Measure-valued solutions of scalar conservation laws with boundary conditions. (English) Zbl 0702.35155

The equation into consideration is the conservation law \[ (1)\quad u_ t+\sum^{d}_{j=1}f_ j(u)_{x_ j}=0 \text{ in } \Omega \times R_+ \] with initial condition (2) u(,0)\(=u_ 0\) on \(\Omega\) and boundary condition \[ (3)\quad (sgn(u(\bar x,t)-k)-sgn a(\bar x,t)- k)\times (f(u(\bar x,t)-f(k))n(\bar x)\geq 0 \] for all \(k\in R\), \((\bar x,t)\in \Gamma \times R_+\). Here \(\Omega\) is a bounded open set of \(R^ d\) with smooth boundary \(\Gamma =\partial \Omega\) and outward unit normal n, u: \(\Omega\times R_+\to R\), \(f=(f_ 1,...,f_ d): R\to R^ d\), \(u_ 0: \Omega \to R\), a: \(\Gamma\times R_+\to R\). The boundary \(\Gamma\) of \(\Omega\) is supposed to be smooth. Smoothness is provided also for \(f,u_ 0\) and a. A measure-valued solution to problem (1)-(3) is defined. A uniqueness theorem is proved. The obtained result is used to prove convergence towards the unique solution for approximate solutions which are uniformly bounded in \(L_{\infty}\), weakly consistent with certain entropy inequalities and strongly consistent with the initial condition, i.e. without using derivative estimates. As an example convergence of a finite element method is demonstrated.
Reviewer: I.Ginchev


35L65 Hyperbolic conservation laws
35A25 Other special methods applied to PDEs
Full Text: DOI


[1] C. Bardos, A. Y. LeRoux & J. C. Nedelec, First order quasilinear equations with boundary conditions, Comm. P.D.E. 4 (9), pp. 1017-1034, 1979. · Zbl 0418.35024
[2] R. J. DiPerna, Measure-valued solutions of conservation laws. Arch. Rational Mech. Anal. 88 (1985), 223-270. · Zbl 0616.35055
[3] F. Dubois & P. LeFloch, C.R. Acad. Sci Paris 304, série I, 1987, p. 75-78.
[4] T. J. R. Hughes & M. Mallet, A new finite element formulation for computational fluid dynamics: IV. a discontinuity-capturing operator for multidimensional advective-diffusive systems, Computer Methods in Applied Mechanics and Engineering 58 (1986) 329-336. · Zbl 0587.76120
[5] C. Johnson, U. Nävert & J. Pitkäranta, Finite element methods for linear hyperbolic problems, Computer Methods in Applied Mechanics and Engineering 45 (1984) 285-312. · Zbl 0537.76060
[6] C. Johnson & J. Saranen, Streamline diffusion methods for problems in fluid mechanics, Math. Comp. 47, 1986, pp. 1-18. · Zbl 0609.76020
[7] C. Johnson & A. Szepessy, On the convergence of a finite element method for a nonlinear hyperbolic conservation law, Math. Comp. 49, N. 180 Oct 1987, p. 427-444. · Zbl 0634.65075
[8] C. Johnson, A. Szepessy & P. Hansbo, On the convergence of shock-capturing streamline diffusion finite element methods for hyperbolic conservation laws, Preprint No. 1987:21, Dep. of Math., Chalmers Univ. of Technology, S-41296 Göteborg, to appear in Math. Comp. · Zbl 0685.65086
[9] P. D. Lax, Shock waves and entropy, in Contributions to Nonlinear Functional Analysis, ed. E. A. Zarantonello, Academic Press (1971), 603-634.
[10] A. Y. le Roux, Étude du problème mixte pour une équation quasi-linéaire du premier ordre C.R. Acad. Sc. Paris 285, Série A-351. · Zbl 0366.35019
[11] A. Y. le Roux, Approximation de quelques problèmes hyperboliques non-linéaires. Thése d’Etat, Rennes, 1979.
[12] A. Szepessy, Convergence of a shock-capturing streamline diffusion finite element method for scalar conservation laws in two space dimensions, Preprint No. 1988-07, Department of Mathematics, Chalmers University of Technology, S-41296 Göteborg, to appear in Math. Comp. Oct. 1989. · Zbl 0679.65072
[13] A. Szepessy, Thesis, 1989, Dept. of Math., Chalmers Univ. of Technology, S-41296 Göteborg.
[14] L. Tartar, The Compensated Compactness Method Applied to Systems of Conservation Laws, J. M. Ball (ed.), Systems of Nonlinear Partial Differential Equations, 263-285. NATO ASI series, C. Reidel Publishing (1983). · Zbl 0536.35003
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.