Exact solution of Navier-Stokes equations - The theory of functions of a complex variable under Dirac-Pauli representation and its application in fluid dynamis. II
[For part I see the review above (Zbl 0627.76032).] The author makes an attempt to obtain the exact solution of Navier-Stokes equations which is an unsolved problem. He uses the theory of complex functions under the Dirac-Pauli representation, and the Kaluza `Ghost' coordinate to transform the Navier-Stokes equation into a pair of nonlinear equations for two unknown complex flow functions \(\psi\) and \({\bar \psi}\). With a suitable transformation and \(\phi =\psi +{\bar \psi}\), this pair reduces to the following nonlinear equation with only one real unknown function \[ (\frac{\partial}{\partial t}-4\nu \frac{\partial^ 2}{\partial y\partial \bar y})\frac{\partial^ 2\phi}{\partial y\partial \bar y}=i(\frac{\partial \phi}{\partial \bar y}\quad \frac{\partial}{\partial y}-\frac{\partial \phi}{\partial y}\quad \frac{\partial}{\partial \bar y})\frac{\partial^ 2\phi}{\partial y\partial \bar y}. \] Since it is the sum of two complex conjugate equations, it turns out after suitable algebraic transformation that \(\phi\) satisfies \[ (\frac{\partial}{\partial t}-4\nu \frac{\partial^ 2}{\partial y\partial \bar y})\phi =-2i\phi \frac{\partial \phi}{\partial y}. \] This can be further simplified into the classical Burgers equation \[ \phi_ t+2i\phi \quad \phi_{\xi}-4\nu \phi_{\xi \xi}=0,\quad \xi =y+\bar y. \] With the aid of the Cole-Hopf transformation \(\phi =4i\nu (1/\omega)(\partial \omega /\partial \xi)\), the Burgers equation becomes \(\omega_ t=4\nu\omega_{\xi \xi}\). This Cole-Hopf transformation can be written as the Bäcklund transformation \[ \omega_{\xi}=- \frac{i}{4\nu}\phi \omega,\quad \omega_ t=(-i\phi_{\xi}-\frac{\phi^ 2}{4\nu})\omega. \] Finally, the author claims that the exact solution of the Navier-Stokes equation can be determined by the linear equation \[ 2(\frac{\partial}{\partial t}-4\nu \frac{\partial^ 2}{\partial y\partial \bar y})\frac{\partial^ 2\psi}{\partial y\partial \bar y}=2i({\bar \phi}\frac{\partial}{\partial \phi}-\phi \quad \frac{\partial}{\partial \bar y})\frac{\partial^ 2\psi}{\partial y\partial \bar y}+i(\frac{\partial {\bar \phi}}{\partial \bar y}\frac{\partial^ 2\psi}{\partial y^ 2}-\frac{\partial \phi}{\partial y}\frac{\partial^ 2\psi}{\partial \bar y^ 2}) \] and the nonlinear Burgers equation \(\phi_ t+2i\phi\phi_{\xi}-4\nu\phi_{\xi \xi}=0\) and its complex conjugate. In the opinion of the reviewer, the solution of the problem is incomplete. The paper contains several printing errors.
scientific article; zbMATH DE number 4020675