18 problems

The relativistic Boltzmann equation in the relaxation-time model describes non-equilibrium dynamics, and its hydrodynamic gradient expansion is generally asymptotic rather than con…

Let $M$ be a compact manifold of dimension $2$, and let the $L^2$-metric be the right-invariant metric on the group of volume-preserving diffeomorphisms induced by the $L^2$-length…

Hydrodynamical Weinstein conjecture. Every smooth rotational Beltrami flow on a Riemannian three-manifold has a closed orbit.

In kinetic theory, consider the factorially divergent gradient series for the hydrodynamic expansion and its Borel resummation. Borel-resummation conjecture. Hydrodynamics can be s…

Let $\mathfrak L_0$ be the space of local observables, let $o,o'\in\mathfrak L_0$, and let $\mathcal T_{\vec\lambda}$ and $\b{\mathcal T}_{\vec\lambda}$ be a twist field and its co…

Let $V_\ell$ denote the eigenspace indexed by $\ell$ in the Zeitlin model on $SU(N)$, and let $\widetilde{r}_\ell(N)$ be the averaged Ricci curvature in that eigenspace. Eventual n…

Optimal mesoscopic-scale conjecture. The optimal choice of mesoscopic length and averaging time is obtained when

Consider an integrable many-body system in a no-shock regime with ballistic scaling $\mathrm{x}=\ell x$ and $\micro t=\ell t$ as $\ell\to\infty$. Let $\xi$ denote the fluctuating-h…

Let $\Phi(\omega)$ be a spectral function with finite, positive zero-frequency value, $0<\Phi(0)<\infty$. Suppose that hydrodynamic tails make $\Phi(\omega)$ nonanalytic at…

The exact hydrodynamic closure of a kinetic model modifies the entropy so as to ensure pure dissipation in the hydrodynamic variables; this is viewed as a non-local extension of Ko…

The fluid is described by Euler-scale hydrodynamics with macroscopic fields whose characteristic modes may be self-interacting through the dependence of their effective propagation…

The present solutions are analytic, parametric solutions of non-relativistic Navier–Stokes hydrodynamics with spherical symmetry, and the authors consider analogous spheroidal, ell…

Finite-dimensionality conjecture. Finite-dimensionality of ${\cal Q}_0$ is intimately related to non-integrability.

Helicity uniqueness conjecture. The helicity is the only Casimir function for $\operatorname{SDiff}(M)$.

Universality conjecture. This diffusion-kernel expression applies to any integrable model, including Galilean and relativistic quantum and classical field theories, quantum chains,…

Universality conjecture. The resulting diffusion matrix and Navier–Stokes correction are universal for every integrable model.

Consider the large-wave-vector regime in which entanglement between hydrodynamic and non-hydrodynamic modes destroys the exact hydrodynamics for the Grad moment equations. Conjectu…

For the Boltzmann equation, let the linearized collision operator have a fivefold-degenerate zero eigenvalue corresponding to the hydrodynamic modes, with the remainder of its spec…