{"title":"Global existence and decay estimates of the classical solution to the compressible Navier-Stokes-Smoluchowski equations in ℝ3","authors":"Leilei Tong","doi":"10.1515/anona-2023-0131","DOIUrl":null,"url":null,"abstract":"\n <jats:p>The compressible Navier-Stokes-Smoluchowski equations under investigation concern the behavior of the mixture of fluid and particles at a macroscopic scale. We devote to the existence of the global classical solution near the stationary solution based on the energy method under weaker conditions imposed on the external potential compared with Chen et al. (Global existence and time–decay estimates of solutions to the compressible Navier-Stokes-Smoluchowski equations, Discrete Contin. Dyn. Syst. 36 (2016), no. 10, 5287–5307). Under further assumptions that the stationary solution <jats:inline-formula>\n <jats:alternatives>\n <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0131_eq_001.png\" />\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\n <m:msup>\n <m:mrow>\n <m:mrow>\n <m:mo>(</m:mo>\n <m:mrow>\n <m:msub>\n <m:mrow>\n <m:mi>ρ</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mi>s</m:mi>\n </m:mrow>\n </m:msub>\n <m:mrow>\n <m:mo>(</m:mo>\n <m:mrow>\n <m:mi>x</m:mi>\n </m:mrow>\n <m:mo>)</m:mo>\n </m:mrow>\n <m:mo>,</m:mo>\n <m:mn>0</m:mn>\n <m:mo>,</m:mo>\n <m:mn>0</m:mn>\n </m:mrow>\n <m:mo>)</m:mo>\n </m:mrow>\n </m:mrow>\n <m:mrow>\n <m:mi>T</m:mi>\n </m:mrow>\n </m:msup>\n </m:math>\n <jats:tex-math>{\\left({\\rho }_{s}\\left(x),0,0)}^{T}</jats:tex-math>\n </jats:alternatives>\n </jats:inline-formula> is in a small neighborhood of the constant state <jats:inline-formula>\n <jats:alternatives>\n <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0131_eq_002.png\" />\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\n <m:msup>\n <m:mrow>\n <m:mrow>\n <m:mo>(</m:mo>\n <m:mrow>\n <m:mover accent=\"true\">\n <m:mrow>\n <m:mi>ρ</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mo>¯</m:mo>\n </m:mrow>\n </m:mover>\n <m:mo>,</m:mo>\n <m:mn>0</m:mn>\n <m:mo>,</m:mo>\n <m:mn>0</m:mn>\n </m:mrow>\n <m:mo>)</m:mo>\n </m:mrow>\n </m:mrow>\n <m:mrow>\n <m:mi>T</m:mi>\n </m:mrow>\n </m:msup>\n </m:math>\n <jats:tex-math>{\\left(\\bar{\\rho },0,0)}^{T}</jats:tex-math>\n </jats:alternatives>\n </jats:inline-formula> at infinity, we also obtain the time decay rates of the solution by the combination of the energy method and the linear <jats:inline-formula>\n <jats:alternatives>\n <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0131_eq_003.png\" />\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\n <m:msup>\n <m:mrow>\n <m:mi>L</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mi>p</m:mi>\n </m:mrow>\n </m:msup>\n </m:math>\n <jats:tex-math>{L}^{p}</jats:tex-math>\n </jats:alternatives>\n </jats:inline-formula>-<jats:inline-formula>\n <jats:alternatives>\n <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0131_eq_004.png\" />\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\n <m:msup>\n <m:mrow>\n <m:mi>L</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mi>q</m:mi>\n </m:mrow>\n </m:msup>\n </m:math>\n <jats:tex-math>{L}^{q}</jats:tex-math>\n </jats:alternatives>\n </jats:inline-formula> decay estimates.</jats:p>","PeriodicalId":3,"journal":{"name":"ACS Applied Electronic Materials","volume":"58 9","pages":""},"PeriodicalIF":4.3000,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Electronic Materials","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/anona-2023-0131","RegionNum":3,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"ENGINEERING, ELECTRICAL & ELECTRONIC","Score":null,"Total":0}
引用次数: 0
Abstract
The compressible Navier-Stokes-Smoluchowski equations under investigation concern the behavior of the mixture of fluid and particles at a macroscopic scale. We devote to the existence of the global classical solution near the stationary solution based on the energy method under weaker conditions imposed on the external potential compared with Chen et al. (Global existence and time–decay estimates of solutions to the compressible Navier-Stokes-Smoluchowski equations, Discrete Contin. Dyn. Syst. 36 (2016), no. 10, 5287–5307). Under further assumptions that the stationary solution (ρs(x),0,0)T{\left({\rho }_{s}\left(x),0,0)}^{T} is in a small neighborhood of the constant state (ρ¯,0,0)T{\left(\bar{\rho },0,0)}^{T} at infinity, we also obtain the time decay rates of the solution by the combination of the energy method and the linear Lp{L}^{p}-Lq{L}^{q} decay estimates.
所研究的可压缩纳维-斯托克斯-斯莫卢霍夫斯基方程涉及流体和粒子在宏观尺度上的混合行为。与 Chen 等人(Global existence and time-decay estimates of solutions to the compressible Navier-Stokes-Smoluchowski equations, Discrete Contin.Dyn.Syst.36 (2016),第 10 期,5287-5307)。进一步假设静止解 ( ρ s ( x ) , 0 , 0 ) T {\left({\rho }_{s}\left(x),0,0)}^{T} 在恒定状态 ( ρ ¯ , 0 , 0 ) 的一个小邻域内。 T {left(\bar{\rho},0,0)}^{T}在无穷远处,我们还可以通过能量法和线性 L p {L}^{p} 的结合得到解的时间衰减率。 - L q {L}^{q} 衰减估计。
期刊介绍:
ACS Applied Electronic Materials is an interdisciplinary journal publishing original research covering all aspects of electronic materials. The journal is devoted to reports of new and original experimental and theoretical research of an applied nature that integrate knowledge in the areas of materials science, engineering, optics, physics, and chemistry into important applications of electronic materials. Sample research topics that span the journal's scope are inorganic, organic, ionic and polymeric materials with properties that include conducting, semiconducting, superconducting, insulating, dielectric, magnetic, optoelectronic, piezoelectric, ferroelectric and thermoelectric.
Indexed/Abstracted:
Web of Science SCIE
Scopus
CAS
INSPEC
Portico