20 problems

Consider the Cauchy problem for the multi-dimensional compressible Euler equations, and call a weak solution entropy/admissible if it satisfies the relevant entropy inequalities. N…

Let $P_{k_0(\lambda)}$ and $Q_{k_0(\lambda)}$ denote the remainder terms in the time asymptotic expansion for the compressible Euler equations with time-dependent damping. All-orde…

Liu's conjecture. Solutions of the Euler equations with damping should behave asymptotically like solutions of the porous media equation.

Generic nonexistence conjecture. Entropy-rate-admissible weak solutions possibly do not exist for a generic set of initial data.

Entropy-production conjecture. The conclusion of Theorem $$ should hold for every

Waiting-time conjecture. When $\alpha=1$, the boundary behavior $c\sim x$ should, after some time, cease to persist; the boundary should then start moving and attain the physical v…

Consider solutions of the compressible Euler system with frictional damping whose density has finite total mass. A Barenblatt self-similar solution is the canonical nonlinear diffu…

For the irrotational compressible Euler equation with a vacuum boundary, let $c_s$ denote the speed of sound and d_{Gamma_t the distance to the boundary of a particle. In the slo…

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…

Positivity conjecture. For every profile $K(x)$, one has $\nu>0$; consequently, the system admits linearly dispersive waves for all wave numbers $k\in\mathbb{R}$.

Existence conjecture. Space- and time-periodic solutions to the Euler equations should exist.

A Guderley solution is a radial self-similar solution of the Euler system, defined at least for all negative times, with a converging shock propagating into a quiescent fluid near…

Wang's sharp local well-posedness conjecture. The sharp local well-posedness results would hold for initial data satisfying

Asymptotic-equivalence conjecture. It is conjectured that the system is time-asymptotically equivalent to the decoupled system

Let $w$ solve the equation denoted by … lambda=1,quad 0<muleq 2,qquadtext{or}qquad lambda1,quad mu0. … will blow up in finite time for a family of smooth initial data…

Sharp local well-posedness conjecture. The solution $(v,\varrho,\mathfrak{w})$ is well-posed for $t\in(0,T]$ for some $T>0$. This conjecture seeks a sharp regularity threshold for…

Let $(\rho,u_r,u_\uptheta)$ be the solution to the two-dimensional isentropic compressible Euler equations given in Theorem general, and let $\Omega(t)$ be the material curve defin…

For $\rho$ the density and $u=(u_r,u_\uptheta)$ the velocity solving the two-dimensional isentropic compressible Euler equations, suppose the solution constructed in Theorem genera…

Let $\bar\rho$ and $u$ be a smooth solution of the isentropic hyperbolic system … … with $p=\bar\rho^\gamma$, $\gamma>1$, and constant $c$. Finite-time vacuum conjecture. If there…

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