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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
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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$67
Individual Members: US$40.20
Institutional Members: US$53.60
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.

Readership

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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