Global Well-Posedness and Determining Nodes of Non-Autonomous Navier–Stokes Equations with Infinite Delay on Bounded Domains
The asymptotic behavior of solutions to nonlinear partial differential equations is an important tool for studying their long-term behavior. However, when studying the asymptotic behavior of solutions to nonlinear partial differential equations with delay, the delay factor <inline-formula><...
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2025-01-01
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| author | Huanzhi Ge Feng Du |
| author_facet | Huanzhi Ge Feng Du |
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| description | The asymptotic behavior of solutions to nonlinear partial differential equations is an important tool for studying their long-term behavior. However, when studying the asymptotic behavior of solutions to nonlinear partial differential equations with delay, the delay factor <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mo>(</mo><mi>t</mi><mo>+</mo><mi>θ</mi><mo>)</mo></mrow></semantics></math></inline-formula> in the delay term may lead to oscillations, hysteresis effects, and other phenomena in the solution, which increases the difficulty of studying the well-posedness and asymptotic behavior of the solution. This study investigates the global well-posedness and asymptotic behavior of solutions to the non-autonomous Navier–Stokes equations incorporating infinite delays. To establish global well-posedness, we first construct several suitable function spaces and then prove them using the Galekin approximation method. Then, by accurately estimating the number of determining nodes, we reveal the asymptotic behavior of the solution. The results indicate that the long-term behavior of a strong solution can be determined by its values at a finite number of nodes. |
| format | Article |
| id | doaj-art-50b177dd439c4042818b474ac8bcfa09 |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-01-01 |
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| spelling | doaj-art-50b177dd439c4042818b474ac8bcfa092025-01-24T13:39:47ZengMDPI AGMathematics2227-73902025-01-0113222210.3390/math13020222Global Well-Posedness and Determining Nodes of Non-Autonomous Navier–Stokes Equations with Infinite Delay on Bounded DomainsHuanzhi Ge0Feng Du1School of Information and Mathematics, Yangtze University, Jingzhou 434023, ChinaSchool of Mathematics and Physics Science, Jingchu University of Technology, Jingmen 448000, ChinaThe asymptotic behavior of solutions to nonlinear partial differential equations is an important tool for studying their long-term behavior. However, when studying the asymptotic behavior of solutions to nonlinear partial differential equations with delay, the delay factor <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mo>(</mo><mi>t</mi><mo>+</mo><mi>θ</mi><mo>)</mo></mrow></semantics></math></inline-formula> in the delay term may lead to oscillations, hysteresis effects, and other phenomena in the solution, which increases the difficulty of studying the well-posedness and asymptotic behavior of the solution. This study investigates the global well-posedness and asymptotic behavior of solutions to the non-autonomous Navier–Stokes equations incorporating infinite delays. To establish global well-posedness, we first construct several suitable function spaces and then prove them using the Galekin approximation method. Then, by accurately estimating the number of determining nodes, we reveal the asymptotic behavior of the solution. The results indicate that the long-term behavior of a strong solution can be determined by its values at a finite number of nodes.https://www.mdpi.com/2227-7390/13/2/222Navier–Stokes equationsinfinite delayglobal well-posednessasymptotic behaviordetermining nodeslong-time behavior |
| spellingShingle | Huanzhi Ge Feng Du Global Well-Posedness and Determining Nodes of Non-Autonomous Navier–Stokes Equations with Infinite Delay on Bounded Domains Mathematics Navier–Stokes equations infinite delay global well-posedness asymptotic behavior determining nodes long-time behavior |
| title | Global Well-Posedness and Determining Nodes of Non-Autonomous Navier–Stokes Equations with Infinite Delay on Bounded Domains |
| title_full | Global Well-Posedness and Determining Nodes of Non-Autonomous Navier–Stokes Equations with Infinite Delay on Bounded Domains |
| title_fullStr | Global Well-Posedness and Determining Nodes of Non-Autonomous Navier–Stokes Equations with Infinite Delay on Bounded Domains |
| title_full_unstemmed | Global Well-Posedness and Determining Nodes of Non-Autonomous Navier–Stokes Equations with Infinite Delay on Bounded Domains |
| title_short | Global Well-Posedness and Determining Nodes of Non-Autonomous Navier–Stokes Equations with Infinite Delay on Bounded Domains |
| title_sort | global well posedness and determining nodes of non autonomous navier stokes equations with infinite delay on bounded domains |
| topic | Navier–Stokes equations infinite delay global well-posedness asymptotic behavior determining nodes long-time behavior |
| url | https://www.mdpi.com/2227-7390/13/2/222 |
| work_keys_str_mv | AT huanzhige globalwellposednessanddeterminingnodesofnonautonomousnavierstokesequationswithinfinitedelayonboundeddomains AT fengdu globalwellposednessanddeterminingnodesofnonautonomousnavierstokesequationswithinfinitedelayonboundeddomains |