Contents -- Preface -- Thomas Y. Hou, Xinwei Yu: Introduction to the Theory of Incompressible Inviscid Flows -- Abstract -- 1 Introduction -- 2 Derivation and exact solutions -- 2.1 Derivation of the Euler equations -- 2.2 The Vorticity-Stream function formulation -- 2.2.1 Vorticity -- 2.2.2 Vorticity-Stream function formulation -- 2.2.3 2D Euler equations -- 2.3 Conserved quantities -- 2.3.1 Local conserved quantities -- 2.3.2 Global conserved quantities -- 2.4 Special flows -- 2.4.1 Axisymmetric flow -- 2.4.2 Radially (circularly) symmetric flow -- 2.4.3 Jets and strains -- 3 Local well-posedness of the 3D Euler equation -- 3.1 Analytical preparations -- 3.1.1 Sobolev spaces -- 3.1.2 Hodge decomposition and the Leray projection -- 3.1.3 The Aubin-Lions lemma -- 3.1.4 Calculus inequalities -- 3.1.5 Gronwall's inequality -- 3.2 Properties of mollifiers -- 3.3 Global existence of the mollified equation -- 3.4 Local existence of the Euler equations -- 4 The BKM blow-up criterion -- 4.1 The Beale-Kato-Majda criterion -- 4.2 Improvements of the BKM criterion -- 5 Recent global existence results -- 5.1 Sufficient conditions by Constantin-FeffermanMajda -- 5.2 Sufficient conditions by Deng-Hou-Yu -- 6 Lower dimensional models for the 3D Euler equations -- 6.1 1-D model -- 6.2 The 2-D QG equation -- 6.2.1 Existence and blow-up criteria -- 6.2.2 Global existence result by Constantin-Majda-Tabak -- 6.2.3 Global existence result by Cordoba and Fefferman -- 6.2.4 Final remarks about the QG equation -- 7 Vortex patch -- 7.1 The contour dynamics equation (eDE) -- 7.2 Levelset formulation and global existence -- References -- Denis Serre: Systems of Conservation Laws. Theory, Numerical Approximation and Discrete Shock Profiles. -- 1 Hyperbolic systems of conservation laws -- 1.1 The Cauchy problem: classical solutions -- 1.1.1 Hyperbolicity -- 1.1.2 Entropies.

1.1.3 Local well-posedness in HS(jRd) -- 1.2 The Cauchy problem: weak solutions -- 1.2.1 Break-down of smooth solutions -- 1.2.2 Weak solutions -- 1.2.3 The Rankine-Hugoniot condition -- 1.2.4 Non-uniqueness of weak solutions -- 1.2.5 Entropy admissibility condition -- 1.2.6 The viscosity approach -- 1.2.7 The scalar case -- 1.3 Shock waves -- 1.3.1 The Hugoniot locus -- 1.3.2 Genuine nonlinearity -- 1.3.3 The Lax shock inequality -- 1.3.4 Viscous shock profiles -- 1.4 The Riemann problem -- 1.4.1 Rarefaction waves -- 1.4.2 Contact discontinuities -- 1.4.3 The theorem of Lax -- 1.5 Existence of viscous shock profiles -- 1.5.1 The scalar case -- 1.5.2 Reduction to a center manifold (bifurcation analysis) -- 1.5.3 Lax shocks -- 1.5.4 Under-compressive shocks -- 2 Finite difference schemes -- 2.1 Conservative schemes -- 2.1.1 Consistency -- 2.1.2 Order of accuracy -- 2.1.3 Linearized L2-stability -- 2.1.4 The Courant-Friedrichs-Lewy condition -- 2.1.5 Entropy-consistent schemes -- 2.2 Examples -- 2.2.1 The naive centered scheme -- 2.2.2 The Lax-Friedrichs scheme -- 2.2.3 The Lax-Wendroff scheme -- 2.2.4 The Godunov scheme -- 2.3 Schemes for scalar equations -- 2.3.1 Monotone schemes -- 2.3.2 Kutznetsov's error estimate -- 3 Discrete shock profiles -- 3.1 DSPs and conservation -- 3.1.1 The function Y -- 3.1.2 Scalar case: monotone schemes -- 3.2 Existence theory for rational N -- 3.2.1 DSPs for small steady Lax shocks -- 3.2.2 DSPs for steady Lax shocks: the Godunov scheme -- 3.2.3 What can go wrong? -- References -- Seiji Ukai, Tong Yang: Kinetic Theory and Conservation Laws: An Introduction. -- Abstract -- 1 Introduction -- 1.1 Overview -- 1.2 Boltzmann equation -- 1.2.1 Intuitive derivation -- 1.2.2 Collision invariants and H functional -- 1.2.3 Assumptions on cross-sections -- 1.2.4 Basic properties of the collision operators.

