52 problems

Vortex filament conjecture. Vorticity initially concentrated around a curve should remain close to a curve evolving according to the binormal-curvature equation above, to leading o…

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…

Strong ill-posedness conjecture. The Euler equations with Riesz forcing are strongly ill-posed in $L^\infty$. The corresponding equations are a model for fluid-dynamical systems co…

Consider the De Gregorio model on $S^1$, given by … where $H$ is the Hilbert transform and smooth initial data are prescribed. Global regularity conjecture for the De Gregorio mode…

Structural stability conjecture. It is conjectured that the steady Mach reflection in $\mathbb{R}^2$ is structurally stable.

Consider the 2D Euler equation on the two-dimensional torus, with the line of fixed points given by $\Omega=\Gamma\cos x_1$ for a nonzero real constant $\Gamma$. A heteroclinic orb…

Let $Omega$ be a compact region in $\mathbb{R}^n$ with smooth boundary. Consider solutions of the Navier–Stokes equations in $Omega$ and solutions of the Euler equations with the s…

The Euler equations describe compressible inviscid flow, and a carbuncle is a numerical instability associated especially with methods based on Riemann solvers. Robinet, Gressier,…

Smooth isentropic implosions are self-similar, radially symmetric solutions of the isentropic Euler system with equation of state $p=\frac{1}{\gamma}\rho^\gamma$, constructed for q…

DiPerna's conjecture. All $\prec_{DiP}$ maximal solutions are weak solutions; equivalently, their energy defect vanishes for almost every time.

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…

Let $d\in\{2,3\}$. Let $u\in C^0([0,1];C^\alpha(\mathbb T^d))\cap C^\alpha([0,1];C^0(\mathbb T^d))$ be a weak solution of the incompressible Euler equations, for some Hölder expone…

Let $omega_0$ be an initial vorticity and let … be its $omega$-limit set. For initial data $omega$ in this set, further mixing means a loss of the conserved-orbit enstrophy, measur…

Let $omega_0$ be an initial vorticity, let $S_t(\omega_0)$ be the corresponding Euler flow, and define the $omega$-limit set by … Here the closure is taken in the weak- topology. S…

Šverák's conjecture. For generic $omega_0\in L^\infty(\mathbb{S}^2)$, the orbit $\{S_t(\omega_0)\mid t\geq 0\}$ is not precompact in $L^2$. This is interpreted as a mathematical fo…

Let $N$ coaxial vortex rings have cylindrical-coordinate positions…

Optimal growth conjecture. There exists $C(s)>0$ such that, for every $t\in[0,t_0)$ and every $s\in(0,\sigma]$,

Positive-growth conjecture. There is no sequence $(a_n)$ in $cmathcal{M}^{\sigma}$ for which $b3_{s,(a_n)}<0$, and data randomly initialized in this manner grow in $H^s$ on average…

Exceptional-support conjecture. Part (i) of the theorem on invariant measures already contains essentially all counterexamples to non-invariance: for…

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.

Hypersonic similarity convergence conjecture. As $\tau$ becomes small, the solutions of the full Euler problem should converge to the solution in $\Omega$ of the hypersonic small-d…

Let $(\omega,u)$ be a smooth solution of the two-dimensional incompressible Euler equation on $\mathbb{T}^2$, with vorticity transported by the velocity field. Sverák's conjecture.…

Let $u^0\not\equiv 0$ belong to $H^s_{as\&df}(\mathbb{R}^3)$ with $s>\frac{5}{2}$, and let $\omega^0$ be its associated vorticity. Assume that $\omega^0$ is odd in $z$ and satisfie…

Let $u^0 otequiv 0$ belong to H^s{as&df}(mathbb{R}^d), where $dgeq 4$ and $s1+frac{d}{2}$, and let $omega^0$ be its associated vorticity. Assume that $omega^0$ is odd in $z$ and…