New Entropy Conditions for the Scalar Conservation Law with Discontinuous Flux

We propose new Kruzhkov type entropy conditions for one dimensional scalar conservation law with a discontinuous flux in rather general form (no convexity or genuine linearity is needed). We prove existence and uniqueness of the entropy admissible weak solution to the corresponding Cauchy problem merely under the assumption that initial data belong to the BVclass. Such initial data enable us to prove that the sequence of solutions to a special vanishing viscosity approximation of the considered equation is, at the same time, the sequence of quasisolutions to a non-degenerate scalar conservation law. This provides existence of the solution admitting strong traces at the interface. The admissibility conditions are chosen so that a kind of crossing condition is satisfied which, together with existence of traces, provides uniqueness of the solution. In the current contribution, we consider the following problem { ∂tu + ∂x (H(x)f(u) + H(−x)g(u)) = 0, (t, x) ∈ IR × IR u|t=0 = u0(x) ∈ BV (IR), x ∈ IR (1) where u is the scalar unknown function; u0 is an integrable initial function of bounded variation such that a ≤ u0 ≤ b, a, b ∈ IR; H is the Heaviside function; and f, g ∈ C 0 (R) are such that f(a) = f(b) = g(a) = g(b) = 0. Problems such as (1) are non-trivial generalization of scalar conservation law with smooth flux, and they describe different physical phenomena (flow in porous media, sedimentation processes, traffic flow, radar shape-from-shading problems, blood flow, gas flow in a variable duct...). Therefore, beginning with eighties (probably from [33]), problems of type (1) are under intensive investigations. As usual in conservation laws, the Cauchy problem under consideration in general does not possess classical solution, and it can have several weak solutions. Since it is not possible to directly generalize standard theory of entropy admissible solutions [22], in order to choose a proper weak solution to (1) many admissibility conditions were proposed. We mention minimal jump condition [16], minimal variation condition and Γ condition [9, 10], entropy conditions [18, 1], vanishing capillary pressure limit [17], admissibility conditions via adapted entropies [5, 7] or via conditions at the interface [2, 11]. But, in every of the mentioned approaches, some structural hypothesis on the flux (such as convexity or genuine nonlinearity) or on the form of the solution (see Date: February 4, 2010. 1991 Mathematics Subject Classification. 35L65.