19 problems

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…

Uniqueness conjecture. The self-similar profile with this asymptotic behavior is unique.

The Porous Medium Equation is expected to provide an approximation of the geometric Hele–Shaw free boundary problem. In particular, an incompressible limit would relate solutions o…

Consider the stochastic porous medium equation in one dimension with large-$h$ behavior $V(h)\sim |h|^s$ for $s>0$, and the renormalization-group equations governing its flow. Let…

Let $m(x,t)$ be a congestion function in the porous medium equation and its pressure formulation, and consider the Hele-Shaw limit obtained through a JKO approximation scheme. The…

Let $T>0$, let $m$ be the porous-medium exponent, and let $g:{\mathbb T}\to[0,1]$ be measurable. For each $\kappa>0$, let…

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

The numerical method is applied to the time-fractional porous medium equation, with fractional order $[?]$ and convergence order measured as the number of subdivisions of the inter…

Volume-growth power conjecture. The correct power-law growth of the support volume is indeed

Equality conjecture. We conjecture that $R=R_c$ when $\kappa=0$.

Let $\rho$ be a solution to the porous medium equation on $[-1,0] \times B_2(0)$. Assume that there is a nonempty open cone $C$ such that $\rho(t,x) \leq \rho(t,y)$ whenever…

Consider the critical nonhomogeneous porous medium equation in dimension $N=2$ with general, not necessarily radially symmetric, initial data. The preceding discussion establishes…

Consider the porous medium equation … with $p=P_m(\rho):=\frac{m}{m-1}\rho^{m-1}$, $G'<0$, and $G(p_M)=0$ for some $p_M>0$. In the stiff pressure limit $m\to\infty$, let…

Let $u_*$ be the solution of the perturbation equation on $(0,S)\times\overline{H}$, where $H$ is the spatial half-space, arising from sufficiently small Lipschitz initial data. Th…

Let $\rho$ be a distributional solution of the porous medium equation whose pressure $\rho^{m-1}$ is smooth up to the relevant parabolic boundary and whose level sets are analytic.…

Consider the porous medium equation referred to as Eq. (1), with initial data $u_0\in L^{\infty}(\mathbb{R}^N)$ satisfying … and allowing a peak in $\{0<|x|<\infty\}$. Conjecture o…

Gaspard and Gilbert's conjecture. The hydrodynamic equation of mesoscopic energy exchange models of a billiard lattice is the porous medium equation with $m=\frac{1}{2}$.

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 compressible Euler equations with damping, denoted by, and let the canonical boundary be characterized by the physical vacuum boundary condition. Long-time behavior co…