18 problems

Approximation-independence conjecture. The set of admissible boundary values, and hence the boundary layer, should be independent of the approximation method for nonlinear systems…

Similarity-form solution conjecture. The correction term $\psi_0$ may be similarity and connected to $x^\alpha f(yx^{-\beta})$ and $\mathcal{U}(x)$. This conjecture concerns extend…

Boundary-layer confinement conjecture. In the diffusive limit, the boundary layer becomes confined to a shrinking region near $\Gamma$, so that the influence of the precise diffusi…

Boundary-layer thickness conjecture. For a general domain, the boundary-layer thickness is proportional to the product of the volume of $\Omega$ and the boundary curvature.

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…

Most-unstable-region conjecture. This region is where the most unstable instabilities may be found. The paper states that instabilities occur in this regime, but gives no resolutio…

Let $\nu$ denote the viscosity, and consider linear instabilities of an exponential shear-layer profile or a Blasius profile. Saturation-size conjecture. These linear instabilities…

Let $\nu$ denote the viscosity, and consider small perturbations of the exponential or Blasius shear-layer profile in the viscous boundary-layer problem. The perturbations generate…

Let $hat{x}inpartialmathcal{D}$ be a boundary point, let $n_{hat{x}}$ be the unit outer normal there, and introduce the rescaled normal coordinate … For $y\in[0,\infty)$, let…

Boundary-layer current-conversion conjecture. The normal current $-\sigma\nabla\phi$ turns into the superconducting current $\rho_{out}^2\nabla\chi_{out}$ within a thin one-dimensi…

Let $E_n$ be the discrete energy, let $x^n_\ast$ denote the discrete configuration under consideration, and let $\rho^n_\ast$ be the empirical measure of the discrete dislocation w…

Let $(\rho,u,\theta)$ solve the compressible Navier–Stokes–Fourier system with viscosity and heat-conductivity parameters tending to zero, subject to the generalized Navier boundar…

Let $u$ solve the two-dimensional Navier–Stokes equations with viscosity $u$, and let $overline{u}$ denote the corresponding Euler solution. Suppose the vanishing viscosity limit…

Generic shear-flow instability conjecture. Generically, shear flows for the Navier–Stokes equations are linearly unstable, and the Prandtl expansion is not valid in Sobolev spaces.

Let $Re$ denote the Reynolds number, and let $\tau_w^{NS}$ be the Navier–Stokes wall shear whose complex singularities are being tracked. The large-scale and small-scale interactio…

For a flow at finite Reynolds number $Re$, the large-scale interaction is the viscous-inviscid interaction occurring over a scale comparable with the characteristic length and velo…

Mean friction velocity conjecture. $u_\star\to\infty$ when $\lVert {\bf f}\rVert_{0,2,\Om_f}\to\infty$. This conjecture asserts that increasing the forcing norm drives the mean fri…

Let $N”$ denote the number of scalar Neumann conditions imposed in the viscous problem and let $N^2_-$ denote the number of outgoing characteristic modes associated with the releva…