Compressible Navier-Stokes equations. Theory and shape optimization.
Résumé
The monograph is devoted to the study of boundary value problems for equations
of viscous gas dynamics, named compressible Navier-Stokes equations. The math-
ematical theory of Navier -Stokes equations is very interesting in its own right, but
its principal significance lies in the central role Navier-Stokes equations now play
in fluid dynamics. Most of this book concentrates on those aspects of the theory
that have proven useful in applications. Although the theory is satisfactory in what concerns local time behavior
and small data, many issues of global behavior of solutions for large data are far
from being understood. There are a vast range of unsolved problems concerning
questions such as regularity of solutions to compressible Navier-Stokes equations,
the theory of weak solutions for small adiabatic exponents, existence theory for
heat conducting fluids. Existence theory.
This issue is important since no progress in the mathematical
theory of Navier-Stokes equations can be made without answering the basic ques-
tions on their well-posedness. We focus on existence results for the inhomogeneous
in/out flow problem, in particular the problem of the flow around a body placed in
a finite domain. Notice that the majority of known results are related to viscous
gas flows in domains bounded by impermeable walls. In/out flow problems are
still poorly investigated. With applications to
shape optimization theory in mind, we consider the problem of the flow around a
body placed in a bounded domain for small Mach and Reynolds numbers.
Stability of solutions with respect to nonsmooth data and domain perturbations.
Propagation of rapid oscillations in compressible fluids.
In compressible viscous
flows, any irregularities in the initial and boundary data are transferred inside
the flow domain along fluid particle trajectories. The transport of singularities in
viscous compressible flows was studied by Hoff [56, 57]. In this book we discuss the
propagation of rapid oscillations of the density, which can be regarded as acoustic
waves. The main idea is that any rapidly oscillating sequence is associated with
some stochastic field named the Young measure (see Tartar and Perthame
for basic ideas). We establish that the distribution function of this stochastic
field satisfies a kinetic equation of a special form, which leads to a rigorous model
for propagation of nonlinear acoustic waves. Notice that oscillations can be induced
not only by oscillations of initial and boundary data, but also by irregularities of
the boundary of the flow domain.
Domain dependence of solutions to compressible Navier-Stokes equations.
This issue is important because of applications to shape optimization theory.
The first global result on domain dependence of solutions to compressible Navier-
Stokes equations is due to Feireisl, who proved that the set of solutions to
compressible Navier-Stokes equations is compact provided the set of flow domains
is compact in the Kuratowski-Mosco topology and their boundaries have “uni-
formly small” volumes. We prove that the compactness result holds true if the
set of flow domains is compact in the Kuratowski-Mosco topology, and also that
some cost functionals, such as the drag and the work of hydrodynamical forces,
are continuous in this topology. With applications to shape optimization in mind,
we consider the shape differentiability of strong solutions and give formulae for
the shape derivative of the drag functional.