Marcati, Pierangelo Approximate solutions to conservation laws via convective parabolic equations. (English) Zbl 0653.35057 Commun. Partial Differ. Equations 13, No. 3, 321-344 (1988). The author presents some applications of the theory of compensated compactness to the convergence of approximate solutions of scalar hyperbolic conservation law \[ (*)\quad u_ t+f(u)_ x=0,\quad t\geq 0,\quad x\in R,\quad u(x,0)=u_ 0(x). \] The first approximation is based on the convective parabolic equation \[ (**)\quad u_ t+f(u)_ x=\epsilon \psi (u)_{xx},\quad t\geq 0,\quad x\in R;\quad \epsilon >0,\quad u(x,0)=u_ 0(x). \] The solutions \(u^{\epsilon}\) of (**) converge weakly star in \(L^{\infty}\) to the weak solution of (*) and if f is not affine, the convergence is strong in L p, \(p<\infty\), with the entropy inequality for the limit solution. The latter approximation is based on the modified Lax-Friedrichs difference scheme - formal approximation to the equation \[ u_ t+f(u)_ x=(\Delta t/\lambda \quad 2)\psi (u)_{xx},\quad \lambda =\Delta x/\Delta t,\quad t\to 0. \] These approximations again converge weakly star in \(L^{\infty}\) to the weak solution of (*), verifying the entropy inequality. Reviewer: A.Doktor Cited in 10 Documents MSC: 35L65 Hyperbolic conservation laws 35K55 Nonlinear parabolic equations 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 35A35 Theoretical approximation in context of PDEs Keywords:compensated compactness; convergence; approximate solutions; conservation law; convective parabolic equation; weak solution; entropy inequality; Lax-Friedrichs difference scheme PDFBibTeX XMLCite \textit{P. Marcati}, Commun. Partial Differ. Equations 13, No. 3, 321--344 (1988; Zbl 0653.35057) Full Text: DOI References: [1] DOI: 10.1007/BF00251724 · Zbl 0519.35054 · doi:10.1007/BF00251724 [2] DOI: 10.1007/BF00752112 · Zbl 0616.35055 · doi:10.1007/BF00752112 [3] Di Perna R.J., Trans. A.M.S 292 pp 383– (1985) · doi:10.1090/S0002-9947-1985-0808729-4 [4] Diaz, J.I. and Kersner, R. ”On a Nonlinear Degenerate Parabolic Equation in infiltration or evaporation through a porous medium.” Preprint · Zbl 0634.35042 [5] Gilding B.H., Ann. Scuola Normale Sup. Pisa 4 pp 393– (1977) [6] Gilding B.H., Arch. Rat, Mech. Analysis 16 pp 127– (1976) [7] DOI: 10.1002/cpa.3160180408 · Zbl 0141.28902 · doi:10.1002/cpa.3160180408 [8] Lax P.D., Contribution to Nonlinear Functional Analysis (1971) [9] Marcati, P. ”Convergence of Approximate Solutions to Scalar Conservation Laws by Degenerate Diffusion.” Preprint, Univ. of L’Aquila – Italy, November, 1985, (submitted) · Zbl 0703.35032 [10] Murat F., Ann. Scuola Normale Sup. Pisa 5 pp 489– (1978) [11] Murat F., J. Math pures et appl 60 pp 309– (1981) [12] DOI: 10.1002/cpa.3160350602 · Zbl 0479.35053 · doi:10.1002/cpa.3160350602 [13] Tartar L., Res. Notes in Math 4 (1979) [14] Van Hove L., Nederl. Akad. Weten 50 pp 18– (1947) [15] Volpert A.I., Math. USSR Sbornik 7 pp 365– (1969) · Zbl 0191.11603 · doi:10.1070/SM1969v007n03ABEH001095 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.