Blowup for $ {{\rm{C}}}^{1} $ solutions of Euler equations in $ {{\rm{R}}}^{N} $ with the second inertia functional of reference
The compressible Euler equations are an elementary model in mathematical fluid mechanics. In this article, we combine the Sideris and Makino-Ukai-Kawashima's classical functional techniques to study the new second inertia functional of reference: $ { H}_{ref}{ (t) = }\frac{1}{2}\int_{\Om...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
AIMS Press
2023-01-01
|
| Series: | AIMS Mathematics |
| Subjects: | |
| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.2023412?viewType=HTML |
| _version_ | 1797905473214611456 |
|---|---|
| author | Manwai Yuen |
| author_facet | Manwai Yuen |
| author_sort | Manwai Yuen |
| collection | DOAJ |
| description | The compressible Euler equations are an elementary model in mathematical fluid mechanics. In this article, we combine the Sideris and Makino-Ukai-Kawashima's classical functional techniques to study the new second inertia functional of reference:
$ { H}_{ref}{ (t) = }\frac{1}{2}\int_{\Omega(t)}\left( { \rho-\bar{\rho}}\right) \left\vert { \vec{x} }\right\vert ^{2}dV{{ , }} $
for the blowup phenomena of $ C^{1} $ solutions $ (\rho, \vec{u}) $ with the support of $ \left({ \rho-\bar{\rho}}, \vec{u}\right) $, and with a positive constant $ { \bar{\rho}} $ for the adiabatic index $ \gamma > 1 $. We find that if the total reference mass
$ M_{ref}(0) = { \int_{{\bf R}^{N}}} (\rho_{0}({ \vec{x}})-\bar{\rho})dV\geq0, $
and the total reference energy
$ E_{ref}(0) = \int_{{\bf R}^{N}}\left( \frac{1}{2}\rho_{0}({ \vec {x}})\left\vert \vec{u}_{0}({ \vec{x}})\right\vert ^{2}+\frac {K}{\gamma-1}\left( \rho_{0}^{\gamma}({ \vec{x}})-\bar{\rho }^{\gamma}\right) \right) dV, $
with a positive constant $ K $ is sufficiently large, then the corresponding solution blows up on or before any finite time $ T > 0 $. |
| first_indexed | 2024-04-10T10:05:53Z |
| format | Article |
| id | doaj.art-930efa74d15b479ea87ad2c53a1f95d3 |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-04-10T10:05:53Z |
| publishDate | 2023-01-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-930efa74d15b479ea87ad2c53a1f95d32023-02-16T01:00:01ZengAIMS PressAIMS Mathematics2473-69882023-01-01848162817010.3934/math.2023412Blowup for $ {{\rm{C}}}^{1} $ solutions of Euler equations in $ {{\rm{R}}}^{N} $ with the second inertia functional of referenceManwai Yuen 0Department of Mathematics and Information Technology, The Education University of Hong Kong, Hong KongThe compressible Euler equations are an elementary model in mathematical fluid mechanics. In this article, we combine the Sideris and Makino-Ukai-Kawashima's classical functional techniques to study the new second inertia functional of reference: $ { H}_{ref}{ (t) = }\frac{1}{2}\int_{\Omega(t)}\left( { \rho-\bar{\rho}}\right) \left\vert { \vec{x} }\right\vert ^{2}dV{{ , }} $ for the blowup phenomena of $ C^{1} $ solutions $ (\rho, \vec{u}) $ with the support of $ \left({ \rho-\bar{\rho}}, \vec{u}\right) $, and with a positive constant $ { \bar{\rho}} $ for the adiabatic index $ \gamma > 1 $. We find that if the total reference mass $ M_{ref}(0) = { \int_{{\bf R}^{N}}} (\rho_{0}({ \vec{x}})-\bar{\rho})dV\geq0, $ and the total reference energy $ E_{ref}(0) = \int_{{\bf R}^{N}}\left( \frac{1}{2}\rho_{0}({ \vec {x}})\left\vert \vec{u}_{0}({ \vec{x}})\right\vert ^{2}+\frac {K}{\gamma-1}\left( \rho_{0}^{\gamma}({ \vec{x}})-\bar{\rho }^{\gamma}\right) \right) dV, $ with a positive constant $ K $ is sufficiently large, then the corresponding solution blows up on or before any finite time $ T > 0 $.https://www.aimspress.com/article/doi/10.3934/math.2023412?viewType=HTMLcompressible euler equationsinitial value problemsblowupsecond inertia functional of referenceenergy method |
| spellingShingle | Manwai Yuen Blowup for $ {{\rm{C}}}^{1} $ solutions of Euler equations in $ {{\rm{R}}}^{N} $ with the second inertia functional of reference AIMS Mathematics compressible euler equations initial value problems blowup second inertia functional of reference energy method |
| title | Blowup for $ {{\rm{C}}}^{1} $ solutions of Euler equations in $ {{\rm{R}}}^{N} $ with the second inertia functional of reference |
| title_full | Blowup for $ {{\rm{C}}}^{1} $ solutions of Euler equations in $ {{\rm{R}}}^{N} $ with the second inertia functional of reference |
| title_fullStr | Blowup for $ {{\rm{C}}}^{1} $ solutions of Euler equations in $ {{\rm{R}}}^{N} $ with the second inertia functional of reference |
| title_full_unstemmed | Blowup for $ {{\rm{C}}}^{1} $ solutions of Euler equations in $ {{\rm{R}}}^{N} $ with the second inertia functional of reference |
| title_short | Blowup for $ {{\rm{C}}}^{1} $ solutions of Euler equations in $ {{\rm{R}}}^{N} $ with the second inertia functional of reference |
| title_sort | blowup for rm c 1 solutions of euler equations in rm r n with the second inertia functional of reference |
| topic | compressible euler equations initial value problems blowup second inertia functional of reference energy method |
| url | https://www.aimspress.com/article/doi/10.3934/math.2023412?viewType=HTML |
| work_keys_str_mv | AT manwaiyuen blowupforrmc1solutionsofeulerequationsinrmrnwiththesecondinertiafunctionalofreference |