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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

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.