Global Mild Solution for a Fractional Chemotaxis–Fluid System Modeling Coral Fertilization With Tensor-Valued Sensitivity

IF 2.1 3区数学 Q1 MATHEMATICS, APPLIED

Mathematical Methods in the Applied Sciences Pub Date : 2025-01-09 DOI:10.1002/mma.10671

Heng Ruan, Zuhan Liu, Chao Jiang

{"title":"Global Mild Solution for a Fractional Chemotaxis–Fluid System Modeling Coral Fertilization With Tensor-Valued Sensitivity","authors":"Heng Ruan, Zuhan Liu, Chao Jiang","doi":"10.1002/mma.10671","DOIUrl":null,"url":null,"abstract":"<div>\n \n In this paper, we consider the following a fractional chemotaxis–fluid system modeling coral fertilization with tensor-valued sensitivity in \n<math>\n <semantics>\n <mrow>\n <msup>\n <mrow>\n <mi>ℝ</mi>\n </mrow>\n <mrow>\n <mn>3</mn>\n </mrow>\n </msup>\n </mrow>\n <annotation>$$ {\\mathbb{R}}&#x0005E;3 $$</annotation>\n </semantics></math>.\n Here the tensor-valued sensitivity function \n<math>\n <semantics>\n <mrow>\n <mi>S</mi>\n <mfenced>\n <mrow>\n <mi>x</mi>\n <mo>,</mo>\n <mi>ρ</mi>\n <mo>,</mo>\n <mi>c</mi>\n </mrow>\n </mfenced>\n </mrow>\n <annotation>$$ S\\left(x,\\rho, c\\right) $$</annotation>\n </semantics></math> satisfies \n<math>\n <semantics>\n <mrow>\n <mo>|</mo>\n <mi>S</mi>\n <mfenced>\n <mrow>\n <mi>x</mi>\n <mo>,</mo>\n <mi>ρ</mi>\n <mo>,</mo>\n <mi>c</mi>\n </mrow>\n </mfenced>\n <mo>|</mo>\n <mo>≤</mo>\n <msub>\n <mrow>\n <mi>C</mi>\n </mrow>\n <mrow>\n <mi>S</mi>\n </mrow>\n </msub>\n </mrow>\n <annotation>$$ \\mid S\\left(x,\\rho, c\\right)\\mid \\le {C}_S $$</annotation>\n </semantics></math>. We show that if \n<math>\n <semantics>\n <mrow>\n <mi>α</mi>\n <mo>∈</mo>\n <mo>(</mo>\n <mfrac>\n <mrow>\n <mn>1</mn>\n </mrow>\n <mrow>\n <mn>2</mn>\n </mrow>\n </mfrac>\n <mo>,</mo>\n <mn>1</mn>\n <mo>)</mo>\n </mrow>\n <annotation>$$ \\alpha \\in \\left(\\frac{1}{2},1\\right) $$</annotation>\n </semantics></math> and the initial data satisfy \n<math>\n <semantics>\n <mrow>\n <msub>\n <mrow>\n <mo>‖</mo>\n <mrow>\n <msub>\n <mrow>\n <mi>ρ</mi>\n </mrow>\n <mrow>\n <mn>0</mn>\n </mrow>\n </msub>\n </mrow>\n <mo>‖</mo>\n </mrow>\n <mrow>\n <msubsup>\n <mrow>\n <mi>L</mi>\n </mrow>\n <mrow>\n <mi>w</mi>\n </mrow>\n <mrow>\n <mfrac>\n <mrow>\n <mn>3</mn>\n </mrow>\n <mrow>\n <mn>2</mn>\n <mi>α</mi>\n </mrow>\n </mfrac>\n </mrow>\n </msubsup>\n <mfenced>\n <mrow>\n <msup>\n <mrow>\n <mi>ℝ</mi>\n </mrow>\n <mrow>\n <mn>3</mn>\n </mrow>\n </msup>\n </mrow>\n </mfenced>\n </mrow>\n </msub>\n <mo>≤</mo>\n <mi>ϵ</mi>\n <mo>,</mo>\n <mo>‖</mo>\n <msub>\n <mrow>\n <mi>m</mi>\n </mrow>\n <mrow>\n <mn>0</mn>\n </mrow>\n </msub>\n <msub>\n <mrow>\n <mo>‖</mo>\n </mrow>\n <mrow>\n <msup>\n <mrow>\n <mi>L</mi>\n </mrow>\n <mrow>\n <mfrac>\n <mrow>\n <mn>3</mn>\n </mrow>\n <mrow>\n <mn>2</mn>\n <mi>α</mi>\n </mrow>\n </mfrac>\n </mrow>\n </msup>\n <mfenced>\n <mrow>\n <msup>\n <mrow>\n <mi>ℝ</mi>\n </mrow>\n <mrow>\n <mn>3</mn>\n </mrow>\n </msup>\n </mrow>\n </mfenced>\n <mo>∩</mo>\n <msup>\n <mrow>\n <mi>L</mi>\n </mrow>\n <mrow>\n <mfrac>\n <mrow>\n <mn>3</mn>\n </mrow>\n <mrow>\n <mn>4</mn>\n <mi>α</mi>\n <mo>−</mo>\n <mn>2</mn>\n </mrow>\n </mfrac>\n </mrow>\n </msup>\n <mfenced>\n <mrow>\n <msup>\n <mrow>\n <mi>ℝ</mi>\n </mrow>\n <mrow>\n <mn>3</mn>\n </mrow>\n </msup>\n </mrow>\n </mfenced>\n </mrow>\n </msub>\n <mo>≤</mo>\n <mi>ϵ</mi>\n <mo>,</mo>\n <msub>\n <mrow>\n <mo>‖</mo>\n <mrow>\n <mo>∇</mo>\n <msub>\n <mrow>\n <mi>c</mi>\n </mrow>\n <mrow>\n <mn>0</mn>\n </mrow>\n </msub>\n </mrow>\n <mo>‖</mo>\n </mrow>\n <mrow>\n <msubsup>\n <mrow>\n <mi>L</mi>\n </mrow>\n <mrow>\n <mi>w</mi>\n </mrow>\n <mrow>\n <mfrac>\n <mrow>\n <mn>3</mn>\n </mrow>\n <mrow>\n <mn>2</mn>\n <mi>α</mi>\n <mo>−</mo>\n <mn>1</mn>\n </mrow>\n </mfrac>\n </mrow>\n </msubsup>\n <mfenced>\n <mrow>\n <msup>\n <mrow>\n <mi>ℝ</mi>\n </mrow>\n <mrow>\n <mn>3</mn>\n </mrow>\n </msup>\n </mrow>\n </mfenced>\n </mrow>\n </msub>\n <mo>≤</mo>\n <mi>ϵ</mi>\n <mo>,</mo>\n <msub>\n <mrow>\n <mo>‖</mo>\n <mrow>\n <msub>\n <mrow>\n <mi>u</mi>\n </mrow>\n <mrow>\n <mn>0</mn>\n </mrow>\n </msub>\n </mrow>\n <mo>‖</mo>\n </mrow>\n <mrow>\n <msubsup>\n <mrow>\n <mi>L</mi>\n </mrow>\n <mrow>\n <mi>w</mi>\n </mrow>\n <mrow>\n <mfrac>\n <mrow>\n <mn>3</mn>\n </mrow>\n <mrow>\n <mn>2</mn>\n <mi>α</mi>\n </mrow>\n </mfrac>\n </mrow>\n </msubsup>\n <mfenced>\n <mrow>\n <msup>\n <mrow>\n <mi>ℝ</mi>\n </mrow>\n <mrow>\n <mn>3</mn>\n </mrow>\n </msup>\n </mrow>\n </mfenced>\n </mrow>\n </msub>\n <mo>≤</mo>\n <mi>ϵ</mi>\n </mrow>\n <annotation>$$ {\\left\\Vert {\\rho}_0\\right\\Vert}_{L_w&#x0005E;{\\frac{3}{2\\alpha }}\\left({\\mathbb{R}}&#x0005E;3\\right)}\\le \\epsilon, {\\left\\Vert {m}_0\\right\\Vert}_{L&#x0005E;{\\frac{3}{2\\alpha }}\\left({\\mathbb{R}}&#x0005E;3\\right)\\cap {L}&#x0005E;{\\frac{3}{4\\alpha -2}}\\left({\\mathbb{R}}&#x0005E;3\\right)}\\le \\epsilon, {\\left\\Vert \\nabla {c}_0\\right\\Vert}_{L_w&#x0005E;{\\frac{3}{2\\alpha -1}}\\left({\\mathbb{R}}&#x0005E;3\\right)}\\le \\epsilon, {\\left\\Vert {u}_0\\right\\Vert}_{L_w&#x0005E;{\\frac{3}{2\\alpha }}\\left({\\mathbb{R}}&#x0005E;3\\right)}\\le \\epsilon $$</annotation>\n </semantics></math> or \n<math>\n <semantics>\n <mrow>\n <msub>\n <mrow>\n <mo>‖</mo>\n <mrow>\n <msub>\n <mrow>\n <mi>ρ</mi>\n </mrow>\n <mrow>\n <mn>0</mn>\n </mrow>\n </msub>\n </mrow>\n <mo>‖</mo>\n </mrow>\n <mrow>\n <msubsup>\n <mrow>\n <mi>L</mi>\n </mrow>\n <mrow>\n <mi>w</mi>\n </mrow>\n <mrow>\n <mfrac>\n <mrow>\n <mn>3</mn>\n </mrow>\n <mrow>\n <mn>2</mn>\n <mi>α</mi>\n </mrow>\n </mfrac>\n </mrow>\n </msubsup>\n <mfenced>\n <mrow>\n <msup>\n <mrow>\n <mi>ℝ</mi>\n </mrow>\n <mrow>\n <mn>3</mn>\n </mrow>\n </msup>\n </mrow>\n </mfenced>\n </mrow>\n </msub>\n <mo>≤</mo>\n <mi>ϵ</mi>\n <mo>,</mo>\n <mo>‖</mo>\n <msub>\n <mrow>\n <mi>m</mi>\n </mrow>\n <mrow>\n <mn>0</mn>\n </mrow>\n </msub>\n <msub>\n <mrow>\n <mo>‖</mo>\n </mrow>\n <mrow>\n <msup>\n <mrow>\n <mi>L</mi>\n </mrow>\n <mrow>\n <mfrac>\n <mrow>\n <mn>3</mn>\n </mrow>\n <mrow>\n <mn>2</mn>\n <mi>α</mi>\n </mrow>\n </mfrac>\n </mrow>\n </msup>\n <mfenced>\n <mrow>\n <msup>\n <mrow>\n <mi>ℝ</mi>\n </mrow>\n <mrow>\n <mn>3</mn>\n </mrow>\n </msup>\n </mrow>\n </mfenced>\n <mo>∩</mo>\n <msup>\n <mrow>\n <mi>L</mi>\n </mrow>\n <mrow>\n <mfrac>\n <mrow>\n <mn>3</mn>\n </mrow>\n <mrow>\n <mn>4</mn>\n <mi>α</mi>\n <mo>−</mo>\n <mn>2</mn>\n </mrow>\n </mfrac>\n </mrow>\n </msup>\n <mfenced>\n <mrow>\n <msup>\n <mrow>\n <mi>ℝ</mi>\n </mrow>\n <mrow>\n <mn>3</mn>\n </mrow>\n </msup>\n </mrow>\n </mfenced>\n </mrow>\n </msub>\n <mo>≤</mo>\n <mi>ϵ</mi>\n <mo>,</mo>\n <msub>\n <mrow>\n <mo>‖</mo>\n <mrow>\n <msub>\n <mrow>\n <mi>c</mi>\n </mrow>\n <mrow>\n <mn>0</mn>\n </mrow>\n </msub>\n </mrow>\n <mo>‖</mo>\n </mrow>\n <mrow>\n <msup>\n <mrow>\n <mi>L</mi>\n </mrow>\n <mrow>\n <mn>1</mn>\n </mrow>\n </msup>\n <mfenced>\n <mrow>\n <msup>\n <mrow>\n <mi>ℝ</mi>\n </mrow>\n <mrow>\n <mn>3</mn>\n </mrow>\n </msup>\n </mrow>\n </mfenced>\n </mrow>\n </msub>\n <mo>≤</mo>\n <mi>ϵ</mi>\n <mo>,</mo>\n <msub>\n <mrow>\n <mo>‖</mo>\n <mrow>\n <msub>\n <mrow>\n <mi>u</mi>\n </mrow>\n <mrow>\n <mn>0</mn>\n </mrow>\n </msub>\n </mrow>\n <mo>‖</mo>\n </mrow>\n <mrow>\n <msubsup>\n <mrow>\n <mi>L</mi>\n </mrow>\n <mrow>\n <mi>w</mi>\n </mrow>\n <mrow>\n <mfrac>\n <mrow>\n <mn>3</mn>\n </mrow>\n <mrow>\n <mn>2</mn>\n </mrow>\n </mfrac>\n </mrow>\n </msubsup>\n <mfenced>\n <mrow>\n <msup>\n <mrow>\n <mi>ℝ</mi>\n </mrow>\n <mrow>\n <mn>3</mn>\n </mrow>\n </msup>\n </mrow>\n </mfenced>\n </mrow>\n </msub>\n <mo>≤</mo>\n <mi>ϵ</mi>\n <mo>,</mo>\n </mrow>\n <annotation>$$ {\\left\\Vert {\\rho}_0\\right\\Vert}_{L_w&#x0005E;{\\frac{3}{2\\alpha }}\\left({\\mathbb{R}}&#x0005E;3\\right)}\\le \\epsilon, {\\left\\Vert {m}_0\\right\\Vert}_{L&#x0005E;{\\frac{3}{2\\alpha }}\\left({\\mathbb{R}}&#x0005E;3\\right)\\cap {L}&#x0005E;{\\frac{3}{4\\alpha -2}}\\left({\\mathbb{R}}&#x0005E;3\\right)}\\le \\epsilon, {\\left\\Vert {c}_0\\right\\Vert}_{L&#x0005E;1\\left({\\mathbb{R}}&#x0005E;3\\right)}\\le \\epsilon, {\\left\\Vert {u}_0\\right\\Vert}_{L_w&#x0005E;{\\frac{3}{2}}\\left({\\mathbb{R}}&#x0005E;3\\right)}\\le \\epsilon, $$</annotation>\n </semantics></math> then this system admits a unique global mild solution, and we also determine the asymptotic behavior and optimal convergence rates of mild solution.\n </div>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"48 6","pages":"6278-6291"},"PeriodicalIF":2.1000,"publicationDate":"2025-01-09","publicationTypes":"Journal 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引用次数: 0

