15 problems

Let two-dimensional turbulence be considered in the infinite-Reynolds-number limit, and let the turbulent cascade be the transfer of enstrophy to small length-scales. Onsager's con…

Within the weak formulation, let $\nu_\mathbb F{}_m$ be the abstract signed Radon measure replacing the generally non-canonical product $F^\vee_m(\mathbb J)\,e_1(\widehat J_m)$, an…

Let a fluid dynamics model with zero initial datum possess infinitely many weak solutions. Couple it with a transport equation and subject the velocity field to multiplicative and…

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

Consider the one-dimensional Cauchy problem for equation … should hold for every … This conjecture concerns the range of parameters for which positive weak solutions exist; the pap…

Let $(\varrho, {\bf v}_m, \mathcal{E}) \in \mathcal{U}[\varrho_0, {\bf v}_{m0}, \mathcal{E}_0]$ be an admissible dissipative solution, meaning that it is minimal with respect to th…

Supercritical nonuniqueness and non-Leray-Hopf conjecture. Then there exist two weak solutions $u,v\in X^{p,q}([0,T];\mathbb{T}^d)$ of the Navier–Stokes equations with

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…

Higher-dimensional convergence conjecture. The sequence $cv_n$ is bounded in $W_0^{1,2}(\Omega)\cap L^\infty(\Omega)$ and therefore converges, up to subsequences, to a bounded nonn…

Let $T>0$) and $R>0$. Let $\mathcal{X}_{01}=(U_{01},U_{11})$ and $\mathcal{X}_{02}=(U_{02},U_{12})$ belong to $\mathcal{H}_0$, with $\mathcal{X}_{01}$ and $\mathcal{X}_{02}$ having…

Let $u_0\in H_P^2(\Omega)$ be initial data and set $w_0:=(I-\Delta)u_0$. The pair $(u,w)\in L^2(0,T;H_P^2(\Omega))\times L^2(0,T;L^2(\Omega))$ is the candidate pair obtained in the…

Let $m$ be a weak solution of the system, with $D$ and $c$ constants, $p[m]$ the pressure associated with $m$, and $r(m)$ defined by the regularization in the system. A slow-soluti…

Consider admissible weak solutions of the compressible barotropic Euler system, and compare their energy dissipation over time. Maximal-dissipation conjecture. The principle of max…

Let $\Omega$ be a finite-volume open convex domain with $C^\infty$ boundary, let $f\in L^2(\Omega)$, and let $\rho_{(\varepsilon)}(x,t)$ solve the regularized problem … where…

Consider the Cauchy problem … for $(t,x)\in\mathbb{R}^+\times\mathbb{R}$, where $\mathcal{L}_\lambda$ is the fractional Laplacian defined by … A non-uniqueness conjecture for weak…