Topic
partial differential equations
The research papers the show has read on partial differential equations.
- Monthly Notices of the Royal Astronomical Society
Finite time blow-up for the hypodissipative Navier Stokes equations with a force in $L^1_t C_x^{1,ε}\cap L^{\infty}_{t}L_{x}^2$
Imagine describing how water or air flows using math equations. The real Navier-Stokes equations (used for actual fluids like water) have a 'friction' term that's known to smooth things out, but mathematicians don't know if solutions can ever go haywire (blow up) in finite time. This paper studies a modified, weaker-friction version of these equations and an added external push (forcing). The researchers mathematically prove that even though the fluid starts out perfectly smooth and calm, if you add a very specific push (like a carefully choreographed sequence of nudges), the fluid's motion can become infinitely 'jagged' or chaotic at some specific future moment, even though before that moment everything looks fine and smooth. It's like proving that a perfectly smooth road can be engineered to suddenly become infinitely bumpy at exactly mile marker 10, if you control the terrain (the forcing) carefully enough.
- Annals of Mathematics
Finite-time singularity formation for C^{1,alpha} solutions to the incompressible Euler equations on R^3
Imagine stirring water in a very idealized, frictionless way (no viscosity) and asking a computer to predict the flow forever using perfect math rules. Since the 1920s, mathematicians knew that if you start with a reasonably smooth swirl of fluid, the equations will behave nicely, at least for some period of time. The big open question was: can the fluid always be predicted this way forever, or can the flow become 'infinitely twisted' in some finite amount of time, effectively breaking the equations? This paper proves that yes, for a genuinely simple 3D swirling flow, the fluid's velocity field can become infinitely 'sharp' (in a mathematical sense related to gradients) in a finite amount of time, even though it started out perfectly smooth. It's like proving a perfectly smooth ripple can, in finite time, spontaneously form a jagged crease with no external push.