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ma8105:2019v:lectures [2019-04-05]
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ma8105:2019v:lectures [2019-04-08] (nåværende versjon)
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| ^ 13 | General Sobolev inequalities, \\ embedding, compactness in \(W^{1,p}\) \\ \\ | | 116-118, 18-19, 112-114 \\ \\ | **Obs:** 1 only one lecture this week. \\ \\ **Rellich-Kondrachov:** In the lectures I give a stronger version of this result than presented in the notes of Holden. I follow the book of Evans, and show compact embedding into Hoelder spaces \(C^{0\,\gamma}\). \\ \\ **The proof** of the first part of Rellich-Kondarchov's compactness theorem follows from extension, Kolmogorov-Riesz compactness theorem, and interpolation in \(L^p\). This way avoids the long regularization + Arzela-Ascoli argument used in the Holden notes (and in PDE book by Evans). In fact the the regularization + Arzela-Ascoli argument is exactly the argument we used in class to proof Kolmogorov-Riesz in the first place (but the proof in the notes of Holden is slightly different). \\ \\ **My lecture notes** from this week: \\ \\ [[http://www.math.ntnu.no/emner/MA8105/2019v/notes/LecNote25.03.2019_GenSobolevIneq_and_RellichKondrachovCompThm.pdf|General Sobolev inequalities and Strong comactness in \(W^{1,p}\)]] \\ \\ | | ^ 13 | General Sobolev inequalities, \\ embedding, compactness in \(W^{1,p}\) \\ \\ | | 116-118, 18-19, 112-114 \\ \\ | **Obs:** 1 only one lecture this week. \\ \\ **Rellich-Kondrachov:** In the lectures I give a stronger version of this result than presented in the notes of Holden. I follow the book of Evans, and show compact embedding into Hoelder spaces \(C^{0\,\gamma}\). \\ \\ **The proof** of the first part of Rellich-Kondarchov's compactness theorem follows from extension, Kolmogorov-Riesz compactness theorem, and interpolation in \(L^p\). This way avoids the long regularization + Arzela-Ascoli argument used in the Holden notes (and in PDE book by Evans). In fact the the regularization + Arzela-Ascoli argument is exactly the argument we used in class to proof Kolmogorov-Riesz in the first place (but the proof in the notes of Holden is slightly different). \\ \\ **My lecture notes** from this week: \\ \\ [[http://www.math.ntnu.no/emner/MA8105/2019v/notes/LecNote25.03.2019_GenSobolevIneq_and_RellichKondrachovCompThm.pdf|General Sobolev inequalities and Strong comactness in \(W^{1,p}\)]] \\ \\ | |
| ^ 14 | Compactness (cont.), \\ Sobolev chain rule, finite differences. \\ \\ **Application:** Application: Convergence of finite difference approximation for the Porous Medium Equation (PME). \\ \\ A. About PME (eq'n, background, self-similar solutions, derivation, well-posedness and a priori estimates). \\ \\ B. The explicit monotone finite difference approximation. \\ \\ C. A priori estimates for the approximation. \\ \\ | | 114-116, 126, 128 | **My lecture notes** from this week: \\ \\ [[http://www.math.ntnu.no/emner/MA8105/2019v/notes/LecNote06.04.2017_Rellich_ChainRule_and_DifferenceQuotients.pdf|Strong comactness, chain rule, and difference quotients]] \\ \\ [[http://www.math.ntnu.no/emner/MA8105/2019v/notes/LecNote05.04.2019_Application_PorousMediumEquation_I.pdf|Porous Medium Equations I: Intro, finite difference approximation, a priori estimates]] \\ \\ | | ^ 14 | Compactness (cont.), \\ Sobolev chain rule, finite differences. \\ \\ **Application:** Application: Convergence of finite difference approximation for the Porous Medium Equation (PME). \\ \\ A. About PME (eq'n, background, self-similar solutions, derivation, well-posedness and a priori estimates). \\ \\ B. The explicit monotone finite difference approximation. \\ \\ C. A priori estimates for the approximation. \\ \\ | | 114-116, 126, 128 | **My lecture notes** from this week: \\ \\ [[http://www.math.ntnu.no/emner/MA8105/2019v/notes/LecNote06.04.2017_Rellich_ChainRule_and_DifferenceQuotients.pdf|Strong comactness, chain rule, and difference quotients]] \\ \\ [[http://www.math.ntnu.no/emner/MA8105/2019v/notes/LecNote05.04.2019_Application_PorousMediumEquation_I.pdf|Porous Medium Equations I: Intro, finite difference approximation, a priori estimates]] \\ \\ | |
| ^ 15 | D. Interpolation in time and compactness. \\ \\ E. Convergence of the method. \\ \\ Plan the oral exam. \\ Dates and organization. \\ \\ | | | **Only one lecture this week - Monday** \\ \\ **Reference group meeting** \\ \\ **Decide on exam date** \\ \\ | | ^ 15 | D. Interpolation in time and compactness. \\ \\ E. Convergence of the method. \\ \\ Plan the oral exam. \\ Dates and organization. \\ \\ | | | **Only one lecture this week - Monday** \\ \\ **Reference group meeting** \\ \\ **Decide on exam date** \\ \\ **My lecture notes** from this week: \\ \\ [[http://www.math.ntnu.no/emner/MA8105/2019v/notes/LecNote08.04.2019_Application_PorousMediumEquation_II.pdf|Porous Medium Equations II: compactness, convergence]] \\ \\ | |