Well-posedness for General 2
نویسندگان
چکیده
We consider the Cauchy problem for a strictly hyperbolic 2 2 system of conservation laws in one space dimension u t + F (u)] x = 0; u(0; x) = u(x); (1) which is neither linearly degenerate nor genuinely non-linear. We make the following assumption on the characteristic elds. If r i (u); i = 1; 2; denotes the i-th right eigenvector of DF (u) and i (u) the corresponding eigenvalue, then the set fu : r i r i (u) = 0g is a smooth curve in the u-plane that is transversal to the vector eld r i (u): Systems of conservation laws that fullll such assumptions arise in studying elastodynamics or rigid heat conductors at low temperature. For such systems we prove the existence of a closed domain D L 1 ; containing all functions with suuciently small total variation, and of a uniformly Lipschitz continuous semigroup S : D 0; +1) ! D with the following properties. Each trajectory t 7 ! S t u of S is a weak solution of (1). Viceversa, if a piecewise Lipschitz, entropic solution u = u(t; x) of (1) exists for t 2 0; T ]; then it coincides with the trajectory of S, i.e. u(t;) = S t 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 satysfying the above assumption.
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