On the initial-boundary value problems for barotropic motions of a viscous gas in a region with permeable boundaries
For the pair of functions \(u(g,t)\), \(\rho(y,t)\) the following system of equations is considered in a rectangle \(Q= \{(y,t)\): \(0<y<Y\), \(0<t<T\}\) \[ \rho \biggl( {{\partial u} \over {\partial t}}+u {{\partial u} \over {\partial y}} \biggr)=\mu {{\partial^ 2 u} \over {\partial y^ 2}}- {\partial \over {\partial y}} (\rho^ \gamma), \qquad {{\partial\rho} \over {\partial t}}+u {{\partial\rho} \over {\partial y}}+\rho {{\partial u} \over {\partial y}} =0 \tag{1} \] under the initial conditions \[ u(y,0) = u_ 0(y), \qquad \rho(y,0)= \rho_ 0(y), \qquad 0\leq y\leq Y,\tag{2} \] and one of the boundary conditions stated below \[ \begin{aligned} u(0,t) &= u_ 1(t)>0, \qquad \rho(0,t)= \rho_ 1(t), \qquad u(Y,t)= u_ 2(t)\geq 0, \tag{3}\\ u(0,t) &= u_ 1(t)>0, \qquad \rho(0,t)= \rho_ 1(t), \qquad \rho(Y,t)= \rho_ 2(t), \tag{4}\\ u(0,t) &= u_ 1(t)<0, \qquad u(Y,t)= u_ 2(t)\geq 0. \tag{5} \end{aligned} \] The solvability of the problems (1), (2), (3); (1), (2), (4) and (1) (2), (5) are proved in Sobolev and in Hölder spaces.
scientific article; zbMATH DE number 701492