有外力项的双极可压NavieR-Stokes-Poisson方程解的整体存在性和最佳衰减率

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1、摘要本学位论文中，我们研究了有外力项的双极可压三维Navier-Stokes-Poisson方程．假设外力项在Sobolev空间中足够小，通过解一个非线性耦合的椭圆系统来建立稳态解的存在性．然后，在给定的初值接近稳态解的条件下，我们证明双极可压的三维Navier-Stokes—Poisson方程初值问题的全局适定性．最后，我们通过对非线性系统作详尽的能量估计和对线性方程产生的半群作L2衰减估计，来得到方程解逼近稳态解的最优驴收敛速度．全文的结构安排如下：第一章是前言，简要介绍问题的物理数学背景和已有的相关研究结果．同时给出本文的主

2、要定理，第二章说明双极等熵可压Navier-Stokes．Poisson方程稳态解的唯一存在性．第三章证明该问题的光滑解的整体存在性．第四章构造该问题光滑解的最佳工2衰减速率．第五章说明双极非等熵可压Navier-Stokes-Poisson方程稳态解的唯一存在性。第六章证明该问题的光滑解的整体存在性．第七章构造该问题光滑解的最佳L2衰减速率．关键词：双极Navier-Stokes．Poisson方程，稳态解，光滑解，能量估计，最佳衰减率．AbstractInthispapeLweconsiderthethreedimension

3、alcompressiblebipolarNavier-Stokes—Poissoneauationswiththepotentialexternalforce．UnderthesmallnessassumptionoftheexternalforeemsomeS0bolevspace，theexistenceofthestationarysolutionisestablishedbysolvinganonlinearcoupledellipticsystern．Next，weshowglobalwell—posednessoft

4、heinitialValueproblemfor也e也reedimensionalcompressiblebipolarNavier-Stokes—Poissonequations，providedtheprescribedi11itialdateisclosetOthestationarysolution．Finally,basedontheelaborateenergyestimate8forthenonlinearsy蝴nandL2-decayestimatesforsemigroupgeneratedbythelinear

5、izedequatlon，wegivemeoptimalL2-convergenceratesofthesolutionstowardsthestationarysollmon‘Thispaperisorganizedasfollows·ChaDterOneisanintroducfion．Irisdevotedtointroducingphysicalbackgrounda11dpreVlou8mathematicalresearchworksonshockreflectionanddiffraction．Themainprob

6、lems，mainresultsinthisdissertationarealsoillustrated．InChapterTwo．wepresenttheuniqueexistenceofthestationarysolutionforthecompress—ibleisen仃opicbipolarNavier-Stokes-Poissonequations．TheglobalexistenceofsmoothsOluuonsforthisDroblemisprovedinChapterThree．InChapterFour,w

7、eestablishtheoptimal∥nmedecavratesofsmoothsolutionsconstructedinChapterThree·InChapterFive，wepresenttheuniqueexistenceofthestationarysolutionforthecomp化88。iblenon—iser血opicbipolarNavier-Stokes—Poissonequations．Theglobalexistenceofsmooth80—lutionsf研thisproblemisprovedi

8、nChapterSix．InChapterSeven，weestablishtheoptimal∥timedecayratesofsmoothsolutionsconstructedinChapterSix·KevWbrds：BipolarNavi