Existence for adaptive porous flow with a negative drag jump

Let $d \ge 2$ and let $\Omega \subset \mathbb{R}^d$ be a bounded connected Lipschitz domain, whose boundary is partitioned, up to surface measure zero, into relatively open pressure and flux parts $\Sigma_p$ and $\Sigma_v$. Let $2 \le r < \infty$ and $s = r/(r-1)$, and let $\phi_1 : [0,1] \to [0,\infty)$ and $\phi_2 : [1,\infty) \to [0,\infty)$ be continuous and nondecreasing with $$c\, a^{(r-2)/2} \le \phi_2(a) \le C\bigl(1 + a^{(r-2)/2}\bigr) \quad (a \ge 1), \qquad 0 < \phi_2(1) < \phi_1(1),$$ for constants $c, C > 0$; the last inequality is a negative jump of the drag (inverse permeability) at the normalized speed threshold one. For $u \in L^r(\Omega; \mathbb{R}^d)$ let $$\Lambda(u) = \Bigl\{ \phi_1(|u|^2)\, u\, \mathbf{1}_{\{|u| < 1\}} + \phi_2(|u|^2)\, u\, \mathbf{1}_{\{|u| > 1\}} + h\, \mathbf{1}_{\{|u| = 1\}} : h \in L^s(\Omega; \mathbb{R}^d) \Bigr\},$$ so that the drag on $\{|u| = 1\}$ is free: neither a branch value nor a prescribed convex combination. Take $q \in L^r(\Omega)$, $f \in L^s(\Omega; \mathbb{R}^d)$ and $u_0 \in L^r(\Sigma_v)$. If $\Sigma_p$ has positive surface measure, prescribe the pressure through an extension $P \in W^{1,s}(\Omega)$ and put $V = \{\psi \in W^{1,s}(\Omega) : \operatorname{Tr} \psi = 0 \text{ on } \Sigma_p\}$. If $\Sigma_p = \varnothing$, put $P = 0$, $V = \{\psi \in W^{1,s}(\Omega) : \int_\Omega \psi = 0\}$ and assume $\int_\Omega q = \int_{\partial\Omega} u_0$. Fumagalli and Patacchini, who introduced this adaptive model, proved existence of solutions in every dimension when $\phi_2(1) \ge \phi_1(1)$, where the underlying energy is convex, and, when $\phi_2(1) < \phi_1(1)$, in dimension one.

Problem. Does every such choice of domain, laws and data admit $u \in L^r(\Omega; \mathbb{R}^d)$, $p \in P + V$ and $\zeta \in \Lambda(u)$ with $$\zeta = f - \nabla p \ \text{ a.e. in } \Omega, \qquad \int_\Omega u \cdot \nabla \psi = -\int_\Omega q \psi + \int_{\Sigma_v} u_0 \psi \quad (\psi \in V)?$$ A weak solution of this law is asked for; it need not minimize an energy.