Mathematics (Analysis of PDEs)
Finite time blow-up for the hypodissipative Navier Stokes equations with a force in $L^1_t C_x^{1,ε}\cap L^{\infty}_{t}L_{x}^2$
Mathematicians Diego Córdoba, Luis Martínez-Zoroa, and Fan Zheng construct explicit forced solutions to the hypodissipative Navier-Stokes equations that blow up in finite time despite remaining smooth beforehand.
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.
- 01
Establishes finite-time blow-up of classical, finite-energy solutions to the forced hypodissipative (fractional) Navier-Stokes equations for dissipation exponents α in [0, α0) with α0=(22-8√7)/9
- 02
Constructs explicit solutions on R^3×[0,T] with external forcing lying in L^1_t C_x^{1,ε} ∩ L^∞_t L^2_x
- 03
Shows the velocity field remains smooth (C^∞∩L^2) for all times before the blow-up time T
- 04
Proves that the time-integral of the gradient of the velocity, ∫_0^t |∇u| ds, diverges as t approaches the blow-up time T
- 05
Extends prior blow-up results for hypodissipative Navier-Stokes equations to a broader range of dissipation strengths and forcing regularity classes
In this work we establish the formation of singularities of classical solutions with finite energy of the forced fractional Navier Stokes equations where the dissipative term is given by $|\nabla|^α$ for any $α\in [0, α_0)$ ($α_0 = \frac{22-8\sqrt7}{9} > 0$). We construct solutions in $\mathbb{R}^3\times [0,T]$ with a finite $T>0$ and with an external forcing which is in $L^1_t([0, T]) C_x^{1,ε}\cap L^{\infty}_{t}L_{x}^2$, such that on the time interval $0 \le t < T$, the velocity $u$ is in the space $C^\infty\cap L^2$ and such that as the time $t$ approaches the blow-up moment $T$, the integral $\int_0^t |\nabla u| ds$ tends to infinity.