Global weak existence for nonlocal quantum drift–diffusion–Poisson

Let $I = (0, 1)$ and let $H_0 = -\partial_{xx}$ on $L^2(I)$ with periodic boundary conditions, $D(H_0) = \{\psi \in H^2(I) : \psi(0) = \psi(1),\ \psi'(0) = \psi'(1)\}$. Write $H^1_{\rm per}(I) = \{v \in H^1(I) : v(0) = v(1)\}$, $H^{-1}_{\rm per}$ for its dual and $\mathcal{J}_1$ for the trace class. The density $n[\rho]$ of a density operator $\rho$ is defined by $\operatorname{Tr}(\varphi\rho) = \int_I \varphi\, n[\rho]\, dx$ for $\varphi \in L^\infty(I)$, and $\mathcal{E}_+$ is the set of density operators $\rho$ with $H_0^{1/2} \rho H_0^{1/2}$ trace class (extended from its natural domain). For nonnegative $n \in H^1_{\rm per}(I)$ put $$\mathcal{K}(n) = \inf \bigl\{ \operatorname{Tr}(\rho \log \rho - \rho) + \operatorname{Tr}(H_0^{1/2} \rho H_0^{1/2}) : \rho \in \mathcal{E}_+,\ n[\rho] = n \bigr\},$$ with $0 \log 0 = 0$ and $\inf \varnothing = +\infty$, let $V[n] \in H^1_0(I)$ solve $-V[n]'' = n$, and put $\mathcal{F}(n) = \mathcal{K}(n) + \frac12 \int_I |V[n]'|^2 \, dx$. The nonlocal quantum drift–diffusion–Poisson system of Degond, Méhats and Ringhofer, with no external potential, is $$\partial_t n + \partial_x \bigl( n\, \partial_x (A - V) \bigr) = 0, \qquad -V'' = n, \qquad n = n\bigl[e^{-(H_0 + A)}\bigr].$$ For $n_0 \in H^1_{\rm per}(I)$, a global weak solution with $n(0) = n_0$ is a triple $(n, A, V)$ with $A$ real such that for every finite $T$ $$\begin{aligned} &n \ge 0, \quad n \in L^\infty(0, T; H^1_{\rm per}) \cap H^1(0, T; H^{-1}_{\rm per}) \cap C([0, T]; L^2(I)), \quad n(0) = n_0, \quad \textstyle\int_I n(t) = 1,\\ &A \in L^2(0, T; H^1_{\rm per}(I)), \quad V \in L^\infty(0, T; H^1_0(I)), \quad -V'' = n,\\ &\rho(t) = \exp[-(H_0 + A(t))] \in L^\infty(0, T; \mathcal{E}_+), \quad H_0 \rho H_0 \in L^2(0, T; \mathcal{J}_1), \end{aligned}$$ for almost every $t$, $n = n[\rho]$ and $\rho(t)$ attains $\mathcal{K}(n(t))$; for every real $\varphi \in C^1([0, T]; H^1_{\rm per}(I))$ with $\varphi(T) = 0$, $$\int_0^T \!\! \int_I n\, \partial_t \varphi \, dx\, dt + \int_I n_0 \varphi(0) \, dx + \int_0^T \!\! \int_I n\, \partial_x (A - V)\, \partial_x \varphi \, dx\, dt = 0;$$ and for almost every $t \ge 0$ $$\mathcal{F}(n(t)) + \int_0^t \!\! \int_I n\, |\partial_x (A - V)|^2 \, dx\, ds \le \mathcal{F}(n_0).$$ Pinaud proved that when the initial relative free energy with respect to the equilibrium is small enough, a weak solution exists on every finite interval, with $n \in L^\infty(0, T; H^1_{\rm per})$, $\partial_t n \in L^2(0, T; H^{-1}_{\rm per})$ and $A$, $V$, $\rho = \exp[-(H_0 + A)]$ in the spaces above, that it satisfies the free-energy identity, and that it converges exponentially fast to the equilibrium (2019).

Problem. For every real $n_0 \in H^1_{\rm per}(I)$ with $\min_{[0, 1]} n_0 > 0$ and $\int_I n_0 = 1$, does a global weak solution with $n(0) = n_0$ exist?