32 problems

Asymptotic expansion conjecture. For a large class of boundary conditions $\mathbf{u}^{*}$, respectively source terms $\boldsymbol{f}$, there exists a solution of the body problem,…

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

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…

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 $\delta\in(0,1)$ 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 $u_0\in L^2$ 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…

Let $\alpha(\{\delta_j\}_{j=1}^n)$ be the exponent associated with the multi-level logarithmically improved criteria, and let $s$ denote the Sobolev regularity exponent. Potential…

Landau-solution classification conjecture. Then $u$ is one of the Landau solutions, which are minus-one homogeneous and axisymmetric without swirl.

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

Let $M\subset\mathbb{R}^2$ be any simply connected domain such that the constant-vorticity Euler solution $u_*$ on $M$ has a single eddy, meaning that its streamfunction has a sing…

Tran–Dritschel conjecture. The enstrophy dissipation satisfies

Let $a>1$ and $\sigma>0$, and let $\mathfrak{L}_1$ be the linear operator for the first nonzero Fourier mode after reducing the eigenvalue problem from $\mathfrak{L}_n$ to $n=1$. P…

Let $a>1$, and let $\mathfrak{L}_0$ be the linear operator arising from the $n=0$ mode of the linearized problem. Positivity conjecture for $\mathfrak{L}_0$. For all $a>1$, all non…

Supercritical nonuniqueness and non-Leray-Hopf conjecture. Then there exist two weak solutions $u,v\in X^{p,q}([0,T];\mathbb{T}^d)$ of the Navier–Stokes equations with

Let $u\in C\left(\left[0,T_{\max}\right);\dot{H}^1_{df}\right)$ be a smooth solution of the Navier–Stokes equation with finite maximal existence time $T_{\max}<+\infty$. Let…

Let $\Omega$ be the rectangular domain described above, and let $L^2_{\operatorname{div}}(\Omega)$ denote the space of square-integrable divergence-free vector fields tangent to…

K41–Onsager conjecture for Navier–Stokes. There exists a finite open interval $I$ and a sequence of suitable weak solutions $v_{\nu_j}$ with $\nu_j\to0$ that is uniformly bounded i…

Consider the steady two-dimensional Navier–Stokes equations with boundary conditions $\boldsymbol{u}^{*}$ and source term $\boldsymbol{f}$, and let $\boldsymbol{F}$ denote the net…

Navier–Stokes global regularity conjecture. For every $\nu>0$ and every such initial data $u_0$, such a global smooth finite-energy solution exists.

Let $(u_0,0,T)$ be a smooth periodic set of data. Global regularity for homogeneous periodic data with normalised pressure. There exists a smooth periodic solution $(u,p,u_0,0,T)$…

Let $(u,p,u_0,0,T,L)$ be a smooth periodic, homogeneous normalised-pressure solution, with $0<T<T_0<\infty$ and periodic data norm ${\mathcal H}^1(u_0,0,T,L)\leq A<\infty$. A prior…

Let $(u_0,f,T)$ be an $H^1$ set of data such that, for every multi-index $\alpha$ and every compact $K\subset\mathbf{R}^3$, … and … Global well-posedness from spatially smooth…

Let $(u,p,u_0,0,T)$ be a smooth homogeneous $H^1$ solution, with $\|u_0\|_{H^1_x(\mathbf{R}^3)}\leq A<\infty$. A priori global homogeneous $H^1$ bound. There exists a function…

Let $(u_0,f,T)$ be $H^1$ data and let an $H^1$ mild solution be a solution with $u\in L^\infty_tH^1_x\cap L^2_tH^2_x$ and the corresponding pressure and Duhamel formulation. Global…