On the operator formulation of a class of problems of linear fluid dynamics (Q1359343)

We propose a new operator formulation of a rather large class of problems of linear fluid dynamics. We give a result on the unique solvability of an evolution problem. In a bounded open set \(\Omega\) in \(\mathbb{R}^3\), we consider the equation \[ \rho u''_t- \rho(u_t'\times\omega)+\nabla p-(\nabla\rho\cdot u)\nabla\Pi= f(t,x),\tag{1} \] where \(u(t,x)\) and \(p(t,x)\) are unknown vector and scalar fields in \(\Omega\) and the functions \(\rho(x)\) and \(\Pi(x)\) and the vector field \(\omega(x)\) are defined in \(\Omega\) and have uniformly continuous first-order derivatives, the function \(\rho(x)\) being positive and bounded away from zero; \(f(t,x)\) is a given nonstationary vector field in \(\Omega\) used to state problems of linear fluid dynamics. For equation (1) we pose a boundary-value problem, state the natural concept of a generalized solution, and derive an operator equation equivalent to the generalized formulation of the problem. The basis of the method used here is the idea of the ``embedding'' of the boundary-value problem into an integral identity. The boundary condition by which the equation is supplemented contains two bilinear forms as parameters. By varying them, one can obtain as special cases a number of problems studied earlier. Moreover, the approach applied here leads rapidly to the goal and makes it possible to avoid excessive auxiliary constructions.