We construct a global smooth approximate solution to a multidimensional scalar conservation law describing the shock wave formation process for initial data with small variation. In order to solve the problem, we modify the method of characteristics by introducing "new characteristics", nonintersecting curves along which the (approximate) solution to the problem under study is constant. The procedure is based on the weak asymptotic method, a technique which appeared to be rather powerful for investigating nonlinear waves interactions. Introduction.In the current paper, we construct an approximate (in a weak sense) solution u ε (t, x), x ∈ R d , t ∈ R + , where ε is a regularization parameter, corresponding to a multidimensional shock wave formation in the case of a Cauchy problem for a scalar conservation law.In order to make the problem precise, assume that R d is divided into three disjoint domains Ω L , Ω 0 and Ω R , i.e., R d = Ω L∪ Ω 0∪ Ω R , where∪ denotes disjoint union. Let Γ L = ∂Ω L = Ω L ∩ Ω 0 and Γ R = ∂Ω R = Ω R ∩ Ω 0 (see Figure 1). Assume that Γ L and Γ R are (d − 1)-dimensional manifolds admitting the following representation:

MSC (2010) 60G35, 35Q99, 41A60Using an idea going back to Madelung, we construct global in time solutions to the transport equation corresponding to the asymptotic solution of the Kolmogorov-Feller equation describing a system with diffusion, potential and jump terms. To do that we use the construction of a generalized delta-shock solution of the continuity equation for a discontinuous velocity field. We also discuss corresponding problem of asymptotic solution construction (Maslov tunnel asymptotics).

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