The limit of vanishing viscosity for doubly nonlinear parabolic equations
We show that solutions of the doubly nonlinear parabolic equation \begin{equation*} \frac{\partial b(u)}{\partial t} - \epsilon \operatorname{div}(a(\nabla u)) + \operatorname{div}(f(u)) = g \end{equation*} converge in the limit $\epsilon \searrow 0$ of vanishing viscosity to an entropy so...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
University of Szeged
2014-03-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Subjects: | |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=2470 |
| _version_ | 1797830650421575680 |
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| author | Ales Matas Jochen Merker |
| author_facet | Ales Matas Jochen Merker |
| author_sort | Ales Matas |
| collection | DOAJ |
| description | We show that solutions of the doubly nonlinear parabolic equation
\begin{equation*}
\frac{\partial b(u)}{\partial t} - \epsilon \operatorname{div}(a(\nabla u)) + \operatorname{div}(f(u)) = g
\end{equation*}
converge in the limit $\epsilon \searrow 0$ of vanishing viscosity to an entropy solution of the doubly nonlinear hyperbolic equation
\begin{equation*}
\frac{\partial b(u)}{\partial t} + \operatorname{div}(f(u)) = g \,.
\end{equation*}
The difficulty here lies in the fact that the functions $a$ and $b$ specifying the diffusion are nonlinear. |
| first_indexed | 2024-04-09T13:39:27Z |
| format | Article |
| id | doaj.art-6e747080ddd84fa8b473fef78512b96a |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:39:27Z |
| publishDate | 2014-03-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-6e747080ddd84fa8b473fef78512b96a2023-05-09T07:53:03ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752014-03-012014811410.14232/ejqtde.2014.1.82470The limit of vanishing viscosity for doubly nonlinear parabolic equationsAles Matas0Jochen Merker1University of West-Bohemia, Department of Mathematics, Univerzitni 22, CZ - 306 14 Pilsen, Czech RepublicUniversity of Rostock, Ulmenstr. 69 (Haus 3), D-18051 Rostock, GermanyWe show that solutions of the doubly nonlinear parabolic equation \begin{equation*} \frac{\partial b(u)}{\partial t} - \epsilon \operatorname{div}(a(\nabla u)) + \operatorname{div}(f(u)) = g \end{equation*} converge in the limit $\epsilon \searrow 0$ of vanishing viscosity to an entropy solution of the doubly nonlinear hyperbolic equation \begin{equation*} \frac{\partial b(u)}{\partial t} + \operatorname{div}(f(u)) = g \,. \end{equation*} The difficulty here lies in the fact that the functions $a$ and $b$ specifying the diffusion are nonlinear.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=2470vanishing viscositydoubly nonlinear evolution equationsconservation lawquasilineardegenerate |
| spellingShingle | Ales Matas Jochen Merker The limit of vanishing viscosity for doubly nonlinear parabolic equations Electronic Journal of Qualitative Theory of Differential Equations vanishing viscosity doubly nonlinear evolution equations conservation law quasilinear degenerate |
| title | The limit of vanishing viscosity for doubly nonlinear parabolic equations |
| title_full | The limit of vanishing viscosity for doubly nonlinear parabolic equations |
| title_fullStr | The limit of vanishing viscosity for doubly nonlinear parabolic equations |
| title_full_unstemmed | The limit of vanishing viscosity for doubly nonlinear parabolic equations |
| title_short | The limit of vanishing viscosity for doubly nonlinear parabolic equations |
| title_sort | limit of vanishing viscosity for doubly nonlinear parabolic equations |
| topic | vanishing viscosity doubly nonlinear evolution equations conservation law quasilinear degenerate |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=2470 |
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