The Navier-Stokes Equations

This book offers an elementary, self-contained approach to the mathematical theory of viscous, incompressible fluid in a domain of the Euclidian space, described by the equations of Navier-Stokes. It is the first to provide a systematic treatment of the subject. It is designed for students familiar with basic tools in Hilbert and Banach spaces, but fundamental properties of, for example, Sobolev spaces, are collected in the first two chapters.

Contenu

I Introduction.- 1 Basic notations.- 1.1 The equations of Navier-Stokes.- 1.2 Further notations.- 1.3 Linearized equations.- 2 Description of the functional analytic approach.- 2.1 The role of the Stokes operator A.- 2.2 The stationary linearized case.- 2.3 The stationary nonlinear case.- 2.4 The nonstationary linearized case.- 2.5 The full nonlinear case.- 3 Function spaces.- 3.1 Smooth functions.- 3.2 Smoothness properties of the boundary ??.- 3.3 Lq-spaces.- 3.4 The boundary spaces Lq (??).- 3.5 Distributions.- 3.6 Sobolev spaces.- II Preliminary Results.- 1 Embedding properties and related facts.- 1.1 Poincaré inequalities.- 1.2 Traces and Green's formula.- 1.3 Embedding properties.- 1.4 Decomposition of domains.- 1.5 Compact embeddings.- 1.6 Representation of functionals.- 1.7 Mollification method.- 2 The operators ? and div.- 2.1 Solvability of div v = g and ?p = f.- 2.2 A criterion for gradients.- 2.3 Regularity results on div v = g.- 2.4 Further results on the equation div v = g.- 2.5 Helmholtz decomposition in L2-spaces.- 3 Elementary functional analytic properties.- 3.1 Basic facts on Banach spaces.- 3.2 Basic facts on Hilbert spaces.- 3.3 The Laplace operator ?.- 3.4 Resolvent and Yosida approximation.- III The Stationary Navier-Stokes Equations.- 1 Weak solutions of the Stokes equations.- 1.1 The notion of weak solutions.- 1.2 Embedding properties of $$\widehat W_{0,\sigma }^{1,2}(\Omega )$$.- 1.3 Existence of weak solutions.- 1.4 The nonhomogeneous case div u = g.- 1.5 Regularity properties of weak solutions.- 2 The Stokes operator A.- 2.1 Definition and properties.- 2.2 The square root $$ A^{\tfrac{1} {2}} $$ of A.- 2.3 The Stokes operator A in ?n.- 2.4 Embedding properties of D(A?).- 2.5 Completion of the space D(A?).- 2.6 The operator $$ A^{ - \tfrac{1} {2}} $$P div.- 3 The stationary Navier-Stokes equations.- 3.1 Weak solutions.- 3.2 The nonlinear term u · ?u.- 3.3 The associated pressure p.- 3.4 Existence of weak solutions in bounded domains.- 3.5 Existence of weak solutions in unbounded domains.- 3.6 Regularity properties for the stationary nonlinear system.- 3.7 Some uniqueness results.- IV The Linearized Nonstationary Theory.- 1 Preliminaries for the time dependent linear theory.- 1.1 The nonstationary Stokes system.- 1.2 Basic spaces for the time dependent theory.- 1.3 The vector valued operator $$ \tfrac{d} {{dt}} $$.- 1.4 Time dependent gradients ?p.- 1.5 A special solution class of the homogeneous system.- 1.6 The inhomogeneous evolution equation u? + Au = f.- 2 Theory of weak solutions in the linearized case.- 2.1 Weak solutions.- 2.2 Equivalent formulation and approximation.- 2.3 Energy equality and strong continuity.- 2.4 Representation formula for weak solutions.- 2.5 Basic estimates of weak solutions.- 2.6 Associated pressure of weak solutions.- 2.7 Regularity properties of weak solutions.- V The Full Nonlinear Navier-Stokes Equations.- 1 Weak solutions.- 1.1 Definition of weak solutions.- 1.2 Properties of the nonlinear term u · ?u.- 1.3 Integral equation for weak solutions and weak continuity.- 1.4 Energy equality and strong continuity.- 1.5 Serrin's uniqueness condition.- 1.6 Integrability properties of weak solutions in space and time,the scale of Serrin's quantity.- 1.7 Associated pressure of weak solutions.- 1.8 Regularity properties of weak solutions.- 2 Approximation of the Navier-Stokes equations.- 2.1 Approximate Navier-Stokes system.- 2.2 Properties of approximate weak solutions.- 2.3 Regularity properties of approximate weak solutions.- 2.4 Smooth solutions of the Navier-Stokes equationswith "slightly" modified forces.- 2.5 Existence of approximate weak solutions.- 2.6 Uniform norm bounds of approximate weak solutions.- 3 Existence of weak solutions of the Navier-Stokes system.- 3.1 Main result.- 3.2 Preliminary compactness results.- 3.3 Proof of Theorem 3.1.1.- 3.4 Weighted energy inequalities and time decay.- 3.5 Exponential decay for domains for which thePoincarü inequality holds.- 3.6 Generalized energy inequality.- 4 Strong solutions of the Navier-Stokes system.- 4.1 The notion of strong solutions.- 4.2 Existence results.