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The semigroup approach to systems of conservation laws. (English) Zbl 0866.35064
This survey paper provides an account of recent developments concerning the uniqueness and continuous dependence of solutions to systems of conservation laws. Progress has been achieved by a new approach, based on the construction of a Lipschitz semigroup compatible with the standard solutions of Riemann problems. We give here an overview of the main ideas and techniques involved in this “semigroup approach”, with the aid of several pictures. Proofs will often be given a rather sketchy and informal way.
Consider the Cauchy problem for a strictly hyperbolic \(n\times n\) system of conservation laws in one space dimension: \[ (1)\quad u_t+ \bigl[f(u) \bigr]_x=0, \qquad (2) \quad u(0,x)= \overline u(x). \] We assume that each characteristic field is either linearly degenerate or genuinely nonlinear. The semigroup approach can be summarized in the following steps: (1) Construct a uniformly Lipschitz semigroup \(S\) of weak solutions of (1), consistent with the standard solutions of the Riemann problems. (2) Show that this semigroup is essentially unique, and that each trajectory \(t\mapsto S_t\overline u\) provides the unique “good” solution of the corresponding Cauchy problem (1), (2). (3) Provide an intrinsic characterization of the trajectories of the semigroup, in terms of local integral estimates.
This approach has two main advantages: (i) It provides the uniqueness of solutions within the same class of BV functions where an existence theorem can be proved. (ii) It yields the Lipschitz continuous dependence of solutions on the initial data, as well as error estimates concerning the \(L^1\) distance between an approximate solution and the exact one.
We present the basic ideas involved in the construction of the semigroup. We first examine how the distance between two “infinitesimally close” solutions changes at times of wave-front interactions. This leads to the introduction of a Riemann type metric, uniformly equivalent to the standard \(L^1\) distance, which is contractive w.r.t. the flow generated by the conservation equations. We describe the key steps in the construction of approximate solutions which depend Lipschitz continuously on the initial data. In the limit, we obtain a “Standard Riemann Semigroup” of solutions of (1). We review a construction of piecewise constant approximate solutions by wave-front tracking. These approximations play a key role in the proof of uniqueness of the semigroup. Finally, we motivate the characterization of semigroup trajectories in terms of local integral estimates.

35L65 Hyperbolic conservation laws
32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces