A transformation of coordinates in a problem with a free boundary
The paper deals with the nonstationary flow of a compressible fluid with power dependency of the tangential tension on the sliding velocity under resilient filtering conditions. At the initial moment the fluid covers the half-space \(x<0\), while in the motion it is bounded by the solid wall \(x=0\). The case is studied when the fluid density decreases interior to the half-space, i.e. the equation \[ u_ t={\partial \over {\partial x}} u_ x^ m, \qquad x<0, \qquad 0<t\leq T \] with initial boundary conditions \(u(x,0)=u_ 0(x)\), \(u_ 0(x)>0\), \(x\in(-1,0]\), \(u_ 0(x)\equiv 0\), \(x\leq-1\), \(u(0,t)=p\), \(t\geq 0\). The problem is solved by means of the transformation of coordinates \((x=X(\xi,t))\) connected with equipotentials, i.e. the evolution of a fluid particle is defined as its motion along the path of the constant density value \(u(X(\xi,t),t)=u_ 0(\xi)\). After the transformation a nonclassical boundary-value problem is obtained for the investigation of which the author uses the theory of partial differential equations: he defines the notion of generalized solution of this boundary-value problem, proves its existence, obtains estimates. Using the obtained results he proves the existence of a solution to the initial problem, its continuity and positiveness.
scientific article; zbMATH DE number 20553