Dafermos, Constantine M. Quasilinear hyperbolic systems with involutions. (English) Zbl 0614.35057 Arch. Ration. Mech. Anal. 94, 373-389 (1986). 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. Reviewer: M.Schneider Cited in 1 ReviewCited in 36 Documents MSC: 35L60 First-order nonlinear hyperbolic equations 35L67 Shocks and singularities for hyperbolic equations 35B65 Smoothness and regularity of solutions to PDEs Keywords:quasilinear hyperbolic systems; classical solution; weak solution; Cauchy problem; entropy inequality PDFBibTeX XMLCite \textit{C. M. Dafermos}, Arch. Ration. Mech. Anal. 94, 373--389 (1986; Zbl 0614.35057) Full Text: DOI References: [1] Ball, J. M., Strict convexity, strong ellipticity and regularity in the calculus of variations. Math. Proc. Cambridge Phil. Soc. 87 (1980), 501-513. · Zbl 0451.35028 [2] Bers, L., F. John, & M. Schechter, Partial Differential Equations. New York: Interscience 1964. · Zbl 0126.00207 [3] Crandall, M. G., & P. H. Rabinowitz, Bifurcation from simple eigenvalues. J. Functional Anal. 8 (1971), 321-340. · Zbl 0219.46015 [4] Dafermos, C. M., The second law of thermodynamics and stability. Arch. Rational Mech. Anal. 70 (1979), 167-179. · Zbl 0448.73004 [5] Dafermos, C. M., The equations of elasticity are special. Trends in Applications of Pure Mathematics to Mechanics. Vol. III, ed. R. J. Knops, London: Pitman 1981, pp. 96-103. [6] Dafermos, C. M., Hyperbolic systems of conservation laws. Systems of Nonlinear Partial Differential Equations, ed. J. M. Ball. NATO ASI Series C, No. 111. Dordrecht: D. Reidel 1983, pp. 25-70. · Zbl 0536.35048 [7] DiPerna, R. J., Uniqueness of solutions to hyperbolic conservation laws. Indiana U. Math. J. 28 (1979), 137-188. · Zbl 0409.35057 [8] DiPerna, R. J., Convergence of approximate solutions to conservation laws. Arch. Rational Mech. Anal. 82 (1983), 27-70. · Zbl 0519.35054 [9] DiPerna, R. J., Compensated compactness and general systems of conservation laws. Trans. Am. Math. Soc. 242 (1985), 383-420. · Zbl 0606.35052 [10] Lax, P. D., Hyperbolic systems of conservation laws. Comm. Pure Appl. Math. 10 (1957), 537-566. · Zbl 0081.08803 [11] Lax, P. D., Shock waves and entropy. Contributions to Functional Analysis, ed. E. A. Zarantonello. New York: Academic Press 1971, pp. 603-634. [12] Liu, T.-P., The entropy condition and the admissibility of shocks. J. Math. Anal. Appl. 53 (1976), 78-88. · Zbl 0332.76051 [13] Malek-Madani, R., Energy criteria for finite hyperelasticity. Arch. Rational Mech. Anal. 77 (1981), 177-188. · Zbl 0478.73025 [14] Tartar, L., Compensated compactness and applications to partial differential equations. Nonlinear Analysis and Mechanics Vol. IV, ed. R. J. Knops, London: Pitman 1979, pp. 136-212. · Zbl 0437.35004 [15] Truesdell, C., & W. Noll, The Nonlinear Field Theories of Mechanics. Handbuch der Physik III/3, ed. S. Flügge. Berlin Heidelberg New York: Springer 1965. · Zbl 0779.73004 [16] Volpert, A. I., The spaces BV and quasilinear equations. Mat. Sbornik 73 (1967), 255-302. English transl. Math. USSR Sbornik 2 (1967), 225-267. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.