13 problems

Vanishing viscosity conjecture. The entropic solutions of the hyperbolic system (1.1) actually coincide with the limits of solutions $u^\varepsilon$ to the parabolic system as the…

Vanishing viscosity conjecture. The entropic solutions of the hyperbolic system (1.1) actually coincide with the limits of solutions to the parabolic system

Let $T_M$ be the minimal time for uniform controllability, defined by … Let $T_0$ denote the optimal time required for every relevant trajectory of the transport field to reach the…

The self-similar axisymmetric Euler equations admit no solutions with velocity discontinuities at finitely many points satisfying the Rankine–Hugoniot conditions … Here $\theta$ is…

Longer-time approximation conjecture. An approximation result of the form $$

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…

Let $E_0$ be the threshold from the assumptions, let $G_{,E}$ be the previously defined constant, and let $\Phi'$ and $\mathcal{T}$ be the functions related by…

K41–Onsager conjecture for Navier–Stokes. There exists a finite open interval $I$ and a sequence of suitable weak solutions $v_{\nu_j}$ with $\nu_j\to0$ that is uniformly bounded i…

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…

Let $\theta_0$ satisfy the hypotheses of the Hölder $1/2$ a priori estimate conjecture, and let $\theta$ be a vanishing-viscosity limit of solutions of … Here…

Let $\theta$ solve the one-dimensional nonlocal transport equation … where $H$ is the Hilbert transform, and let $C^{1/2}$ denote the Hölder class with exponent $1/2$. A Hölder reg…

Let $M\ne0$, $\varepsilon\in(0,1)$, and consider the transport-diffusion equation on $(0,T)\times(0,L)$, … with boundary control at $x=0$, homogeneous boundary condition at $x=L$,…

Strong zero-dissipation limit conjecture. A general weak entropy solution to the inviscid flow should be the strong limit of the solution to the corresponding viscous flows with th…