Young-regime well-posedness conjecture for locally monotone equations

Let $V\to H\to V^*$ be the compact Gelfand triple from the assumptions, let $A$ satisfy the stated local monotonicity and coercivity hypotheses, and let $\theta\frac{1}{\theta}:=(1-\theta)\cdot 1+\theta\cdot 0$. For $\theta\frac{1}{\theta}$ and $V_\theta$ an interpolation space of $V$ and $V^*$, define

$$\frac{1}{\alpha_\theta}:=\frac{1-\theta}{\alpha'}+\frac{\theta}{\alpha},\qquad r_\theta:=(1-\theta)\cdot1+\theta\cdot0.$$

If $Z\in B^{r_\theta+}_{\alpha_\theta,\infty}V_\theta$, then Young-regime well-posedness conjecture. The equation

$$\mathrm{d}u=A(t,u)\,\mathrm{d}t+\mathrm{d}Z,\qquad u(0)=u_0\in H,$$

is well-posed.

This conjecture proposes that the sufficient temporal regularity obtained by interpolating between the known regimes extends to the full family of locally monotone abstract Young equations. The paper establishes existence under stronger regularity, but uniqueness is known only for additive drivers and linear multiplicative spatially homogeneous noise; the general well-posedness statement remains open.