The Navier-Stokes Equations, Softcover reprint of the original 1st ed. 2001
An Elementary Functional Analytic Approach
Birkhäuser Advanced Texts Basler Lehrbücher Series

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.