# Distribution Solutions of Nonlinear Systems of Conservation Laws

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*Michael Sever*

The local structure of solutions of initial value problems for
nonlinear systems of conservation laws is considered. Given large
initial data, there exist systems with reasonable structural
properties for which standard entropy weak solutions cannot be
continued after finite time, but for which weaker solutions,
valued as measures at a given time, exist. At any given time, the
singularities thus arising admit representation as weak limits of
suitable approximate solutions in the space of measures with
respect to the space variable.

Two distinct classes of singularities have emerged in this
context, known as delta-shocks and singular shocks.
Notwithstanding the similar form of the singularities, the
analysis of delta-shocks is very different from that of singular
shocks, as are the systems for which they occur. Roughly speaking,
the difference is that for delta-shocks, the density
approximations majorize the flux approximations, whereas for
singular shocks, the flux approximations blow up faster. As
against that admissible singular shocks have viscous structure.

#### Table of Contents

# Table of Contents

## Distribution Solutions of Nonlinear Systems of Conservation Laws

- Contents vii8 free
- Chapter 1. General distribution solutions 110 free
- Chapter 2. Delta-shocks 4453
- 1. The structural conditions 4453
- 2. Definition of solutions 5059
- 3. Delta-shocks and Lagrangian coordinates 5463
- 4. Continuation of delta- shock solutions 5968
- 5. Nonhyperbolic systems with delta-shock solutions 6776
- 6. Strictly hyperbolic, linearly degenerate pairs 7079
- 7. Additional examples 7584

- Chapter 3. Singular shocks 8190
- Bibliography 160169