Nonlinear Conservation Laws, Fluid Systems and Related Topics

The following sections are included:

• Preface

• Contents

In this chapter, we consider the 3D incompressible Euler equations. We present classical and recent results on the issue of global existence/finite time singularity. We also introduce the theories of lower dimensional model equations of the 3D Euler equations and the vortex patch problem.

The following sections are included:

• Hyperbolic systems of conservation laws
  • The Cauchy problem: classical solutions
    • Hyperbolicity
    • Entropies
    • Local well-posedness in H^s(ℝ^d)
  • The Cauchy problem: weak solutions
    • Break-down of smooth solutions
    • Weak solutions
    • The Rankine-Hugoniot condition
    • Non-uniqueness of weak solutions
    • Entropy admissibility condition
    • The viscosity approach
    • The scalar case
  • Shock waves
    • The Hugoniot locus
    • Genuine nonlinearity
    • The Lax shock inequality
    • Viscous shock profiles
  • The Riemann problem
    • Rarefaction waves
    • Contact discontinuities
    • The theorem of Lax
  • Existence of viscous shock profiles
    • The scalar case
    • Reduction to a center manifold (bifurcation analysis)
    • Lax shocks
    • Under-compressive shocks

• Finite difference schemes
  • Conservative schemes
    • Consistency
    • Order of accuracy
    • Linearized L²-stability
    • The Courant-Friedrichs-Lewy condition
    • Entropy-consistent schemes
  • Examples
    • The naive centered scheme
    • The Lax-Friedrichs scheme
    • The Lax-Wendroff scheme
    • The Godunov scheme
  • Schemes for scalar equations
    • Monotone schemes
    • Kutznetsov's error estimate

• Discrete shock profiles
  • DSPs and conservation
    • The function Y
    • Scalar case: monotone schemes
  • Existence theory for rational η
    • DSPs for small steady Lax shocks
    • DSPs for steady Lax shocks: the Godunov scheme
    • What can go wrong?

• References

Both the areas of the kinetic equations and conservation laws are extremely large and contain enormous theories and challenging open problems. Based on this thinking, this notes serves only as a short course aiming at some particular parts in these two areas. In other words, many interesting and important theories and phenomena in these two areas are not covered here. However, we will try to make this notes to be self-contained and cover some selected topics according to the authors' interest and to the length of this course. Therefore, it can be viewed as an introduction to these two areas and somehow it is also expected to lead the readers to the frontier of the research in these areas.

The following sections are included:

• Introduction
  • Why statistical description
  • What characterizes statistical behavior
  • Relative stability of statistical solution over single trajectories

• Stationary statistics
  • Invariant measures and stationary statistical solutions
  • Definition, existence
  • Ergodicity
  • Invariant measure, stationary statistical solution and attractor
  • Dependence on parameters
  • Regular perturbation
  • Singular perturbation
  • Remarks on direct applications
    • An application to NSE: energy dissipation rate per unit mass
    • An application to RBC: heat transfer in the vertical direction (Nusselt number)
  • Maximum entropy principle
  • Application to ODEs
  • Application to basic geophysical systems

• Remarks on time dependent statistics
  • Definition, existence
  • Applications to NSE
    • Reynolds equation for the average flow
    • Moment closure problem
    • Fluctuation dissipation theory (FDT)

• Appendix: some useful theorems

• Acknowledgement

• References

We analyze the self-similar isentropic irrotational Euler system via the hodograph transformation. We diagonalize the system of equations in the phase space. We use these equations to analyze the binary interaction of arbitrary planar rarefaction waves, which includes the classical problem of a wedge of gas expanding into vacuum.