Operator approach to linear problems of hydrodynamics. Vol. 2: Nonself-adjoint problems for viscous fluids.
[For the review of Vol. 1 (2001) see Zbl 0979.76002.] Applications of the theory of linear operators are considered for linearized Navier-Stokes equations describing the flows of viscous incompressible fluids. The velocity vector field \(u(t,x)\) is treated as a time-dependent function \(u(t)\) which takes its values in the Hilbert space \(J_0 (\Omega)\), a set of functions of variable \(x\in\Omega\), where \(\Omega\) is a bounded domain in \(\mathbb{R}^3\). The space \(J_0 (\Omega)\) includes all the functions \(v(x)\in L^2 (\Omega)^3\) such that \(\text{div}v=0\) in the distribution sense. After the linearization near a steady velocity field \(v_0 (x)\), the Navier-Stokes equations can be written as the ODE in Banach space \(J_0 (\Omega)\): \(du/dt= -\nu A_0 u - Ru+P_0 f,\) \(u(0)=u_0.\) Here \(P_0 :L^2 (\Omega)^3 \to J_0 (\Omega)\) is the orthoprojection operator, \(f(t,x)\) is the force vector field, \(\nu\) is the viscosity, \(A_0\) is the Stokes operator, and \(Ru=P_0 [(v_0 \cdot\nabla )u+(u\cdot\nabla)v_0 ]\). Restricted to smooth functions, the Stokes operator \(A_0\) coincides with the operator \(-P_0 \Delta\). A part of presentation is devoted to normal oscillations, which are the flows with the velocity law \(u(t,x)=e^{-\nu t\lambda}v(x)\). When \(v_0 (x)=0\) and \(f=0\), one arrives at the eigenvalue problem \(A_0 v=\lambda v\). Nonself-adjoint problems dominate in the volume. As a rule, these problems include time derivatives of unknown functions at the boundary \(\partial\Omega\). As far as the normal oscillations are concerned, the spectral problems include the spectral parameter both in the equation and in the boundary conditions. In their study the authors widely use the theory of nonself-adjoint operators acting in Hilbert space, and also the theory of operator pencils. In particular, the methods of operator pencil factorization and methods of operator theory in a space with indefinite metric find a wide application. The authors address both the classical problems on oscillations of a homogeneous viscous fluid in an open container and a new set of problems on oscillations of partially dissipative hydrodynamics systems, and problems on oscillations of viscoelastic or relaxing fluids.
scientific article; zbMATH DE number 2001178