# Well-Posedness for General \(2\times 2\) Systems of Conservation Laws

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*Fabio Ancona; Andrea Marson*

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.

#### Table of Contents

# Table of Contents

## Well-Posedness for General $2x2$ Systems of Conservation Laws

- Contents vii8 free
- 1. Introduction 112 free
- 2. Preliminaries 1021 free
- 3. Outline of the proof 1829
- 4. The algorithm 3748
- 5. Basic interaction estimates 5162
- 6. Bounds on the total variation and on the interaction potential 6071
- 7. Estimates on the number of discontinuities 97108
- 8. Estimates on shift differentials 112123
- 9. Completion of the proof 157168
- 10. Conclusion 162173
- Bibliography 169180