A periodic boundary value problem for the equation of transonic gas dynamics
The author poses the following boundary value problem in the cylindrical domain \(Q = D \times (0,L)\), \(D \subset \mathbb{R}^2\) (\(D\) is a bounded domain of the \((y_1,y_2)\)-plane with smooth boundary): Find a solution to the equation \[ u_xu_{xx}-u_{y_1y_1}-u_{y_2y_2}+ au_x=f(x,y)\quad (a>0) \tag{1} \] satisfying the conditions \[ u|{}_{x=0} = u|{}_{x=L}, \quad u_x|{}_{x=0} = u_x|{}_{x=L}, \quad \frac {\partial u}{\partial \vec{n}}\biggr|{}_{\partial D \times (0,L)} = 0, \] where \(\vec{n}\) is the normal to \(\partial D\). Under the conditions \[ D^j_xf\big|{}_{x=0}=D^j_xf\big|{}_{x=L},\quad j=0,1,2, \quad \int_Qf dxdy=0, \] \[ |{}|{}|{}f|{}|{}|{}\equiv \biggl( \sum^{3}_{j=0} \bigl\|{}D^j_x f\bigr\|{}^2_{L_2(Q)} +\|{}\nabla_y f_x\|{}^2_{L_2(Q)}\biggl)^{1\over 2}<\alpha, \] where \(\alpha>0\) is some positive number depending on \(a\) and the cylinder \(Q\), the problem is proven to have a regular solution. Moreover, this solution is unique, provided that some additional conditions are satisfied. A characteristic feature of the problem is its ``indifference'' to the type of equation \((1)\), i.e., to the sign of the function \(u_x\).
scientific article; zbMATH DE number 1285614