Simple partially invariant solutions

The continuous medium models of hydrodynamic type admit 11th dimensional Lie algebra of Galilean group enlarged by uniform dilatation of all independent variables. All subalgebras of this Lie algebra are listed up to inner automorphisms. We consider invariant submodels for subalgebras of the small dimensions from 1 to 3. For 4-th dimensional subalgebras, the invariant solutions are the simple solutions depending on finite numbers constants. We formulate a problem on finding partially invariant solutions of the minimal rank. For all 48 types of 4th dimensional subalgebras we calculate the bases of point invariants in terms of the variables convenient for further calculations. This allows us to consider simplest partially invariant solutions of rank 1 defect 1. In addition, both regular and irregular partially invariant submodels are obtained.
We consider three of the 4-th dimensional subalgebras producing regular partially invariant solutions in the Cartesian, cylindrical and spherical coordinates, respectively. We obtain a solution depending on an arbitrary function of two variables in Cartesian coordinates. In the cylindrical coordinates, a submodel is reduced to a first order ordinary differential equation. In the spherical coordinates, we generalize invariant solutions of spherical vortex constructed by a rotation group.
We consider two of 4-th dimensional subalgebras producing irregular partially invariant solutions. The arising overdetermined systems are reduced into an involution. The compatibility conditions give a series of exact solutions depending on arbitrary functions, so-called simple waves. We obtain solutions with a level surface of invariant functions in the form of a moving plane with the constant normal but a varying velocity. For stationary motions with a rotation, we obtain the series of exact solutions depending on arbitrary functions.