Nonlinear Analysis: Theory, Methods & Applications
On singularities of continuity equation solutions
Section snippets
One-dimensional continuity equation
The appearance of the -shock type solutions to continuity equations can easily be explained.
For example, let , It is clear that, for , the solution of Eq. (0.2) with this initial data is defined for all and for . If the solution satisfies the conservation law (0.4), then it is clear that where . 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 -dimensional surface moving in is determined by the equation where , , in a domain in where we work.
Obviously, determining a surface by the equation is equivalent to determining a surface by an equation of the form (, ) under the condition that 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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