Asymptotic Profile for Diffusion Wave Terms of the Compressible Navier–Stokes–Korteweg System
The asymptotic profile for diffusion wave terms of solutions to the compressible Navier–Stokes–Korteweg system is studied on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-...
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MDPI AG
2021-03-01
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| author | Takayuki Kobayashi Masashi Misawa Kazuyuki Tsuda |
| author_facet | Takayuki Kobayashi Masashi Misawa Kazuyuki Tsuda |
| author_sort | Takayuki Kobayashi |
| collection | DOAJ |
| description | The asymptotic profile for diffusion wave terms of solutions to the compressible Navier–Stokes–Korteweg system is studied on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mn>2</mn></msup></semantics></math></inline-formula>. The diffusion wave with time-decay estimate was studied by Hoff and Zumbrun (1995, 1997), Kobayashi and Shibata (2002), and Kobayashi and Tsuda (2018) for compressible Navier–Stokes and compressible Navier–Stokes–Korteweg systems. Our main assertion in this paper is that, for some initial conditions given by the Hardy space, asymptotic behaviors in space–time <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>2</mn></msup></semantics></math></inline-formula> of the diffusion wave parts are essentially different between density and the potential flow part of the momentum. Even though measuring by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>2</mn></msup></semantics></math></inline-formula> on space, decay of the potential flow part is slower than that of the Stokes flow part of the momentum. The proof is based on a modified version of Morawetz’s energy estimate, and the Fefferman–Stein inequality on the duality between the Hardy space and functions of bounded mean oscillation. |
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| last_indexed | 2024-03-10T13:01:03Z |
| publishDate | 2021-03-01 |
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| spelling | doaj.art-7cf9c940e680425b98e29ccfbf670b502023-11-21T11:33:14ZengMDPI AGMathematics2227-73902021-03-019668310.3390/math9060683Asymptotic Profile for Diffusion Wave Terms of the Compressible Navier–Stokes–Korteweg SystemTakayuki Kobayashi0Masashi Misawa1Kazuyuki Tsuda2Department of Systems Innovation, Graduate School of Engineering Science, Osaka University, 1-3, Machikaneyamacho, Toyonakashi 560-8531, JapanDepartment of Science, Faculty of Science, Kumamoto University, 2-39-1, Kurokami, Chuo-ku, Kumamoto 860-8555, JapanFaculty of Science and Engineering, Kyushu Sangyo University, 3-1, Matsukadai 2-Chome, Higashi-ku, Fukuoka 813-8503, JapanThe asymptotic profile for diffusion wave terms of solutions to the compressible Navier–Stokes–Korteweg system is studied on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mn>2</mn></msup></semantics></math></inline-formula>. The diffusion wave with time-decay estimate was studied by Hoff and Zumbrun (1995, 1997), Kobayashi and Shibata (2002), and Kobayashi and Tsuda (2018) for compressible Navier–Stokes and compressible Navier–Stokes–Korteweg systems. Our main assertion in this paper is that, for some initial conditions given by the Hardy space, asymptotic behaviors in space–time <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>2</mn></msup></semantics></math></inline-formula> of the diffusion wave parts are essentially different between density and the potential flow part of the momentum. Even though measuring by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>2</mn></msup></semantics></math></inline-formula> on space, decay of the potential flow part is slower than that of the Stokes flow part of the momentum. The proof is based on a modified version of Morawetz’s energy estimate, and the Fefferman–Stein inequality on the duality between the Hardy space and functions of bounded mean oscillation.https://www.mdpi.com/2227-7390/9/6/683compressible Navier–Stokes–Korteweg systemasymptotic profilediffusion wavehardy space |
| spellingShingle | Takayuki Kobayashi Masashi Misawa Kazuyuki Tsuda Asymptotic Profile for Diffusion Wave Terms of the Compressible Navier–Stokes–Korteweg System Mathematics compressible Navier–Stokes–Korteweg system asymptotic profile diffusion wave hardy space |
| title | Asymptotic Profile for Diffusion Wave Terms of the Compressible Navier–Stokes–Korteweg System |
| title_full | Asymptotic Profile for Diffusion Wave Terms of the Compressible Navier–Stokes–Korteweg System |
| title_fullStr | Asymptotic Profile for Diffusion Wave Terms of the Compressible Navier–Stokes–Korteweg System |
| title_full_unstemmed | Asymptotic Profile for Diffusion Wave Terms of the Compressible Navier–Stokes–Korteweg System |
| title_short | Asymptotic Profile for Diffusion Wave Terms of the Compressible Navier–Stokes–Korteweg System |
| title_sort | asymptotic profile for diffusion wave terms of the compressible navier stokes korteweg system |
| topic | compressible Navier–Stokes–Korteweg system asymptotic profile diffusion wave hardy space |
| url | https://www.mdpi.com/2227-7390/9/6/683 |
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