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Well-Posedness for General \(2\times 2\) Systems of Conservation Laws
Fabio Ancona, University of Bologna, Italy, and Andrea Marson, University of Padova, Italy

Memoirs of the American Mathematical Society
2004; 170 pp; softcover
Volume: 169
ISBN-10: 0-8218-3435-5
ISBN-13: 978-0-8218-3435-0
List Price: US$71
Individual Members: US$42.60
Institutional Members: US$56.80
Order Code: MEMO/169/801
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We consider the Cauchy problem for a strictly hyperbolic \(2\times 2\) system of conservation laws in one space dimension \(u_t+[F(u)]_x=0, u(0,x)=\bar u(x),\) which is neither linearly degenerate nor genuinely non-linear. We make the following assumption on the characteristic fields. If \(r_i(u), \ i=1,2,\) denotes the \(i\)-th right eigenvector of \(DF(u)\) and \(\lambda_i(u)\) the corresponding eigenvalue, then the set \(\{u : \nabla \lambda_i \cdot r_i (u) = 0\}\) is a smooth curve in the \(u\)-plane that is transversal to the vector field \(r_i(u)\).

Systems of conservation laws that fulfill such assumptions arise in studying elastodynamics or rigid heat conductors at low temperature.

For such systems we prove the existence of a closed domain \(\mathcal{D} \subset L^1,\) containing all functions with sufficiently small total variation, and of a uniformly Lipschitz continuous semigroup \(S:\mathcal{D} \times [0,+\infty)\rightarrow \mathcal{D}\) with the following properties. Each trajectory \(t \mapsto S_t \bar u\) of \(S\) is a weak solution of (1). Vice versa, if a piecewise Lipschitz, entropic solution \(u= u(t,x)\) of (1) exists for \(t \in [0,T],\) then it coincides with the trajectory of \(S\), i.e. \(u(t,\cdot) = S_t \bar u.\)

This result yields the uniqueness and continuous dependence of weak, entropy-admissible solutions of the Cauchy problem (1) with small initial data, for systems satisfying the above assumption.


Graduate students and research mathematicians interested in partial differential equations.

Table of Contents

  • Introduction
  • Preliminaries
  • Outline of the proof
  • The algorithm
  • Basic interaction estimates
  • Bounds on the total variation and on the interaction potential
  • Estimates on the number of discontinuities
  • Estimates on shift differentials
  • Completion of the proof
  • Conclusion
  • Bibliography
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