Quasilinear hyperbolic systems with involutions
The author considers quasilinear hyperbolic systems \[ (1)\quad \partial_ tU+\sum^{m}_{\alpha =1}\partial_{\alpha}G_{\alpha}(U)=0 \] where \(x\in {\mathbb{R}}^ m\), the vector U(x,t) takes values in an open subset \({\mathcal O}\subset {\mathbb{R}}^ n\) and \(G_{\alpha}: {\mathcal O}\to {\mathbb{R}}^ n\) are given smooth functions. A classical solution of (1) is a uniformly Lipschitz continuous function \(U=U({\mathbb{R}}^ m\times [0,\tau))\) which satisfies (1) almost everywhere. A weak solution of the class of functions with bounded variation (BV) is a bounded measurable function \(U({\mathbb{R}}^ m\times [0,\tau))\) with distributional derivatives \(\partial_ tU\), \(\partial_{\alpha}U\) which satisfies (1) in the sense of distribution. It can be shown that the Cauchy problem for (1) may have several solutions of class BV even after imposing the additional requirement that they satisfy an entropy inequality \[ (2)\quad \partial_ t\eta (U)+\sum^{m}_{\alpha =1}\partial_{\alpha}q_{\alpha}(U)\leq 0 \] with \(\eta\) convex (\(\eta\) entropy, q entropy flux). The author shows, whenever a classical solution exists the entropy inequality (2) with \(\eta\) strictly K-convex (see Def. 2.1 in the paper) manages to rule out all other weak solutions of class BV with shocks of moderate strength.
scientific article; zbMATH DE number 3994129
scientific article