An open mapping theorem for nonlinear operator equations associated with the Dolbeault complex

Let $ \{\overline \partial^{p,q},\Lambda^{p,q}\} $, $n\geq 1$, be the Dolbeault complex over the complex space ${\mathbb C}^n $ where $\Lambda^{p,q}$ be the trivial vector bundle of the exterior differential forms of bidegree $(p,q)$. Denote by $(\overline \partial^{p,q})^*$ the formal adjoint differential operator for the Dolbeault differential $\overline \partial^{p,q}$ related to the standard Hermitian structure of the Lebesgue space $L^2 _{\Lambda^{p,q}}({\mathbb C}^n)$ consisting of the forms of the corresponding bi-degree. Then the Laplacians $\Delta^{p,q} = (\overline \partial^{p,q})^* \overline \partial^{p,q} + \overline \partial^{p,q-1} (\overline \partial^{p,q-1})^*$ of the complex are strongly elliptic and, with any fixed number $\mu>0$, the operators $\partial_t + \mu \Delta^{p,q} $, acting on the induced bundle $\Lambda^{p,q} (t)$ consisting of differential forms with coefficients, depending on the variable $t \in [0,T]$, $T>0$, as a parameter, are parabolic. We consider the following initial problem: given section $ f $ of the induced bundle $ \Lambda^{p,q} (t) $ and $(p,q)$-form $ u_0 $, find sections $ u,v $ of the bundles $ \Lambda^{p,q} (t) $ and $ \Lambda^{p,q-1} (t) $, respectively, such that
\begin{equation} \label{eq.NS.Dolbeault} \begin{cases} \partial_t u + \mu \Delta^{p,q} u + N^{p,q} (u) + \overline \partial^{p,q-1} v = f& \text{in } {\mathbb C}^n \times (0,T),\\ (\overline \partial^{p,q-1})^* u =0 , \, (\overline \partial^{p,q-2})^* v =0 , & \text{in } {\mathbb C}^n \times [0,T],\\
u(x,0) = u_0& \text{in } {\mathbb C}^n, \end{cases} \end{equation}
where the non-linearity $ N^{p,q} u = M_{p,q,1} ( \overline \partial^{p,q}u, u) + \overline \partial^{p,q-1} M_{p,q,2}(u, u)$ is generated by a rather wide class of linear bi-differential operators $M_{p,q,j}$ of zero order, in specially constructed Bochner-Sobolev type spaces over ${\mathbb C}^n$. Using the standard technique adapted to study Navier-Stokes' type equations (the energy type estimates, the interpolation Gagliardo-Nirenberg inequalities and the Faedo-Galerkin method), see, for instance, [1], we prove that, under reasonable assumptions regarding the nonlinear term, the Frechét derivative $ \mathcal{A}_{p,q}' $ of the nonlinear mapping $ \mathcal{A}_{p,q} $, induced by \eqref{eq.NS.Dolbeault}, is continuously invertible and the mapping $ \mathcal{A}_{p,q} $ itself is open and injective in chosen spaces. However for an Existence Theorem related to even weak (distributional) solutions to \eqref{eq.NS.Dolbeault} one should necessarily assume that the bilinear operators $M_{p,q,1}$ have additional rather restrictive but natural properties.
The thesis is based on the joint results with A.N. Polkovnikov (Siberian Federal University). The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics "BASIS".