IRROTATIONAL FLOW OF FRICTIONLESS FLUIDS, MOSTLY OF INVARIABLE DENSITY

By V. GAENO

CONTENTS INCOMPRESSIBLE MULTIPOLAR FLUID DYNAMICS INTRODUCTION: THE STOKES CONSTITUTIVE LAW AND THE NAVIER–STOKES EQUATIONS MULTIPOLAR FLUID DYNAMICS INTRODUCTION BALANCE EQUATIONS AND THE CLAUSIUS-DUHEM INEQUALITY THE GENERAL CONSTITUTIVE EQUATION FOR MULTIPOLAR VISCOUS FLUIDS CONSEQUENCES OF THE CLAUSIUS-DUHEM INEQUALITY LINEAR MULTIPOLAR VISCOUS FLUIDS THE LINEAR BIPOLAR FLUID THE NONLINEAR BIPOLAR FLUID THE CONSTITUTIVE RELATION FOR A VISCOUS, INCOMPRESSIBLE NONLINEAR BIPOLAR FLUID THE STRUCTURE OF THE HIGHER-ORDER BOUNDARY CONDITIONS 1: VIRTUAL WORK WITHOUT THE INCOMPRESSIBILITY CONSTRAINT STRUCTURE OF THE HIGHER-ORDER BOUNDARY CONDITIONS 2: VIRTUAL WORK SUBJECT TO THE INCOMPRESSIBILITY CONSTRAINT ELEMENTARY EXAMPLES OF INCOMPRESSIBLE NONLINEAR BIPOLAR FLUID FLOW STEADY PLANE POISEUILLE FLOW STEADY (PROPER) POISEUILLE FLOW IN A CIRCULAR CYLINDER PLANE COUETTE FLOW OTHER EXTENSIONS AND GENERALIZATIONS OF THE NAVIER–STOKES MODEL SOME EXAMPLES OF VISCOUS INCOMPRESSIBLE FLOW DESCRIBED BY THE DIPOLAR, CAMASSA-HOLM, AND GREEN AND NAGHDI MODELS EXAMPLES OF VISCOUS INCOMPRESSIBLE FLOW FOR DIPOLAR EQUATIONS SOME APPLICATIONS OF THE VISCOUS CAMASSA-HOLM EQUATIONS FLUID DYNAMICS PROBLEMS FOR THE EXTENDED GREEN-NAGHDI VISCOUS FLOW MODEL A CATALOG OF EXPERIMENTAL RESULTS WHICH ARE INCONSISTENT WITH THE STOKES’ HYPOTHESIS COUETTE FLOW POISEUILLE FLOW PLANE POISEUILLE FLOW OF INCOMPRESSIBLE BIPOLAR VISCOUS FLUIDS EXISTENCE, UNIQUENESS, AND CONTINUOUS DEPENDENCE FOR STEADY POISEUILLE FLOW ESTIMATES AND GENERALIZED REYNOLDS NUMBERS FOR STEADY PLANE POISEUILLE FLOW GENERALIZED REYNOLDS NUMBERS FOR PLANE POISEUILLE FLOW OF A BIPOLAR FLUID THE POISEUILLE FLOW FOR D 1 D ESTIMATES FOR THE POISEUILLE FLOW UYI ; 1/ WHEN ; 1 ¤ UNIQUENESS OF STEADY POISEUILLE FLOW IN THE CLASS OF EQUILIBRIUM FLOWS BETWEEN PARALLEL PLATES THE CONDITION W 2 H 4A/ KEY LEMMAS FOR NONLINEAR VISCOSITY AND THE POISEUILLE FLOW IN A UNIQUENESS OF SOLUTIONS EXISTENCE AND ASYMPTOTIC STABILITY OF TIME-DEPENDENT POISEUILLE FLOWS SOME PRELIMINARY ESTIMATES FOR AN ASSOCIATED PARABOLIC PROBLEM EXISTENCE OF WEAK SOLUTIONS UNIQUENESS AND STABILITY OF SOLUTIONS INCOMPRESSIBLE BIPOLAR FLUID DYNAMICS: