{"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":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":3.2000,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Nonlinear Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/anona-2023-0131","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","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:
Advances in Nonlinear Analysis (ANONA) aims to publish selected research contributions devoted to nonlinear problems coming from different areas, with particular reference to those introducing new techniques capable of solving a wide range of problems. The Journal focuses on papers that address significant problems in pure and applied nonlinear analysis. ANONA seeks to present the most significant advances in this field to a wide readership, including researchers and graduate students in mathematics, physics, and engineering.