**Memoirs of the American Mathematical Society**

2007;
163 pp;
Softcover

MSC: Primary 35;

Print ISBN: 978-0-8218-3990-4

Product Code: MEMO/190/889

List Price: $72.00

Individual Member Price: $43.20

**Electronic ISBN: 978-1-4704-0495-6
Product Code: MEMO/190/889.E**

List Price: $72.00

Individual Member Price: $43.20

# 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