In this thesis, we consider the non-uniqueness of weak solutions to the compressible isentropic Euler equations on a bounded domain. We establish a criterion for the existence of infinitely many solutions with the same initial data. Using this criterion, we prove that for general smooth initial density and velocity, there exists infinitely many weak solutions. Furthermore, given smooth initial density bounded away from zero, we can construct a initial velocity field such taht there exists infinitely many weak solutions with this initial data, and these solutions satisfy the entropy inequality for a positive finite time. The results also generalizes to the free boundary value problems. / 本論文考慮可壓等熵歐拉方程在有界區域上其弱解的非唯一性。我們建立了關於存在無窮多個有同一初始狀態的弱解的判別準則。利用此準則，我們證明了對於一般的光滑的初始密度和速度場，存在無窮多個弱解:進一步的，對於光滑有正下界的初始密度場，我們構造了一小初始速度場，證明存在滿足這樣初始資料的無窮多個弱解，並且，在一個有限時間內，這些弱解滿足熵不等式。這些結果可以推廣到自由邊界情形。 / Detailed summary in vernacular field only. / Luo, Tianwen. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 55-58). / Abstracts also in Chinese. / Introduction --- p.6 / Chapter 1 --- Preliminaries --- p.18 / Chapter 1.1 --- Notational Conventions and Function Spaces --- p.18 / Chapter 1.2 --- Elementary Inequalities --- p.19 / Chapter 1.3 --- Fundamental Lemmas --- p.21 / Chapter 2 --- Formulation and Main Results --- p.22 / Chapter 2.1 --- Mathematical Formulation --- p.23 / Chapter 2.2 --- Main Theorems and the Criterion --- p.24 / Chapter 2.3 --- The Space of Subsolutions --- p.27 / Chapter 3 --- Proof of the Main Theorems --- p.30 / Chapter 3.1 --- Proof of the Subsolution Lemma --- p.31 / Chapter 3.2 --- The Perturbation Property --- p.34 / Chapter 3.3 --- Construction of Suitable Initial Data --- p.39 / Chapter 3.4 --- Convex Integration --- p.43 / Chapter 4 --- Free Boundary Problem --- p.47 / Chapter 4.1 --- Formulation and Results --- p.48 / Chapter 4.2 --- Remarks on the Proofs --- p.51 / Chapter 5 --- Discussions and Future Work --- p.54 / Bibliography --- p.55
| Identifer | oai:union.ndltd.org:cuhk.edu.hk/oai:cuhk-dr:cuhk_328581 |
| Date | January 2012 |
| Contributors | Luo, Tianwen., Chinese University of Hong Kong Graduate School. Division of Mathematics. |
| Source Sets | The Chinese University of Hong Kong |
| Language | English, Chinese |
| Detected Language | English |
| Type | Text, bibliography |
| Format | electronic resource, electronic resource, remote, 1 online resource (58 leaves) |
| Rights | Use of this resource is governed by the terms and conditions of the Creative Commons “Attribution-NonCommercial-NoDerivatives 4.0 International” License (http://creativecommons.org/licenses/by-nc-nd/4.0/) |
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