavior of more complex flows. The other asks how fast two solutions can drift apart when they start from the same conditions.
Why it matters
Simulations of tornadoes, turbulent air over aircraft wings and climate rely on fluid equations that may be unstable or have more than one solution. Clarifying how fast predictions can diverge, and when simple steady flows describe complex ones, could help forecasters, meteorologists and engineers judge when models may miss extreme or catastrophic events.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Camillo De Lellis
Princeton University, NJ·Mathematics & Statistics
Prove the existence of Navier-Stokes solutions whose kinetic energy dissipation stays positive as viscosity goes to zero, without adding spurious oscillations to the initial data, giving rigorous examples of anomalous dissipation.
Theodore Drivas
Stony Brook University, NY·Mathematics & Statistics·Early-career
Analyze the long-time behavior of the two-dimensional Euler equations to explain how ideal fluids form lasting, hurricane-like vortices through vortex merging and mixing.
Mimi Dai
University of Illinois Chicago, IL·Mathematics & Statistics
Construct weak solutions of fluid equations with unphysical properties, such as loss of energy balance, non-uniqueness or singularity formation, using convex integration to show limits of the weak-solution notion.
The school
University of Arkansas: 12 labs on PI Pages, $231M a year on research, 179 research doctorates a year.
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