EXAMPLES OF OTHER FLOWS AND GEOMETRIES FLOW BETWEEN ROTATING CYLINDERS THE NONLINEAR BOUNDARY-VALUE PROBLEM FOR PROPER COUETTE FLOW OF AN INCOMPRESSIBLE BIPOLAR FLUID THE PROPER COUETTE FLOW: D 1 D EXISTENCE, UNIQUENESS, AND CONTINUOUS DEPENDENCE BUBBLE STABILITY IN AN INCOMPRESSIBLE NON-NEWTONIAN VISCOUS FLUID DYNAMICS OF A SPHERICAL VAPOR CAVITY IN AN INCOMPRESSIBLE BIPOLAR, VISCOUS FLUID: D 1 D LINEARIZED DYNAMICS FOR A SPHERICAL CAVITY IN A NON-NEWTONIAN FLUID LIAPUNOV THEORY AND NONLINEAR STABILITY EXTERIOR FLOW OF A BIPOLAR VISCOUS FLUID IN THE PLANE FORMULATION OF THE EXTERIOR FLOW PROBLEM FOR A BIPOLAR FLUID SOLUTION OF THE EXTERIOR PROBLEM IN A TRUNCATED DOMAIN SOLUTION OF THE EXTERIOR PROBLEM IN AN UNBOUNDED DOMAIN EXISTENCE OF A DRAG FORCE FLOW OVER NON-SMOOTH BOUNDARIES EXISTENCE AND UNIQUENESS OF SOLUTIONS PERTURBATION OF DOMAIN RESULTS REGULARITY RESULTS IN POLYGONAL DOMAINS SOME SPECI?C REGULARITY RESULTS GENERAL EXISTENCE AND UNIQUENESS THEOREMS FOR INCOMPRESSIBLE BIPOLAR AND NON-NEWTONIAN FLUID FLOW EXISTENCE, UNIQUENESS, AND STABILITY OF SOLUTIONS TO THE INITIAL-BOUNDARY VALUE PROBLEM FOR BIPOLAR VISCOUS FLUIDS EXISTENCE OF A WEAK SOLUTION REGULARITY OF THE SOLUTION UNIQUENESS OF THE WEAK SOLUTION STABILITY OF THE SOLUTION WEAK AND MEASURE-VALUED SOLUTIONS FOR INCOMPRESSIBLE BIPOLAR FLUIDS WITH 1 D YOUNG MEASURE-VALUED SOLUTIONS YOUNG MEASURES ARE DIRAC AND THE WEAK SOLUTIONS ARE REGULAR FOR P 2, N D YOUNG MEASURES ARE DIRAC FOR 3=2 < P < 2, N D YOUNG MEASURES ARE DIRAC AND THE WEAK SOLUTIONS ARE REGULAR FOR 7=3 P < 6, N D YOUNG MEASURES ARE DIRAC AND THE WEAK SOLUTIONS ARE REGULAR FOR 11=5 P < 7=3, N D YOUNG MEASURES ARE DIRAC FOR 2 P < 11=5, N D YOUNG MEASURES ARE DIRAC FOR 9=5 < P < 2, N D EXISTENCE AND UNIQUENESS FOR INCOMPRESSIBLE FLOW IN AN UNBOUNDED CHANNEL FORMULATION OF THE PROBLEM IN HILBERT SPACE AND SOME PRELIMINARY LEMMAS THE EXISTENCE PROBLEM FOR A SEQUENCE OF APPROXIMATIONS EXISTENCE OF INCOMPRESSIBLE, BIPOLAR, VISCOUS FLOW IN A UNIQUENESS OF BIPOLAR FLOW IN A RELATED WORK ON EXISTENCE AND UNIQUENESS FOR NAVIER–STOKES AND SOME GENERALIZATIONS EXISTENCE AND UNIQUENESS FOR THE INCOMPRESSIBLE NAVIER–STOKES EQUATIONS EXISTENCE AND UNIQUENESS FOR SOLUTIONS OF THE LADYZHENSKAYA MODEL EQUATIONS