Systems of Conservation Laws

1 Problems.- 1.0 Outline.- 1.1 Some models.- 1.2 Basic problems.- 1.2.1 Probing problems.- 1.3 Some solutions.- 1.4 von Neumann paradoxes.- 1.5 End notes.- I Basics in One Dimension.- 2 One-dimensional Scalar Equations.- 2.1 The 1-D Burgers equation.- 2.2 Discontinuities and weak solutions.- 2.3 Rankine—Hugoniot relation.- 2.4 Nonuniqueness and entropy conditions.- 2.5 Some existence and uniqueness results.- 2.6 Some simple numerical schemes.- Exercises.- 3 Riemann Problems.- 3.1 The isentropic Euler system.- 3.1.1 Rarefaction waves.- 3.1.2 Discontinuous solutions.- 3.1.3 Entropy conditions.- 3.2 The adiabatic Euler system for polytropic gases.- 3.2.1 Rarefaction waves.- 3.2.2 Discontinuity.- 3.2.3 The entropy condition.- 3.2.4 Solutions.- 3.3 Lax’s Riemann solutions.- 3.3.1 Hyperbolicity and genuine nonlinearity.- 3.3.2 The Riemann problem.- 3.3.3 Continuous solutions.- 3.3.4 Discontinuous solutions.- 3.3.5 Lax’s entropy condition.- 3.3.6 Complete solutions.- 3.4 Nonconvex equations and viscous profiles.- 3.4.1 Nonconvex scalar equations.- 3.4.2 Viscous profiles.- 3.4.3 Stable viscous profiles.- 3.5 End notes and further references.- 4 Cauchy Problems.- 4.1 Smooth solutions.- 4.1.1 A new proof of blow-up in the scalar case.- 4.1.2 Systems of two equations and Riemann invariants.- 4.1.3 Blow-up and smooth solutions in systems of two equations.- 4.1.4 Remarks.- 4.2 Wave interactions.- 4.2.1 Scalar elementary wave interactions.- 4.2.2 The isentropic Euler system.- 4.3 Glimm’s scheme.- 4.3.1 Glimm’s scheme.- 4.3.2 Estimates.- 4.3.3 Compactness.- 4.3.4 Consistency.- 4.3.5 An example of single shocks.- 4.3.6 An example with large data (Nishida’s result).- 4.4 Generalized Riemann problems.- 4.4.1 Convex scalar equations.- 4.4.2 Nonconvex scalar equations.- 4.5 2.- 7.6.2 Inner-field equations for ? ? 2.- 7.6.3 Inner-field solutions for ? = 2.- 7.6.4 Inner-field solutions for 1 > ? > 2.- 7.6.5 The case ? = 1.- 7.7 Intermediate field solutions for u0 0 u0 = 0, ? ? 1.- 7.10.3.A ? = 2.- 7.10.3.B ? > 2.- 7.10.3.C 1 0 u0 > 0, ? = 2.- 7.10.5 ?0>0 u0 > 0, ? > 2.- 7.10.6 ?0>0 u0 > 0, 1 0 u0 > 0, ? = 1.- 7.10.8 ?0>0 u0 0 u0 2.- 7.10.10 ?0>0 u0 0 u0