38 problems

For the incompressible Euler equations on boundary-free three-dimensional space, $\partial_t u+(u\cdot\nabla)u=-\nabla p$, $\nabla\cdot u=0$ on $\mathbb{R}^3\times[0,\infty)$, with…

Eyink's conjecture. Both limits

Let $I$ be an open interval, and let $(v,p)$ be a weak solution of the incompressible Euler equations on $I \times \mathbb{T}^3$. Assume that $v \in L_t^\infty C_x^{1/3}$ and that…

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 $\delta>0$, and let $rhob_{\delta}$ and $Vb_{\delta}$ be the density and velocity profiles defined by the sharp stratification and shear-flow setup. Let $\mathcal C_{\delta}(k)$…

Let $u$ be an inviscid incompressible flow on $\mathbb{R}^2$ with scalar vorticity $\omega=\partial_1u^2-\partial_2u^1$, recovered from $\omega$ by the Biot–Savart law…

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…

Consider the 3D incompressible Euler equations on $[0,T]\times \mathbb{T}^3$. A weak solution is understood in the usual distributional sense, and $C^{\beta}$ denotes the Hölder sp…

Consider a vortex patch whose boundary has a single corner of acute angle, and define the order of a cusp at $x=x_0$ by … where $f_1$ and $f_2$ represent the graphs of the two side…

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

Let $S$ be a mild solution of the Navier–Stokes strain equation without advection, with S otagin Cleft([0,T{\max});L^2{st}\right), where $L^2{st}$ denotes the space of square-integ…

Let $S_{\mu,1,1}$ denote the set of maximizers associated with impulse $\mu$, and let $\mu$ be the impulse parameter. Small-impulse uniqueness conjecture. There exists a universal…

Let $\Omega$ be the bounded open set defining a Sadovskii vortex patch as described in Theorem 1. Detailed-shape conjecture. The set $\Omega$ is connected, convex, and makes a righ…

Low-frequency spectral asymptotics.

Let $u\in C\left([0,T_{\max}); \dot{H}^s_{\mathcal{M}}\right)$, with $s>\frac{5}{2}$, be an odd, permutation symmetric, $\sigma$-mirror symmetric solution of the Fourier-restricted…

Hamel–Nadirashvili's weaker conjecture. If $z\in D$ is the only stagnation point of $\mathbf{v}$ in $\overline{D}$, meaning that $|\mathbf{v}(z)|=0$ and $|\mathbf{v}|>0$ in…

Hamel–Nadirashvili's conjecture. If $|\mathbf{v}|>0$ in $\overline{D}\backslash\{z\}$, then $z$ is the origin and $\mathbf{v}$ is a circular flow.

LDP under the weak Kato hypothesis. The statement of the main large-deviations theorem holds true if the strong Kato hypothesis is replaced with the weak Kato condition.

Let solutions of the two-dimensional Euler equation evolve as $t\to\pm\infty$. A solution is called generic in the informal sense intended by the conjecture, and its weak limit set…

Let $(\omega,u)$ be a solution of the two-dimensional incompressible Euler equation on $\mathbb{T}^2$, and consider its weak limit set as $t\to\pm\infty$. A solution lies on a comp…

Tran–Dritschel conjecture. The enstrophy dissipation satisfies

Let $M\subset\mathbb{R}^2$ be a bounded planar domain, let $\omega_0\in L^{\infty}(M)$ be initial vorticity, and let $\{\omega(t)\}_{t\in\mathbb{R}}$ be its inviscid incompressible…

Euler-$\alpha$ Hamiltonian dichotomy conjecture. Weak solutions obey the following dichotomy: if $s>1$, every weak solution in $C_t^0H_x^1\cap L_t^3B_{3,\infty}^s$ conserves…

Dissipation-minimization conjecture. The parameters $(V,A_s,A_f)$ are chosen in such a way that the dissipation $R$ is minimal. The claim concerns the selection of parameters for a…