{"title":"Global Existence and Large Time Behavior of Classical Solutions to the Incompressible Inhomogeneous Kinetic-Fluid Model With Energy Exchanges","authors":"Fucai Li, Jinkai Ni, Man Wu","doi":"10.1111/sapm.70077","DOIUrl":null,"url":null,"abstract":"<div>\n \n <p>In this paper, we concentrate on a kinetic-fluid model with energy exchanges, which contains a Vlasov–Fokker–Planck equation for the particles and an incompressible inhomogeneous Navier–Stokes system with heat conductivity for the fluid. The two parts in the model twist together via the momentum and energy exchanges revealing the interaction between the fluid and the particles. For the small initial data near the given equilibrium state, we obtain the global existence, uniqueness, and optimal decay rates of classical solutions in the three-dimensional whole space. Furthermore, we obtain the optimal decay rates of the gradients in <span></span><math>\n <semantics>\n <mi>x</mi>\n <annotation>$x$</annotation>\n </semantics></math> of solutions. For the periodic domain case, the decay rates are exponential. The proofs of our results are mainly based on the new macro–micro decomposition and modified energy-spectrum method. We also introduce some new ideas and develop a new energy method to enclose the a priori estimates. More precisely, we first exploit the new macro–micro decomposition and the dissipation rate on the gradient of the pressure <span></span><math>\n <semantics>\n <mi>p</mi>\n <annotation>$p$</annotation>\n </semantics></math> instead of the density <span></span><math>\n <semantics>\n <mi>ρ</mi>\n <annotation>$\\rho$</annotation>\n </semantics></math> in <span></span><math>\n <semantics>\n <msup>\n <mi>H</mi>\n <mn>2</mn>\n </msup>\n <annotation>$H^2$</annotation>\n </semantics></math> norm to enclose the estimates <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mrow>\n <mo>∥</mo>\n <mi>f</mi>\n <mo>∥</mo>\n </mrow>\n <msubsup>\n <mi>H</mi>\n <mrow>\n <mi>x</mi>\n <mo>,</mo>\n <mi>ξ</mi>\n </mrow>\n <mn>3</mn>\n </msubsup>\n </msub>\n <mo>+</mo>\n <msub>\n <mrow>\n <mo>∥</mo>\n <mi>u</mi>\n <mo>∥</mo>\n </mrow>\n <msubsup>\n <mi>H</mi>\n <mi>x</mi>\n <mn>3</mn>\n </msubsup>\n </msub>\n </mrow>\n <annotation>$\\Vert f\\Vert _{H^3_{x,\\xi }} + \\Vert u\\Vert _{H^3_x}$</annotation>\n </semantics></math>. Then, by applying Gronwall's inequality and establishing the <span></span><math>\n <semantics>\n <msup>\n <mrow>\n <mo>(</mo>\n <mn>1</mn>\n <mo>+</mo>\n <mi>t</mi>\n <mo>)</mo>\n </mrow>\n <mrow>\n <mo>−</mo>\n <mn>5</mn>\n <mo>/</mo>\n <mn>4</mn>\n </mrow>\n </msup>\n <annotation>$(1+t)^{- {5}/{4}}$</annotation>\n </semantics></math> decay rate in the whole space and the exponential decay rate on the periodic domain of <span></span><math>\n <semantics>\n <msub>\n <mrow>\n <mo>∥</mo>\n <mo>∇</mo>\n <mi>u</mi>\n <mo>∥</mo>\n </mrow>\n <msubsup>\n <mi>H</mi>\n <mi>x</mi>\n <mn>2</mn>\n </msubsup>\n </msub>\n <annotation>$\\Vert \\nabla u \\Vert _{H^2_x}$</annotation>\n </semantics></math>, we enclose the estimate <span></span><math>\n <semantics>\n <msub>\n <mrow>\n <mo>∥</mo>\n <mi>ρ</mi>\n <mo>∥</mo>\n </mrow>\n <msubsup>\n <mi>H</mi>\n <mi>x</mi>\n <mn>3</mn>\n </msubsup>\n </msub>\n <annotation>$\\Vert \\rho \\Vert _{H^3_x}$</annotation>\n </semantics></math> in the a priori estimates. Finally, based on the modified energy-spectrum method, and the fact that the transport equation with divergence free velocity field is <span></span><math>\n <semantics>\n <msup>\n <mi>L</mi>\n <mn>1</mn>\n </msup>\n <annotation>$L^1$</annotation>\n </semantics></math> conserved, we obtain the optimal decay rates of the solutions <span></span><math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <mi>f</mi>\n <mo>,</mo>\n <mi>u</mi>\n <mo>,</mo>\n <mi>θ</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$(f,u,\\theta)$</annotation>\n </semantics></math> and their gradients.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"155 1","pages":""},"PeriodicalIF":2.6000,"publicationDate":"2025-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Studies in Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1111/sapm.70077","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
In this paper, we concentrate on a kinetic-fluid model with energy exchanges, which contains a Vlasov–Fokker–Planck equation for the particles and an incompressible inhomogeneous Navier–Stokes system with heat conductivity for the fluid. The two parts in the model twist together via the momentum and energy exchanges revealing the interaction between the fluid and the particles. For the small initial data near the given equilibrium state, we obtain the global existence, uniqueness, and optimal decay rates of classical solutions in the three-dimensional whole space. Furthermore, we obtain the optimal decay rates of the gradients in of solutions. For the periodic domain case, the decay rates are exponential. The proofs of our results are mainly based on the new macro–micro decomposition and modified energy-spectrum method. We also introduce some new ideas and develop a new energy method to enclose the a priori estimates. More precisely, we first exploit the new macro–micro decomposition and the dissipation rate on the gradient of the pressure instead of the density in norm to enclose the estimates . Then, by applying Gronwall's inequality and establishing the decay rate in the whole space and the exponential decay rate on the periodic domain of , we enclose the estimate in the a priori estimates. Finally, based on the modified energy-spectrum method, and the fact that the transport equation with divergence free velocity field is conserved, we obtain the optimal decay rates of the solutions and their gradients.
期刊介绍:
Studies in Applied Mathematics explores the interplay between mathematics and the applied disciplines. It publishes papers that advance the understanding of physical processes, or develop new mathematical techniques applicable to physical and real-world problems. Its main themes include (but are not limited to) nonlinear phenomena, mathematical modeling, integrable systems, asymptotic analysis, inverse problems, numerical analysis, dynamical systems, scientific computing and applications to areas such as fluid mechanics, mathematical biology, and optics.