Pure transport limit for ascent and descent stability

Let $\Omega$ be a domain, let $\boldsymbol{\beta}\in [L^\infty(\Omega)]^d$ be divergence-free, and let $L$ be the transport operator with inflow boundary conditions,

$$L u=\boldsymbol{\beta}\cdot\nabla u.$$

For a discrete scheme, let $T_h$ be the resulting operators, let $G(A)$ denote the graph of an operator $A$, let $\gamma_h$ be the reduced minimum modulus associated with $T_h-\lambda$, and let $\operatorname{asc}$ and $\operatorname{dsc}$ denote ascent and descent. Pure transport limit. If

$$G(T_h-\lambda)\xrightarrow{\mathrm{gap}}G(L-\lambda)$$

as $h\to0$ and

$$\liminf_{h\to0}\gamma_h>0,$$

then, for all sufficiently small $h$,

$$\operatorname{asc}(T_h-\lambda)=\operatorname{asc}(L-\lambda),\qquad \operatorname{dsc}(T_h-\lambda)=\operatorname{dsc}(L-\lambda).$$

The claim concerns preservation of ascent and descent under graph-gap convergence when the reduced minimum modulus stays uniformly positive; the supplied text does not establish whether this statement is proved, conjectural, or open.