A nonclassical generalization, symmetry properties, and exact solutions of the system of Navier-Stokes equations (Q2740272)

The authors consider the following generalized Navier-Stokes equations NEWLINE\[NEWLINE\begin{aligned} &\vec{u}_t+(\vec{\lambda }\vec{u})\vec{u}_x+mu_{xx}=-\frac{\vec{\gamma }}{\rho } \partial_xf(\rho), \\ &\rho_t+\partial_x(\rho \cdot \vec{\lambda }\vec{u})=0. \end{aligned}\tag{1} NEWLINE\]NEWLINE Here \((t,x)\in \mathbb{R}^2, \vec{u}\in \mathbb{R}^n\) (note that in the classical case \(\dim\vec{u}= \dim\vec{x}\)), \(\vec{\lambda }, \vec{\gamma }\) are constant vectors, \(f(\rho)\) is a nonlinear function.NEWLINENEWLINENEWLINEBy means of the Lie method, maximal invariance algebras for (1) are found in three cases: 1) \(f(\rho)\) is arbitrary; 2) \(f(\rho)=c\rho^{k+2}\), \(c, k\) are arbitrary constants; 3) \(f=c\rho^3, \vec{\gamma }=\vec{\lambda }/3c\). In the latter case the equation possesses the widest symmetry.NEWLINENEWLINENEWLINEWhen \(n=2\) the authors construct ansatzes generated by the basic operators of system's invariance algebras, reduce equation (1) to systems of ODE, and find some exact solutions.