Mathematical Analysis and Numerical Methods for Science and Technology [electronic resource] : Volume 6 Evolution Problems II / by Robert Dautray, Jacques-Louis Lions
- XIX. The Linearised Navier-Stokes Equations -- §1. The Stationary Navier-Stokes Equations: The Linear Case -- §2. The Evolutionary Navier-Stokes Equations: The Linear Case -- §3. Additional Results and Review -- XX. Numerical Methods for Evolution Problems -- §1. General Points -- §2. Problems of First Order in Time -- §3. Problems of Second Order in Time -- §4. The Advection Equation -- §5. Symmetric Friedrichs Systems -- §6. The Transport Equation -- §7. Numerical Solution of the Stokes Problem -- XXI. Transport -- §1. Introduction. Presentation of Physical Problems -- §2. Existence and Uniqueness of Solutions of the Transport Equation -- §3. Spectral Theory and Asymptotic Behaviour of the Solutions of Evolution Problems -- §4. Explicit Examples -- §5. Approximation of the Neutron Transport Equation by the Diffusion Equation -- Perspectives -- Orientation for the Reader -- List of Equations -- Table of Notations -- Cumulative Index of Volumes 1-6 -- of Volumes 1-5.
- 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.
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