{"title":"Global existence and boundedness of classical solutions in chemotaxis-(Navier-)Stokes system with singular sensitivity and self-consistent term","authors":"Yuying Wang, Liqiong Pu, Jiashan Zheng","doi":"10.1016/j.aml.2025.109518","DOIUrl":null,"url":null,"abstract":"<div><div>This paper addresses the global existence and boundedness of classical solutions to the Neumann-Neumann-Dirichlet value problem for the chemotaxis system, as described by <span><span><span>(*)</span><span><math><mfenced><mrow><mtable><mtr><mtd><msub><mrow><mi>n</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><mi>u</mi><mi>⋅</mi><mo>∇</mo><mi>n</mi><mo>=</mo><mi>Δ</mi><mi>n</mi><mo>−</mo><mi>χ</mi><mo>∇</mo><mi>⋅</mi><mfenced><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mn>1</mn><mo>+</mo><mi>c</mi></mrow></mfrac><mo>∇</mo><mi>c</mi></mrow></mfenced><mo>+</mo><mo>∇</mo><mi>⋅</mi><mrow><mo>(</mo><mi>n</mi><mo>∇</mo><mi>ϕ</mi><mo>)</mo></mrow><mo>,</mo><mspace></mspace><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>c</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><mi>u</mi><mi>⋅</mi><mo>∇</mo><mi>c</mi><mo>=</mo><mi>Δ</mi><mi>c</mi><mo>−</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup><mi>c</mi><mo>,</mo><mspace></mspace><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><mo>∇</mo><mi>P</mi><mo>=</mo><mi>Δ</mi><mi>u</mi><mo>−</mo><mi>n</mi><mo>∇</mo><mi>ϕ</mi><mo>+</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>1</mn><mo>+</mo><mi>c</mi></mrow></mfrac><mo>∇</mo><mi>c</mi><mo>,</mo><mspace></mspace><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><mo>∇</mo><mi>⋅</mi><mi>u</mi><mo>=</mo><mn>0</mn><mo>,</mo><mspace></mspace><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mi>t</mi><mo>></mo><mn>0</mn></mtd></mtr><mtr><mtd><mfrac><mrow><mi>∂</mi><mi>n</mi></mrow><mrow><mi>∂</mi><mi>ν</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>∂</mi><mi>c</mi></mrow><mrow><mi>∂</mi><mi>ν</mi></mrow></mfrac><mo>=</mo><mn>0</mn><mo>,</mo><mi>u</mi><mo>=</mo><mn>0</mn><mo>,</mo><mspace></mspace><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mi>∂</mi><mi>Ω</mi><mo>,</mo><mi>t</mi><mo>></mo><mn>0</mn></mtd></mtr><mtr><mtd><mi>n</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>,</mo><mi>c</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>0</mn></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>,</mo><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>,</mo><mspace></mspace><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi></mtd></mtr></mtable></mrow></mfenced></math></span></span></span>in a smoothly bounded domain <span><math><mrow><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mrow><mo>(</mo><mi>N</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>)</mo></mrow></mrow></math></span>, where <span><math><mrow><mi>α</mi><mo>></mo><mn>0</mn></mrow></math></span> and the gravitational potential function <span><math><mrow><mi>ϕ</mi><mo>∈</mo><msup><mrow><mi>W</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>∞</mi></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></math></span>. It is confirmed that for any selection of <span><math><mi>χ</mi></math></span> satisfying <span><span><span><math><mrow><mn>0</mn><mo><</mo><mi>χ</mi><mo>≤</mo><mfrac><mrow><msqrt><mrow><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><msubsup><mrow><mi>δ</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><msup><mrow><mo>ln</mo></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mn>1</mn><mo>+</mo><msub><mrow><mo>‖</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>‖</mo></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></msub><mo>)</mo></mrow><mo>+</mo><mi>k</mi><msup><mrow><mrow><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mo>ln</mo></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mn>1</mn><mo>+</mo><msub><mrow><mo>‖</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>‖</mo></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></msub><mo>)</mo></mrow><mo>)</mo></mrow></mrow></msqrt><mo>−</mo><mi>k</mi><msub><mrow><mi>δ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>ln</mo><mrow><mo>(</mo><mn>1</mn><mo>+</mo><msub><mrow><mo>‖</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>‖</mo></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></msub><mo>)</mo></mrow></mrow><mrow><mi>k</mi><mrow><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mo>ln</mo><mrow><mo>(</mo><mn>1</mn><mo>+</mo><msub><mrow><mo>‖</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>‖</mo></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></msub><mo>)</mo></mrow></mrow></mfrac></mrow></math></span></span></span>with <span><math><mrow><mi>k</mi><mo>→</mo><mfrac><mrow><mi>N</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></math></span> and <span><math><mrow><msub><mrow><mi>δ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>→</mo><mi>k</mi><mo>−</mo><mn>1</mn></mrow></math></span>, then the corresponding problem <span><span>(*)</span></span> admits a globally and uniformly bounded classical solution via an approach to introduce a trigonometric-type weight function.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"166 ","pages":"Article 109518"},"PeriodicalIF":2.9000,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Mathematics Letters","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0893965925000680","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
This paper addresses the global existence and boundedness of classical solutions to the Neumann-Neumann-Dirichlet value problem for the chemotaxis system, as described by (*)in a smoothly bounded domain , where and the gravitational potential function . It is confirmed that for any selection of satisfying with and , then the corresponding problem (*) admits a globally and uniformly bounded classical solution via an approach to introduce a trigonometric-type weight function.
期刊介绍:
The purpose of Applied Mathematics Letters is to provide a means of rapid publication for important but brief applied mathematical papers. The brief descriptions of any work involving a novel application or utilization of mathematics, or a development in the methodology of applied mathematics is a potential contribution for this journal. This journal''s focus is on applied mathematics topics based on differential equations and linear algebra. Priority will be given to submissions that are likely to appeal to a wide audience.