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The object ofthis chapter is to present a certain number ofresults on the linearised Navier-Stokes equations. The Navier-Stokes equations, which describe the motion of a viscous, incompressible fluid were introduced already, from the physical point of view, in §1 of Chap. IA. These equations are nonlinear. We study here the equations that emerge on linearisation from the solution (u = 0, p = 0). This is an interesting exercise in its own right. It corresponds to the case of a very slow flow, and also prepares the way for the study of the complete Navier-Stokes equations. This Chap. XIX is made up of two parts, devoted respectively to linearised stationary equations (or Stokes' problem), and to linearised evolution equations. Questions of existence, uniqueness, and regularity of solutions are considered from the variational point of view, making use of general results proved elsewhere. The functional spaces introduced for this purpose are themselves of interest and are therefore studied comprehensively.

Inhaltsverzeichnis

Frontmatter

Chapter XIX. The Linearised Navier-Stokes Equations

Abstract
The object of this chapter is to present a certain number of results on the linearised Navier-Stokes equations. The Navier-Stokes equations, which describe the motion of a viscous, incompressible fluid were introduced already, from the physical point of view, in §1 of Chap. IA. These equations are nonlinear. We study here the equations that emerge on linearisation from the solution (u = 0, p = 0). This is an interesting exercise in its own right. It corresponds to the case of a very slow flow, and also prepares the way for the study of the complete Navier-Stokes equations.
Robert Dautray, Jacques-Louis Lions

Chapter XX. Numerical Methods for Evolution Problems

Abstract
Chapter XX is dedicated to the approximation of evolution problems of the following type, where H is a Hilbert space:
$$ \left\{ \begin{array}{l} find u \in C^0 ([0,T];H) satisfying, in a weak sense \\ (or strong if u(t) \in D(A)), \\ \frac{{du(t)}}{{dt}} + Au(t) = f(t) \\ u(0) = u_0 , \\ \end{array} \right. $$
(1.1)
where for example, A being a partial differential operator with domain D(A) ⊂ H.
Robert Dautray, Jacques-Louis Lions

Chapter XXI. Transport

Abstract
The problems of neutron transport have been presented in Chap. 1A, §5. We recall the essentials below.
Robert Dautray, Jacques-Louis Lions

Backmatter

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