Mimi Dai
University of Illinois Chicago, IL·Mathematics & Statistics
The equations that describe moving fluids, from water to air, were written down almost two hundred years ago. Physicists and engineers use them every day, yet mathematicians still cannot say whether smooth, well-behaved ('classical') solutions always exist and are unique.
- Fields
Mathematics & Statistics
- Topics
fluid equations
Navier-Stokes equations
weak solutions
- Grants
- 1 known grant, through May 2027
What they work on
- 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.
- Provide counterexamples to well-posedness (guaranteed unique, stable solutions) of fluid equations in supercritical spaces through explicit pathological constructions.
- Prove existence of physically realistic solutions that display Kolmogorov's energy cascade, the transfer of energy to smaller scales in turbulence, using convex integration and new efficient-mixing methods.
More about the research
To make progress, mathematicians often work with 'weak solutions', a looser kind of solution that is known to exist. This project probes what weak solutions can and cannot tell us about real fluids, including turbulence.
Why it matters
The work could clarify which mathematical solutions of fluid equations really describe physical flows. That understanding underpins how physicists and engineers model fluids. Proving that solutions with a Kolmogorov energy cascade exist would give a rigorous mathematical basis for a central feature of turbulence.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
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.
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.
Zachary Bradshaw
University of Arkansas, AR·Mathematics & Statistics
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.
The school
University of Illinois Chicago: 42 labs on PI Pages, $532M a year on research, 359 research doctorates a year.
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