<p>We consider the Cauchy problem for a system of nonlinear differential equations similar to the classical evolutional Navier–Stokes equations for an incompressible liquid. The main difference of this system is that it is generated not by the standard gradient, divergence, and curl operators, but by the multidimensional Cauchy–Riemann operator, its the compatibility complex (which is usually called the Dolbeault complex), and its formally adjoint operator. The similarity of the structure makes it possible to prove the theorem of the existence of weak solutions for this problem 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.</p>