Generalized Navier-Stokes equations, associated with the Dolbeault complex
Shlapunov Alexander, Polkovnikov Alexander
TL;DR
The paper develops a Dolbeault-complex–based framework for generalized Stokes/Navier–Stokes equations in $\mathbb{C}^n$, formulating a parabolic system driven by the Dolbeault Laplacians $\Delta^q$ and a bilinear nonlinearity $\mathcal{N}^q$. It constructs a Bochner–Sobolev scale for $(0,q)$-forms and introduces a projection $P^q$ with a pressure reconstruction scheme, enabling analysis of existence, uniqueness (generally not guaranteed for weak solutions), and regularity. The authors prove the existence of weak solutions and an open mapping (stability) theorem for the nonlinear problem, and provide criteria under which strong solutions with higher regularity exist. The results extend Navier–Stokes–type analysis to elliptic complexes, with pressure determined up to Dolbeault-cohomology constants and a clear path to higher regularity under suitable a priori bounds.
Abstract
We consider the Cauchy problem in the band $\mathbb{C}^{n}\times[0, T], n>1,T>0$, for a system of nonlinear differential equations structurally similar to the classical Navier-Stokes equations for an incompressible fluid. The main difference of this system is that it is generated not by the standard gradient operators $\nabla$, divergence div and rotor rot, but by the multidimensional Cauchy-Riemann operator $\overline{\partial}$ in $\mathbb{C}^{n}$, its formally adjoint operator $\overline{\partial}^{*}$ and the compatibility complex for $\overline{\partial}$, which is usually called the Dolbeault complex. The similarity of the structure makes it possible to prove for this problem the theorem of the existence of weak solutions and the open mapping theorem on the scale of specially constructed Bochner-Sobolev spaces. In addition, a criterion for the existence of a ``strong'' solution in these spaces is obtained.