23 problems

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…

Let $(t,x)\in\mathbb{R}_+\times\mathbb{R}^3$. For each $N$, let $\mathbf{u}_N$ denote the corresponding approximating solution, and let $U_{t,x}$ be a neighborhood of $(t,x)$. Loca…

Drift-elimination conjecture. In a gauge-theoretical setting, and by further applying stochastic analysis, one could do away with the drift in any dimension other than $1$.

Critical recurrence diagnostic conjecture. There exist relative filter scales $\ell_n\in[\ell_-,\ell_+]\subset(0,1)$, a subsequence, and a constant $c>0$ such that

The convergence conjecture. As $\delta\to0$, the sequence ${\bf x}_n$ will converge to the approximating solutions obtained by the continuous-in-time nudging method given as Algori…

For the perturbation variables $(\phi,\varphi,\zeta):=(\rho-\bar{\rho},u-\bar{u},\theta-\bar{\theta})$ of the planar entropy wave, the presence of a diffusion wave is expected to d…

Let $w(t)$ be the global solution considered above, satisfying the decay estimates in particular for $\|w(t)\|_\infty$ as $t\to\infty$. Optimal $L^\infty$ decay conjecture. One wou…

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…

Supercritical non-uniqueness conjecture. Non-uniqueness should hold throughout the whole supercritical regime determined by the Ladyzhenskaya–Prodi–Serrin condition.

Cheskidov–Luo's conjecture. Non-uniqueness of weak solutions should hold throughout the full supercritical regime

Let $f$ and \text{\boldmath u} be a large weak solution of the kinetic Cucker–Smale model coupled with the three-dimensional incompressible Navier–Stokes equations, with initia…

Let the incompressible Navier–Stokes equations be denoted by INSE, and consider finite-time blow-up of enstrophy in the inviscid Euler limit. Buaria et al.'s conjecture. Such finit…

The DNN-MG method uses a neural network trained on oscillatory flow data to predict corrections for individual patches in a multigrid solver. Patchwise network design conjecture. T…

The paper considers three Eulerian and one arbitrary Lagrangian–Eulerian finite element schemes for fitted approximation of two-phase incompressible Navier–Stokes flow. The Euleria…

A physical phenomenon or effect is modeled as an object whose behavior can be simulated to arbitrary precision. Physical Church–Turing hypothesis. Every physical phenomenon or effe…

Let $Q_{N_i,\nu_i}$ be probability laws associated with sequences $\nu_i\to 0$ and $N_i\to\infty$, and suppose that $\{Q_{N_i,\nu_i}\}_i$ is weakly convergent to a probability meas…

Let $v$ be the suitable weak solution and let $H(t,s)$ be the quantity defined by the iterated limits in the preceding existence theorem, for $t>s\geq 0$. The quantity $H(t,s)$ is…

Let $X_T$ be the set of states reachable by solutions of … from admissible initial data and forcings in $\mathcal{E}$ during some time $0\leq t^*\leq T$. Assume that $V$ solves the…

Improved spatial regularity conjecture. The pressure $\pi$ has better spatial regularity than membership in $\bH_{\perp}^{-3}$ suggests. The preceding lemma and system formulation…

Controller-dimension conjecture. In the $d$-dimensional case, $d\in\{2,3\}$, it should be enough to take a number $M$ of internal controls proportional to

Let $\mathcal H=\{u\in L^2_{per}(\mathbb T^2)^2: \operatorname{div}u=0,\ \int_{\mathbb T^2}u\,dx=0\}$ be the phase space for the two-dimensional periodic Navier–Stokes equations, a…

Foias–Saut conjecture. Non-resonance should be generic for Dirichlet boundary conditions.