The Cauchy problem for a special class of 2_2 systems of conservation laws with data in L 1 & L is considered. In the strictly hyperbolic case we prove the existence of a weak solution which depends continuously on the initial data with respect to the L 1 -norm. This solution can be characterized in terms of a Kruz$ kov-type entropy condition, which is introduced here.
In this paper we present an improved version of the front-tracking algorithm for systems of conservation laws. The formulation and the theoretical analysis are here somewhat simpler than in previous algorithms. At the same time, our version leads to a more efficient numerical scheme. ᮊ
We establish existence of global-in-time weak solutions to the one dimensional, compressible Navier-Stokes system for a viscous and heat conducting ideal polytropic gas (pressure p = Kθ/τ , internal energy e = cvθ), when the viscosity µ is constant and the heat conductivity κ depends on the temperature θ according to κ(θ) =κθ β , with 0 ≤ β < 3 2 . This choice of degenerate transport coefficients is motivated by the kinetic theory of gasses.Approximate solutions are generated by a semi-discrete finite element scheme. We first formulate sufficient conditions that guarantee convergence to a weak solution. The convergence proof relies on weak compactness and convexity, and it applies to the more general constitutive relations µ(θ) =μθ α , κ(θ) =κθ β , with α ≥ 0, 0 ≤ β < 2 (μ,κ constants). We then verify the sufficient conditions in the case α = 0 and 0 ≤ β < 3 2 . The data are assumed to be without vacuum, mass concentrations, or vanishing temperatures, and the same holds for the weak solutions.
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