
Nonlinear problems of hydrodynamics
Syllabus
 Theoretical mechanics of an ideal fluid
 The least action principle in hydrodynamics. Lagrangian and Eulerian description
of fluid motion. The continuity equation. Functionals and variational derivatives. The
variational EulerLagrange equation for fluid elements. The relabeling symmetry of a fluid
medium and actual dependence of the Lagrangian functional on the Eulerian fields only.
Examples of Lagrangians: the usual hydrodynamics, the relativistic hydrodynamics, the two
fluid nonrelativistic plasma model, the electron magnetic hydrodynamics. General structure
of the equations of motion in the Eulerian description  the generalized Euler equation.
 Hamiltonian formalism in the fluid mechanics. Definition of the canonical momentum
of a fluid element. The Hamiltonian functional. Degenerated noncanonical Poisson
bracket. The helicity functional  an example of a Casimir invariant. The frozenin property
of the generalized vorticity. The Kelvin theorem about conservation of circulation and
the Cauchy's invariant as consequences of the relabeling symmetry. Variational principles
determining the dynamics of frozenin vortex structures of a given topology. The Clebsch
variables as an example of canonical variables. The Lagrangian of the system in the vortex
line representation.
 Potential flows.
 The sound. The Hamiltonian of sound waves. Quadratic approximation and the
normal variables. Nonlinear effects, calculation of the matrix elements of nonlinear wave
interaction.
 Onedimensional gas dynamics. Characteristics. The Riemann invariants.
The Hodograph method. Breaking of simple waves. Shock waves. Discontinuities in the initial conditions.
The "shallow water" theory. The Burgers equation.
 Waves at the free surface. General form of the Lagrangian for incompressible potential
flows with a free surface. Bubbles and drops. Canonical variables η and ψ . The dispersion
relation for surface waves. Asymptotic expansion of the Hamiltonian in powers of the small
nonlinearity parameter. Competition between dispersion and nonlinearity in traveling wave,
the KortewegdeVries equation, solitons. Conformal variables in the planar problem with a
free boundary. The problem of exciting of waves by the wind.
 Nonlinear resonant wave interaction. Reduction of Hamiltonians.
The nwave problem. Explosive instability. Nonlinear Schroedinger equation for wave envelope of weakly
nonlinear quasimonochromatic wave. Wave collapses. Weakly supercritical instabilities,
formation of structures. Wave turbulence. Kinetic equation. Cascades of energy and wave
action.
 Vortex structures in ideal fluid. Vortex sheets, instability of a tangent discontinuity.
Twodimensional flows with piecewise constant vorticity. Thin vortex filaments in 3D space
and point vortices in 2D plane. The Hamiltonian and dynamics of point vortices, application
of the theory of analytical functions. The Hasimoto transform and integrable local
induction approximation in a single vortex filament dynamics. The dispersion law for small
perturbations of a straight vortex filament. The Crow instability for two antiparallel vortex
filaments. The problem of finitetime singularity formation from smooth initial data in the
Euler equation.
 Viscous fluid
 Laminar flows. Equations of motion with taking into account dissipative processes 
the NavierStokes equations. The energy dissipation. The flow in pipe. The fluid motion
between two rotating cylinders. The similarity law. LowReynoldsnumber flows. Flow around
sphere, the Stokes formula. Flow around cylinder, Oseen equation. Laminar trail. Laminar
boundary layer, Prandtl equations. Thermal conductivity in a fluid. Free convection.
Convective instability of fluid.
 Turbulence. The problem of stability of stationary fluid flow. Chaos in dynamical
systems, strange attractor. Developed turbulence. The Kolmogorv's ideas about cascade.
Correlation functions of the velocity field. Abnormal dimensionality.
 Different examples of hydrodynamical phenomena
 Superfluid hydrodynamics. Spectrum of elementary excitations in a quantum Bose
fluid and the superfluidity phenomenon. Twovelocity hydrodynamics. Quantized vortex
filaments. Vortex grid, Tkachenko waves. Superfluid turbulence. Weaklynonideal Bose
gas at zero temperature, the GrossPitaevskii equation. Instability of the condensate and
collapse of the wave function in the attraction case.
 Plasma dynamics.
 Kinetic plasma description. Collisionless plasma.
Selfconsistent field. Vlasov equation. Various kinds of waves in plasma. Landau damping. Relaxation of
an initial perturbation. "Echo" in plasma. Adiabatic capture of electrons. Particle collisions in plasma.
Instabilities in plasma.
 Hydrodynamic plasma description. Manycomponent hydrodynamical plasma models
and their different limit cases: quasineutral plasma, magnetic hydrodynamics (MHD), Hall
MHD, electron MHD.
