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Source-solutions and asymptotic behavior in conservation laws. (English) Zbl 0545.35057
The authors consider a scalar conservation law \(u_ t+\phi(u)_ x=0\) in a single space variable. The initial data is taken to be a finite measure \(\mu\) ; hence the initial condition is stated in the weak form \(\lim_{t\to 0}<\psi(\cdot),u(\cdot,t)>=<\psi(\cdot),d\mu>,\) for all bounded continuous \(\psi\). The motivation for this study comes from the fact that the case where \(\mu\) is the Dirac measure is relevant to the study of the asymptotic behavior of u when \(u(\cdot,0)\) is taken to be an integrable function. The results are different depending on whether u is non-negative, or not, and whether \(\phi\) is odd or convex. In fact, if u is negative somewhere, the problem is not well posed in general, for data which are measures. These differences do not occur if the data is a bounded integrable function. This is an interesting paper having lots of new ideas and it opens up a new area for study.
Reviewer: J.Smoller

35L65 Hyperbolic conservation laws
35L67 Shocks and singularities for hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
Full Text: DOI
[1] \scPh. Bénilan and M. G. Crandall, “Regularizing Effects of Homogeneous Evolution Equations,” Mathematics Research Center Technical Summary Report No. 2076, University of Wisconsin-Madison.
[2] \scK. S. Cheng, Asymptotic behavior of solutions of a conservation law without convexity condition, J. Differential Equations, in press.
[3] Crandall, M.G, The semigroup approach to first order quasilinear equations in several space variables, Israel J. math., 12, 2, 108-132, (1972) · Zbl 0246.35018
[4] \scM. G. Crandall and M. Pierre, “Regularizing Effects for ut + Aϑ(u) = 0 in L1,” Mathematics Research Center Technical Summary Report No. 2187, University of Wisconsin-Madison.
[5] \scC. M. Dafermos, Asymptotic behavior in hyperbolic balance laws. · Zbl 0464.35060
[6] Diperna, R.J, Decay and asymptotic behavior of solutions to nonlinear hyperbolic systems of conservation laws, Indiana univ. math. J., 24, 1047-1071, (1975) · Zbl 0309.35050
[7] Friedman, A; Kamin, S, The asymptotic behavior of gas in an n-dimensional porous medium, Trans. amer. math. soc., 262, 2, 551-563, (1980) · Zbl 0447.76076
[8] Kamin, S, Source-type solutions for equations of nonstationary filtration, J. math. anal. appl., 63, (1978) · Zbl 0387.76083
[9] Kamin, S, Similar solutions and the asymptotics of filtration equations, Arch. rat. mech. anal., 60, 171-183, (1976) · Zbl 0336.76036
[10] Krǔckov, S.N, First order quasilinear equations in several independent variables, Math. USSR-sb., 10, 217-243, (1970) · Zbl 0215.16203
[11] Lax, P, Hyperbolic systems of conservation laws, II, Comm. pure appl. math., 10, 537-566, (1957) · Zbl 0081.08803
[12] Liu, T.-P, Decay to N-waves of solutions of general systems of nonlinear hyperbolic conservation laws, Comm. pure appl. math., 30, 585-610, (1977) · Zbl 0357.35059
[13] \scM. Pierre, “Uniqueness of the Solutions of ut − Δϑ(u) = 0 with Initial Datum a Measure,” Mathematics Research Center Technical Summary Report No. 2171, University of Wisconsin-Madison (J. Nonlinear Anal., inpress). · Zbl 0484.35044
[14] Varadarajan, V.S, Measures on topological spaces, Amer. math. soc. trans. ser. 2, 48, 161-228, (1965)
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