On singularities of continuity equation solutions

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Abstract

Generalized solutions to the continuity equation in one-dimensional and multidimensional cases are constructed in the case of a discontinuous velocity field.

Section snippets

One-dimensional continuity equation

The appearance of the δ-shock type solutions to continuity equations can easily be explained.

For example, let xR1, u={1,x<0,0,x=0,1,x<0,ρ|t=0=ρ0C(R1),suppρ0[1,1]. It is clear that, for x0, the solution of Eq. (0.2) with this initial data is defined for all t>0 and ρ|x0=0 for t>1. If the solution satisfies the conservation law (0.4), then it is clear that ρ=Cδ(x)for t>1, where C=R1ρ0(x). The situation considered in this example is generalized in the following definition, where we assume

Conservation equation and concentration in the multidimensional case

In this section, we generalize the results of the preceding section to the multidimensional case.

Let us recall several facts and formulas. We assume that an n1-dimensional surface γt moving in Rxn is determined by the equation γt={x;t=ψ(x)}, where ψ(x)C1(Rn), ψ0, in a domain in Rxn where we work.

Obviously, determining a surface by the equation t=ψ is equivalent to determining a surface by an equation of the form S(x,t)=0 (S(x,t)C1, x,tS|S=00) under the condition that St0. The last

Acknowledgements

This research was supported by the Russian Foundation for Basic Research, grant no. 05-01-00912, and by DFG Project no. 436 RUS 113/895/0-1. The author is grateful to V.M. Shelkovich who pointed out several misprints in the text from the preprint server.

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