Abstract

In this paper, we consider the following a fractional chemotaxis–fluid system modeling coral fertilization with tensor-valued sensitivity in $ℝ^{3}$ .

Here the tensor-valued sensitivity function $S (x, ρ, c)$ satisfies $| S (x, ρ, c) | \leq C_{S}$ . We show that if $α \in (\frac{1}{2}, 1)$ and the initial data satisfy ${‖ ρ_{0} ‖}_{L_{w}^{\frac{3}{2 α}} (ℝ^{3})} \leq ϵ, ‖ m_{0} ‖_{L^{\frac{3}{2 α}} (ℝ^{3}) \cap L^{\frac{3}{4 α - 2}} (ℝ^{3})} \leq ϵ, {‖ \nabla c_{0} ‖}_{L_{w}^{\frac{3}{2 α - 1}} (ℝ^{3})} \leq ϵ, {‖ u_{0} ‖}_{L_{w}^{\frac{3}{2 α}} (ℝ^{3})} \leq ϵ$ or ${‖ ρ_{0} ‖}_{L_{w}^{\frac{3}{2 α}} (ℝ^{3})} \leq ϵ, ‖ m_{0} ‖_{L^{\frac{3}{2 α}} (ℝ^{3}) \cap L^{\frac{3}{4 α - 2}} (ℝ^{3})} \leq ϵ, {‖ c_{0} ‖}_{L^{1} (ℝ^{3})} \leq ϵ, {‖ u_{0} ‖}_{L_{w}^{\frac{3}{2}} (ℝ^{3})} \leq ϵ,$ then this system admits a unique global mild solution, and we also determine the asymptotic behavior and optimal convergence rates of mild solution.

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来源期刊

Mathematical Methods in the Applied Sciences 数学-应用数学

CiteScore

4.90

自引率

6.90%

发文量

798

审稿时长

6 months

期刊介绍： Mathematical Methods in the Applied Sciences publishes papers dealing with new mathematical methods for the consideration of linear and non-linear, direct and inverse problems for physical relevant processes over time- and space- varying media under certain initial, boundary, transition conditions etc. Papers dealing with biomathematical content, population dynamics and network problems are most welcome. Mathematical Methods in the Applied Sciences is an interdisciplinary journal: therefore, all manuscripts must be written to be accessible to a broad scientific but mathematically advanced audience. All papers must contain carefully written introduction and conclusion sections, which should include a clear exposition of the underlying scientific problem, a summary of the mathematical results and the tools used in deriving the results. Furthermore, the scientific importance of the manuscript and its conclusions should be made clear. Papers dealing with numerical processes or which contain only the application of well established methods will not be accepted. Because of the broad scope of the journal, authors should minimize the use of technical jargon from their subfield in order to increase the accessibility of their paper and appeal to a wider readership. If technical terms are necessary, authors should define them clearly so that the main ideas are understandable also to readers not working in the same subfield.