9 problems

Consider the incompressible MHD system … in $\mathbb R^2$, with $\alpha=1$ and a nontrivial background magnetic field. Resistivity-independent dissipation conjecture. The energy of…

Let $\rho$ solve the advection--diffusion equation on the two-dimensional torus $\mathbb{T}^2=[-\pi,\pi]^2$, with divergence-free velocity field $\mathbf{u}$, diffusivity $\mu$, an…

Let $\beta$ denote the exponent governing the asymptotic stability threshold for a symmetric flow, under either the non-slip or Navier-slip boundary condition. Asymptotic stability…

Beck–Wayne conjecture. When $\nu$ is sufficiently small, the perturbation decays at the rate

Let a perturbation of the Kolmogorov flow evolve in the stable case, with viscosity coefficient $\nu>0$ sufficiently small. Beck–Wayne's enhanced dissipation conjecture. The pertur…

Let $\mathcal M=\mathbb T^d$ and consider the stochastic partial differential equation referred to as … decay exponentially, with an exponential rate arbitrarily close to $-\infty$…

Enhanced-dissipation rate conjecture. For stable monotone shear flows, the enhanced dissipation rate should be $\nu^{1/3}$, and for stable shear flows with a non-degenerate critica…

Let $\omega$ solve the full linearized Navier–Stokes equation around the Kolmogorov flow $V_{NS}=(-a_0e^{-\nu t}\cos y,0)$, namely … with … The vorticity satisfies the same enhance…

Let $p=p(n_0)\in(0,1)$ be the exponent in the decay rate from the preceding theorem, where $n_0$ denotes the order of the first nonvanishing derivative of the shear profile. The en…