The monotonicity and continuity conjecture for generalized Constantin–Lax–Majda fixed points

Let $a\leq 1$ and let $f_a\in\mathbb{D}$ be a fixed point of $\bm{R}_a$, where $\mathbb{D}$ is the class of admissible profiles and $\bm{R}_a$ is the fixed-point map. The space $\mathbb{D}$ is equipped with the $L_\rho^\infty$-norm.

Monotonicity and continuity conjecture. For any $a_1\leq a_2\leq 1$,

$$f_{a_1}(x)\geq f_{a_2}(x),\quad x\in\mathbb{R}.$$

Moreover, there is a family of fixed points $\{f_a:f_a=\bm{R}_a(f_a)\}_{a\leq 1}\subset\mathbb{D}$ such that $f_a$ depends continuously on $a$ in the $L_\rho^\infty$-norm.

The conjecture concerns how the fixed-point profiles vary with the parameter $a$. The paper proves existence of at least one fixed point for every $a\leq 1$, but does not establish uniqueness for general $a$; the asserted monotonicity and continuous selection therefore remain open and are supported by numerical evidence.