On a Nonlinear Degenerate Parabolic Transport-Diffusion Equation with a Discontinuous Coefficient
| dc.contributor.author | Karlsen, Kenneth H. | |
| dc.contributor.author | Risebro, Nils H. | |
| dc.contributor.author | Towers, John D. | |
| dc.date.accessioned | 2020-08-21T19:46:08Z | |
| dc.date.available | 2020-08-21T19:46:08Z | |
| dc.date.issued | 2002-10-27 | |
| dc.description.abstract | We study the Cauchy problem for the nonlinear (possibly strongly) degenerate parabolic transport-diffusion equation ∂tu + ∂x (γ(x)ƒ(u)) = ∂2x A(u), A'(·) ≥ 0, where the coefficient γ(x) is possibly discontinuous and ƒ(u) is genuinely non-linear, but not necessarily convex or concave. Existence of a weak solution is proved by passing to the limit as ε ↓ 0 in a suitable sequence {uε}ε > 0 of smooth approximations solving the problem above with the transport flux γ(x)ƒ(·) replaced by γε(x)ƒ(·) and the diffusion function A(·) replaced by Aε(·), where γε(·) is smooth and A'ε(·) > 0. The main technical challenge is to deal with the fact that the total variation |uε|BV cannot be bounded uniformly in ε, and hence one cannot derive directly strong convergence of {uε}ε > 0. In the purely hyperbolic case (A' ≡ 0), where existence has already been established by a number of authors, all existence results to date have used a singular mapping to overcome the lack of a variation bound. Here we derive instead strong convergence via a series of a priori (energy) estimates that allow us to deduce convergence of the diffusion function and use the compensated compactness method to deal with the transport term. | |
| dc.description.department | Mathematics | |
| dc.format | Text | |
| dc.format.extent | 23 pages | |
| dc.format.medium | 1 file (.pdf) | |
| dc.identifier.citation | Karlsen, K. H., Risebro, N. H., & Towers, J. D. (2002). On a nonlinear degenerate parabolic transport-diffusion equation with a discontinuous coefficient. <i>Electronic Journal of Differential Equations, 2002</i>(93), pp. 1-23. | |
| dc.identifier.issn | 1072-6691 | |
| dc.identifier.uri | https://hdl.handle.net/10877/12450 | |
| dc.language.iso | en | |
| dc.publisher | Southwest Texas State University, Department of Mathematics | |
| dc.rights | Attribution 4.0 International | |
| dc.rights.holder | This work is licensed under a Creative Commons Attribution 4.0 International License. | |
| dc.rights.uri | https://creativecommons.org/licenses/by/4.0/ | |
| dc.source | Electronic Journal of Differential Equations, 2002, San Marcos, Texas: Southwest Texas State University and University of North Texas. | |
| dc.subject | Degenerate parabolic equation | |
| dc.subject | Nonconvex flux | |
| dc.subject | Weak solution | |
| dc.subject | Discontinuous coefficient | |
| dc.subject | Viscosity method | |
| dc.subject | a priori estimates | |
| dc.subject | Compensated compactness | |
| dc.title | On a Nonlinear Degenerate Parabolic Transport-Diffusion Equation with a Discontinuous Coefficient | |
| dc.type | Article |