Abstract. We propose a general framework for the study of L1 contractive semigroups of solutions to conservation laws with discontinuous flux: f ut + f(x, u)x = 0, f(x, u) = l (u), x < 0, f r (CL) (u), x> 0, where the fluxes f l, f r are mainly assumed to be continuous. Developing the ideas of a number of preceding works (Baiti and Jenssen [14], Audusse and Perthame [12], Garavello et al. [35], Adimurthi et al. [3], Bürger et al. [21]), we claim that the whole admissibility issue is reduced to the selection of a family of “elementary solutions”, which are piecewise constant weak solutions of the form c(x) = c l 1l {x<0} + c r 1l {x>0}. We refer to such a family as a “germ”. It is well known that (CL) admits many

Abstract. 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

This note is devoted to a characterization of the vanishing viscosity limit for multi-dimensional conservation laws of the form ut + div f(x, u) = 0, u|t=0 = u0 in the domain R + × R N. The flux f = f(x, u) is assumed locally Lipschitz continuous in the unknown u and piecewise constant in the space variable x; the discontinuities of f(·, u) are contained in the union of a locally finite number of sufficiently smooth hypersurfaces of R N. We define “GV V-entropy solutions ” (this formulation is a particular case of the one of [3]); the definition readily implies the uniqueness and the L 1 contraction principle for the GV V-entropy solutions. Our formulation is compatible with the standard vanishing viscosity approximation u ε t + div (f(x, u ε)) = ε∆u ε, u ε |t=0 = u0, ε ↓ 0, of the conservation law. We show that, provided u ε, ε> 0, enjoy a uniform L ∞ bound and the flux f(x, ·) is non-degenerately nonlinear, vanishing viscosity

Abstract. We prove that a family of solutions to a Cauchy problem for a two dimensional scalar conservation law with a discontinuous smoothed flux and the vanishing viscosity is strongly L1 loc – precompact under a new genuine nonlinearity condition, weaker than in previous works on the subject. 1.

Abstract. By discovering that solutions of the vanishing viscosity approximation (but without flux regularization) to a scalar conservation law with discontinuous flux are equal to a flux crossing point at the interface, we derive entropy conditions which provide well-posedness to a corresponding Cauchy problem. We assume that the flux is such that the maximum principle holds, but we allow multiple flux crossings and we do not need any kind of genuine nonlinearity conditions. Proposed concept is a proper generalization to the standard Kruzhkov entropy conditions and it does not involve transformation of the equation or use of adapted entropies. The subject of the paper is the following Cauchy problem ∂tu + ∂x (H(x)f(u) + H(−x)g(u)) = 0, (t, x) ∈ IR + × IR u|t=0 = u0(x) ∈ L ∞ (IR), x ∈ IR where u is the scalar unknown function; u0 is a function such that a ≤ u0 ≤ b, a, b ∈ IR; H is the Heaviside function; and f, g ∈ C 1 (R) are such that f(a) = g(a) = c1,

Abstract. We prove existence and uniqueness of an entropy solution to a multidimensional scalar conservation law with discontinuous flux. The proof is based on the corresponding kinetic formulation of the considered equation and a ”smart ” change of an unknown function. 1.

Abstract. The activated sludge process (ASP), found in most wastewater treatment plants, consists basically of a biological reactor followed by a sedimentation tank, which has one inlet and two outlets. The purpose of the ASP is to reduce organic material and dissolved nutrients (substrate) in the incoming wastewater by means of activated sludge (microorganisms). The major part of the discharged flow through the bottom outlet of the sedimentation tank is recirculated to the reactor, so that the biomass is reused. Only two material components are considered; the soluble substrate and the particulate sludge. The biological reactions are modelled by two nonlinear ordinary differential equations and the continuous sedimentation process by two hyperbolic partial differential equations (PDEs), which have coefficients that are discontinuous functions in space due to the inlet and outlets. In contrast to previously published modelling-control aspects of the ASP, the theory for such PDEs is utilized. It is proved that the most desired steady-state solutions can be parameterized by a natural control variable; the ratio of the recirculating volumetric flow to the input flow. This knowledge is a key ingredient in a two-variable regulator, with which the effluent dissolved nutrients concentration and the concentration profile in the sedimentation tank are controlled. Theoretical results are supported by simulations. 1.