On the Theory of Discontinuous Solutions to Some Strongly Degenerate Parabolic Equations

Details On the Theory of Discontinuous Solutions to Some Strongly Degenerate Parabolic Equations On the Theory of Discontinuous Solutions to Some Strongly Degenerate Parabolic Equations

2001-05-22

arxiv.org/abs/math/0105174v1

Russian Journal of Mathematical Physics, vol.7, No.3, pp. 341-356, 2000

Mathematics

Analysis of PDEs

26 pages

Scientific paper

It is studied the Cauchy problem for the equations of Burgers' type but with bounded dissipation flux. Such equations degenerate to hyperbolic ones as the velocity gradient tends to infinity. Thus the discontinuous solutions are permitted. In the paper the definition of the generalized solution is given and the existence theorem is established in the classes of functions close to ones of bounded variation. The main feature of used a priori estimates is the fact that one needs to estimate only the diffusion flux, which allows to have in fact arbitrary local growth of the velocity gradient. The uniqueness theorem is proven for essentially narrower class of piecewise smooth functions with regular behavior of discontinuity lines.

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