119 problems

Koch–Nadirashvili–Seregin–Šverák conjecture. Every bounded mild ancient solution of the axi-symmetric Navier–Stokes equations is constant.

Let $v$ be a solution of the forced Navier–Stokes equations on $\mathbb{R}^3$, and suppose that $v$ first loses smoothness at time $T^*>0$. For a fixed constant $c>0$, consider the…

In the setting of fluid mechanics, consider solutions of the three-dimensional Navier–Stokes equation arising from initial data that are sufficiently large in an appropriate sense.…

Let $u_0\in L^2_\sigma$ be divergence-free initial data, let $u$ be an associated Leray solution of the three-dimensional Navier–Stokes equations, and consider the stochastic diffe…

Effective diffusivity conjecture. The sequence $u^N$ converges weakly to the stationary solution $u$ of

Finn's conjecture. If $a$ is sufficiently small, the starting problem admits a solution that tends to a stationary solution as $t\to\infty$.

Scale-invariant solutions are solutions of fluid equations whose spatial behavior is unchanged under the relevant scaling; the surrounding discussion concerns solutions with scale-…

Let $R$ be the Reynolds number for a periodically forced Navier–Stokes fluid, let $\mu_R$ be the probability distribution describing the time-average statistics of the irreversible…

Perelman's conjecture. Every smooth rotated backward self-similar profile with $\alpha\neq 0$ and the stated decay is trivial.

Consider the stochastic two-dimensional Navier–Stokes equations on the torus with additive noise applied only to a degenerate collection of Fourier modes. The viscosity is denoted…

One-component regularity conjecture. If

Let $r_n=2^{-n}$ and define the renormalized orbit … For a relative filter scale $\ell\in(0,1)$, let $F_{n,\ell}^{\rm abs}$ denote the absolute coarse flux activity, let…

Let $u$ solve the incompressible Navier–Stokes equations on $[0,1]\times\mathbb{R}^3$, … For any $\alpha\in[0,1/2)$, consider initial data $u_0\in H^\alpha\cap L^{3,\infty}$ and so…

Korolev–Šverák's endpoint conjecture. The same asymptotic expansion should hold with $\gamma=2$:

Let $u$ be a small steady solution of the three-dimensional Navier–Stokes equation in an exterior domain, and suppose its leading asymptotic term at infinity is a Landau solution.…

Let $\mathcal{L}$ be the operator governing the steady state in the self-similar variables, and let $\bar{\Omega}$ denote its kernel profile. Let $Ai$ be the Airy function, let…

Let $V_N$ be the orbit-level interaction matrix, let $\rho(V_N)$ denote its spectral radius, and let $\lvert V_N\rvert$ be the entrywise absolute-value matrix. Assume the proven sc…

Let $\Lambda_N^{\mathrm{sph}}$ denote the spherical truncation and let $S^{\mathrm{sph}}(N)$ be its associated incidence quantity. Spherical truncation incidence-growth conjecture.…

Let ${\bf u}_1({\bf r},t)$ and ${\bf u}_2({\bf r},t)$ denote solutions of the Navier–Stokes equations with almost identical initial conditions satisfying … for arbitrarily small…

Let $\psi_1(x,y,t)$ and $\psi_2(x,y,t)$ be solutions of the two-dimensional Navier–Stokes equation for turbulent Kolmogorov flow with an even forcing wavenumber $n_K$, subject to t…

Let $\delta\in(0,1)$ and $p>1$. On the two-dimensional torus $\mathbb{T}^2$, let $u_0$ be a divergence-free vector field in $L^2(\mathbb{T}^2)$ with vorticity…

Let $U_1({\bf r})$ be an initial condition with a spatial symmetry, and let … be another initial condition with a different spatial symmetry or without spatial symmetry, where…

Let $u_0\in L^2$ be initial data, and consider sequences of Leray–Hopf weak solutions of the Navier–Stokes equations whose inviscid limits are denoted by $u$. The critical Besov cl…

Fluctuation conjecture. Singular solutions to 3D incompressible Navier–Stokes equations generically are largely fluctuated.

Small-viscosity approximation conjecture. For sufficiently small $u$, the dynamical behavior of the solutions to the viscous equation should sufficiently approximate that of the c…