EXISTENCE AND UNIQUENESS RESULTS FOR VISCOUS FLOW MODELS WITH ARTI?CIAL VISCOSITY EXISTENCE, UNIQUENESS, AND STABILITY THEOREMS FOR MULTIPOLAR FLUIDS OF GRADE GLOBAL REGULARITY OF SOLUTIONS TO THE VISCOUS CAMASSA-HOLM EQUATIONS ATTRACTORS FOR INCOMPRESSIBLE BIPOLAR AND NON-NEWTONIAN FLOWS: BOUNDED DOMAINS AND SPACE PERIODIC PROBLEMS LINEARIZED STABILITY OF VISCOUS INCOMPRESSIBLE BIPOLAR EQUATIONS LINEARIZED BIPOLAR EQUATIONS BASIC ESTIMATES FOR THE RATE OF DEFORMATION A SUF?CIENT CONDITION FOR LINEARIZED STABILITY BOUNDS FOR THE DIMENSIONS OF ATTRACTORS FOR NONLINEAR BIPOLAR FLUID FLOW (0 ? < 1) SOME PRELIMINARY CONCEPTS ABSORBING SETS IN H AND H 2/ AND EXISTENCE OF A GLOBAL COMPACT ATTRACTOR A1 THE UNIFORM DIFFERENTIABILITY OF S 1T/ UNIFORM BOUNDEDNESS OF L1TI U0/ HAUSDORFF AND FRACTAL DIMENSIONS OF THE GLOBAL ATTRACTOR A1 ATTRACTORS FOR THE BIPOLAR AND NON-NEWTONIAN PROBLEMS (1 < ? < 0) EXISTENCE OF ABSORBING SETS AND MAXIMAL COMPACT GLOBAL ATTRACTORS FOR S 1T/, 1 UNIFORM DIFFERENTIABILITY OF S ON THE MAXIMAL COMPACT GLOBAL ATTRACTOR A1, 1 > UNIFORM BOUNDEDNESS OF THE FRÉCHET DIFFERENTIAL L1TI U0/, U0 2 A1 BOUNDS FOR DHA1/ AND DF A1/, 1 > LOWER SEMICONTINUITY OF THE ATTRACTORS FOR NONLINEAR BIPOLAR EQUATIONS THE CONVERGENCE PROBLEM A BASIC ESTIMATE FOR SOLUTIONS OF THE SPACE-PERIODIC BIPOLAR PROBLEM CONVERGENCE OF A1 TO A0 AS 1 ! INERTIAL MANIFOLDS, ORBIT SQUEEZING, AND ATTRACTORS FOR BIPOLAR FLOW IN UNBOUNDED CHANNELS INERTIAL MANIFOLDS FOR INCOMPRESSIBLE BIPOLAR VISCOUS FLOW AN OUTLINE OF THE METHODOLOGY FOR PROVING THE EXISTENCE OF AN INERTIAL MANIFOLD THE LIPSCHITZ PROPERTY THE SQUEEZING PROPERTY FOR THE ORBITS OF S A FIXED POINT THEOREM EXISTENCE OF THE INERTIAL MANIFOLD M THE L2 SQUEEZING PROPERTY FOR BIPOLAR VISCOUS FLUIDS AN ESTIMATE FOR NONLINEAR VISCOSITY AN ESTIMATE FOR THE QUOTIENT JJU VJJV =JJU VJJL2/ THE L2Œ0; T I H 3// NORM OF U: AN ALTERNATE DERIVATION OF THE ESTIMATE THE L2 SQUEEZING PROPERTY FOR THE ORBITS OF S EXISTENCE OF MAXIMAL COMPACT ATTRACTORS FOR INCOMPRESSIBLE VISCOUS BIPOLAR FLUIDS IN UNBOUNDED CHANNELS THE ABSORBING SETS IN NH AND H EXISTENCE OF A GLOBAL ATTRACTOR FOR FLOW IN AN UNBOUNDED CHANNEL SOME RELATED WORK ON ATTRACTORS AND INERTIAL MANIFOLDS FOR INCOMPRESSIBLE BIPOLAR AND NON-NEWTONIAN FLOW