17 problems

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.…

Benjamin–Lighthill conjecture. The Bernoulli constant satisfies

Let Problem 3.1 be the quasi-variational–hemivariational inequality whose unknown is a flow velocity…

Consider the stationary Navier–Stokes system in $\mathbb{R}^3$: … Here $u$ is the velocity vector field, $\pi$ is the pressure, and $\dot{H}^1(\mathbb{R}^3)$ denotes the homogeneou…

Let $D$ be an open disk centered at the origin. Let $z\in D$ and let $\mathbf{v}\in C^2(\overline{D}\backslash \\{z\\})$ be a bounded flow solving the Euler equations on…

Let $L_c$ be the macroscale characteristic length and let $alpha_\text{max}$ denote the maximum Brinkman penalization value. The numerical study considers their relation through a…

The Brinkman penalization term is studied through the maximum velocity in solid regions and state-variable errors in fluid regions, with $alpha_\text{max}$ denoting the maximum pen…

Pólya's conjecture. The optimal profile in three dimensions is a prolate spheroid with sharp ends of angle $120$ degrees.

Logarithmic-radius vortex-ring conjecture. The limiting motion of the vorticity centers is governed by a dynamical system that is a combination of the point vortex dynamics and sim…

For a global curve of rotating vortex-pair solutions, with the vortices either corotating or counter-rotating, the limiting configuration is expected to consist of vortex patches i…

Let … and time horizon $T$. Their rescaled copies $(1/m)(mx)$, $x[0,1]$, for $m=2,3,$, are also local maximizers but attain smaller enstrophy. Global-maximizer conjecture. At least…

Consider $N$ vortex rings with radii of order $|\log\varepsilon|^\alpha$ as $\varepsilon\to0$, where $\alpha>2$, and let $z_i(t)$ denote the positions of the corresponding vortices…

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…

In the plane, consider steady Navier–Stokes solutions with velocity tending to zero at infinity and nonzero net force $\boldsymbol{F}$. Let…

Assume the hypotheses of Proposition. Let $\omega^\nu(x,t)$ solve the rotating viscous vorticity equation, and let $\epsilon>0$ be sufficiently small. Let $\omega_\epsilon(x)$ be a…

Let $w_*$ be a radially symmetric vorticity profile of the form given by the source, satisfying the stated assumptions on $q$. Let $w$ and $v$ be the asymptotic vortex-pair profile…

In a two-vortex system, the vortex profiles are considered before the vortices merge. Gaussian relaxation conjecture. The profiles relax to a pair of Gaussian vortices before mergi…