Nonlinear hyperbolic integrodifferential equations and their applications in hydrodynamics

We study the nonlinear mathematical models of shear flows of ideal liquid in open channels and the kinetic equations of quasineutral collisionless plasma. Emphasis is placed on the characteristic properties of the equations of motion, the construction of exact solutions and their physical interpretation. Some results of numerical modeling are also presented. Theoretical analysis of these models is based on the proposed by V. M. Teshukov concept of hyperbolicity and characteristics for systems of equations with operator coefficients. A distinctive feature of integrodifferential models is the presence of both discrete and continuous spectrum of characteristic velocities. This is due to the fact that disturbances in shear flows are transmitted through the surface and internal waves.
Open channel flows of ideal incompressible fluid with velocity shear are considered in the long wave approximation. Nonlinear integrodifferential models of shallow flow with continuous vertical or horizontal velocity distribution are derived. It is shown that mathematically the models are equivalent and, consequently, the obtained early results for two-dimensional open channel flows with a vertical shear can be applied to the spatial flows with horizontally nonhomogeneous velocity field. Necessary and sufficient conditions of generalized hyperbolicity for the equations of motion are formulated, and the characteristic form of the system is calculated. In the case of a channel of constant width, the model reduces to the Riemann integral invariants which are conserved along the characteristics. Stability of shear flows in terms of hyperbolicity of the governing equations is studied. It is shown that the type of the equations of motion can change during the evolution of the flow, which corresponds to the long wave instability for a certain velocity field.
The concepts of sub- and supercritical flows are introduced for the long wave approximation model describing the steady-state horizontal-shear motions of an ideal incompressible fluid with a free boundary in a channel of variable cross-section. Fluid layer flows developed in a local channel contraction or expansion are analyzed. Continuous and discontinuous exact solutions describing different flow regimes are constructed and their properties are studied. Analytical solutions for flows with the formation of recirculation zones are obtained.
For the nonlinear kinetic equation describing the one-dimensional motion of quasineutral collisionless plasma, perturbation velocities are determined and conditions of generalized hyperbolicity are formulated. An example of verification of the hyperbolicity conditions is given, and an analogy with the well-known stability criterion for shear flows is noted. Exact (in particular, periodical) solutions of the model are constructed and interpreted physically for the class of traveling waves. It is shown that traveling waves are stable in the linear approximation only in the case of an insignificant change in the electric potential. Differential conservation laws approximating the basic integrodifferential equation are proposed. These laws are used to perform numerical calculations of wave propagation, which show the possibility of turnover of the kinetic distribution function.
These results are obtained jointly with V.Yu. Liapidevskii and A.K. Khe.