The main topics reflect the fields of mathematics in which
Professor O.A. Ladyzhenskaya obtained her most influential results.
One of the main topics considered in the volume is the
Navier-Stokes equations. This subject is investigated in many
different directions. In particular, the existence and uniqueness
results are obtained for the Navier-Stokes equations in spaces of
low regularity. A sufficient condition for the regularity of
solutions to the evolution Navier-Stokes equations in the
three-dimensional case is derived and the stabilization of a
solution to the Navier-Stokes equations to the steady-state
solution and the realization of stabilization by a feedback
boundary control are discussed in detail. Connections between the
regularity problem for the Navier-Stokes equations and a backward
uniqueness problem for the heat operator are also clarified.
Generalizations and modified Navier-Stokes equations modeling
various physical phenomena such as the mixture of fluids and
isotropic turbulence are also considered. Numerical results for the
Navier-Stokes equations, as well as for the porous medium equation
and the heat equation, obtained by the diffusion velocity method
are illustrated by computer graphs.
Some other models describing various processes in continuum
mechanics are studied from the mathematical point of view. In
particular, a structure theorem for divergence-free vector fields
in the plane for a problem arising in a micromagnetics model is
proved. The absolute continuity of the spectrum of the elasticity
operator appearing in a problem for an isotropic periodic elastic
medium with constant shear modulus (the Hill body) is established.
Time-discretizationproblems for generalized Newtonian fluids are
discussed, the unique solvability of the initial-value problem for
the inelastic homogeneous Boltzmann equation for hard spheres, with
a diffusive term representing a random background acceleration is
proved and some qualitative properties of the solution are studied.
An approach to mathematical statements based on the Maxwell model
and illustrated by the Lavrent'ev problem on the wave formation
caused by explosion welding is presented. The global existence and
uniqueness of a solution to the initial boundary-value problem for
the equations arising in the modelling of the tension-driven
Marangoni convection and the existence of a minimal global
attractor are established. The existence results, regularity
properties, and pointwise estimates for solutions to the Cauchy
problem for linear and nonlinear Kolmogorov-type operators arising
in diffusion theory, probability, and finance, are proved. The
existence of minimizers for the energy functional in the Skyrme
model for the low-energy interaction of pions which describes
elementary particles as spatially localized solutions of nonlinear
partial differential equations is also proved.
Several papers are devoted to the study of nonlinear elliptic
and parabolic operators. Versions of the mean value theorems and
Harnack inequalities are studied for the heat equation, and
connections with the so-called growth theorems for more general
second-order elliptic and parabolic equations in the divergence or
nondivergence form are investigated. Additionally, qualitative
properties of viscosity solutions of fully nonlinear partial
differential inequalities of elliptic and degenerate elliptic type
areclarified. Some uniqueness results for identification of
quasilinear elliptic and parabolic equations are presented and the
existence of smooth solutions of a class of Hessian equations on a
compact Riemannian manifold without imposing any curvature
restrictions on the manifold is established.
General
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