2 Expansions and their unification -- 2.1 Classical expansions -- 2.2 Unification by decomposition -- 3 Detour to hyperbolic conservation laws -- 3.1 Scalar conservation laws -- 3.2 Riemann problem for systems -- 3.3 Well-posedness theory for systems -- 3.3.1 Existence -- 3.3.2 Stability and uniqueness -- 3.4 Vanishing viscosity -- 4 Spectral analysis on the linearized Boltzmann operator -- 4.1 Smoothing properties of etA -- 4.2 Spectral properties of B -- 4.3 Decay rates of etB in Xf3 -- 4.4 Effect of external force -- 5 Global existence and convergence rates -- 5.1 Global existence -- 5.2 Optimal convergence rates -- 5.3 External force, revisited -- References -- Xiaoming Wang: Elementary Statistical Theories with Applications to Fluid Systems. -- 1 Introduction -- 1.1 Why statistical description -- 1.2 What characterizes statistical behavior -- 1.3 Relative stability of statistical solution over single trajectories -- 2 Stationary statistics -- 2.1 Invariant measures and stationary statistical solutions -- 2.2 Definition, existence -- 2.3 Ergodicity -- 2.4 Invariant measure, stationary statistical solution and attractor -- 2.5 Dependence on parameters -- 2.6 Regular perturbation -- 2.7 Singular perturbation -- 2.8 Remarks on direct applications -- 2.8.1 An application to NSE: energy dissipation rate per unit mass -- 2.8.2 An application to RBC: heat transfer in the vertical direction (N usselt number) -- 2.9 Maximum entropy principle -- 2.10 Application to ODEs -- 2.11 Application to basic geophysical systems -- 3 Remarks on time dependent statistics -- 3.1 Definition, existence -- 3.2 Applications to NSE -- 3.2.1 Reynolds equation for the average flow -- 3.2.2 Moment closure problem -- 3.2.3 Fluctuation dissipation theory (FDT) -- Appendix: some useful theorems -- References -- Yuxi Zheng: The Compressible Euler System in Two Space Dimensions.

Abstract -- Introduction -- 1 Physical phenomena and mathematical problems -- 1.1 Euler system in n dimensions -- 1.2 Phenomena -- 1.3 Mathematical treatment -- 1.4 Paradoxes -- 1.5 The 8th millennium priceless problem -- 2 Characteristic decomposition of the pseudo-steady case -- 2.1 Riemann problems -- 2.2 Isentropic system -- 2.3 Some explicit solutions -- 2.4 A characteristic decomposition -- 2.4.1 Introduction to the method of characteristics -- 2.4.2 Decomposition -- 3 The hodograph transformation and the interaction of rarefaction waves -- 3.1 Primary system -- 3.2 The concept of hodograph transformation -- 3.2.1 The hodograph transformation for the pseudo-steady Euler -- 3.2.2 Steady Euler -- 3.2.3 Similarity to one-dimensional problems -- 3.3 Characteristics in both planes -- 3.4 Phase space system of equations -- 3.5 Planar rarefaction waves -- 3.6 The gas expansion problem -- 3.6.1 Planar rarefaction waves in a given direction -- 3.6.2 A wedge of gas -- 3.6.3 A wedge of gas in the hodograph plane -- 3.6.4 Local existence -- 3.6.5 Statement of main existence -- 3.6.6 The maximum norm estimate on (o:,(3,c) -- 3.6.7 Gradient estimates and the proof of Theorem 3.3 -- 3.6.8 Inversion -- 3.6.9 Proof of Theorem 3.4 -- 3.6.10 Properties of the solutions -- 3.7 Summary remarks -- 3.8 Appendix A: simple waves -- 3.8.1 Concept of simple waves -- 3.8.2 Simple waves for pseudo-steady Euler equations -- Appendix B: convertibility -- 4 Local solutions for quasilinear systems -- 4.1 Introduction -- 4.2 Existence of solutions to the Cauchy problem -- 4.2.1 Primary representations -- 4.2.2 Primary estimates -- 4.2.3 Estimates on modulus of continuity -- 4.2.4 Lipschitz data -- 4.3 Goursat problem -- 4.4 Mixed initial-boundary value problem -- 4.5 Application to 2-D Euler -- 5 Invariant regions for systems -- 5.1 Basic theorems -- 5.2 Examples.

6 The pressure gradient system -- 6.1 Introduction -- 6.1.1 Derivation -- 6.1.2 Progress of research -- 6.1.3 One-dimensional planar waves -- 6.2 Two-dimensional Riemann problems -- 6.3 Subsonic region -- 6.4 Four-wave interaction -- 7 Open problems -- Epilogue: Stories -- References.