The equations of fluid mechanics. Continuous description of a fluid; The continuous medium assumption : density; Lagrangian and Eulerian coordinates ; The transport theorem ; Evolution equations; Balance equations; Cauchy's stress theorem; Evolution equations revisited ; Fundamental laws : Newtonian fluids and thermodynamics laws; Fluids at rest; Newton's hypothesis; Consequences of the second law of thermodynamics; Equation for the specific internal energy; Formulation in entropy and temperature ; Summary of the equations ; Incompressible models; The incompressibility assumption; Overview of the incompressible models ; Some exact steady solutions; Poiseuille flow in a pipe; Planar shear flow; Couette flow between two cylinders
Analysis tools. Main notation ; Fundamental results from functional analysis; Banach spaces; Weak and weak-* convergences; Lebesgue spaces; Partitions of unity; A short introduction to distribution theory; Lipschitz continuous functions ; Basic compactness results; Compact sets in function spaces; Compact maps; The Schauder fixed-point theorem ; Functions of one real variable; Differentiation and antiderivatives; Differential inequalities and Gronwall's lemma ; Spaces of Banach-valued functions; Definitions and main properties; Regularity in time; Compactness theorems; Banach-valued Fourier transform ; Some results in spectral analysis of unbounded operators; Definitions; Elementary results of spectral theory; Applications to the semigroup theory
Sobolev spaces. Domains; General definitions; Lipschitz domains ; Sobolev spaces on Lipschitz domains; Definitions; Mollifying operators and Friedrichs commutator estimates; Change of variables; Extension operator; Trace and trace lifting operators; Duality theory for Sobolev spaces; Translation estimates; Sobolev embeddings; Poincaré and Hardy inequalities; Domains of first-order differential operators ; Calculus near the boundary of domains; Local charts description of the boundary; Distance to the boundary : projection on the boundary; Regularised distance; Parametrisation of a neighborhood of ...; Tangential Sobolev spaces; Differential operators in tangential/normal coordinates ; The Laplace problem; Dirichlet boundary conditions; Neumann boundary conditions
Steady Stokes equations. Nečas inequality; Proof of the inequality; Related Poincaré inequalities ; Characterisation of gradient fields : De Rham's theorem ; The divergence operator and related spaces; Right-inverse for the divergence; The space H div ...; Divergence-free vector fields : Leray decomposition ; The curl operator and related spaces; Poincaré's theorems; The space H curl ...; Kernel and image of the curl operator; The div/curl problem ; The Stokes problem; Well-posedness of the Stokes problem; Stokes operator; The unsteady Stokes problem; Penalty approximation of the Stokes problem ; Regularity of the Stokes problem; First degree of regularity; Higher-order regularity; Lq theory of the Stokes problem; Regularity for the div/curl problem ; The Stokes problem with stress boundary conditions; The Stokes-Neumann problem; Regularity properties; Stress boundary conditions ; The interface Stokes problem; Existence and uniqueness; Regularity of the solution ; The Stokes problem with vorticity boundary conditions; Preliminaries; A vector Laplace problem; The Stokes problem
Navier-Stokes equations for homogeneous fluids. Leray's theorem; Properties of the inertia term; Weak formulations of the Navier-Stokes equations; Existence and uniqueness of weak solutions; Kinetic energy evolution; Existence and regularity of the pressure ; Strong solutions; New estimates; The two-dimensional case; The three-dimensional case; Parabolic regularity properties; Regularisation over time ; The steady Navier-Stokes equations; The case of homogeneous boundary conditions; The case of nonhomogeneous boundary conditions; Uniqueness for small data; Asymptotic stability of steady solutions
Nonhomogeneous fluids. Weak solutions of the transport equation; Setting of the problem; Trace theorem : renormalisation property; The initial- and boundary-value problem; Stability theorem ; The nonhomogeneous incompressible Navier-Stokes equations; Main result; Approximate problem; Estimates for the approximate solution; End of the proof of the existence theorem; The case without vacuum
Boundary conditions modelling. Outflow boundary conditions; Setting up the model; Existence and uniqueness ; Dirichlet boundary conditions through a penalty method; A simple example of a boundary layer; Statement of the main result; Formal asymptotic expansion; Well-posedness of profile equations; Convergence of the asymptotic expansion
A, Classic differential Operators. Tin- scalar and vector cases; Definitions; Useful formulas ; Extension to second-order tensors
B, Thermodynamics supplement. Heat capacity ; The first law of thermodynamics : internal energy ; The second law of thermodynamics; Entropy; Internal energy calculation ; Specific variables