各向异性Boussinesq方程的整体适定性

【作者】 郑孝信；

【导师】 苗长兴；

【作者基本信息】 中国工程物理研究院 ， 应用数学， 2013， 博士

【摘要】 流体动力学方程(组)作为刻画物质运动的宏观模型,是我们认识与理解自然现象的一类重要的非线性偏微分方程.它一直占据数学物理界的核心研究领域.如：Boussi-nesq方程能够描述大气与海洋运动中最显著的两个特征：旋转和分层.它有着极强的物理背景和数学意义,引起了研究者的广泛关注.近年来,Boussinesq方程的研究取得了很大进展,尤其二维次临界和临界情形的研究已经达到了一个很满意的状态.最近,通过建立速度场的增长估计和利用方程的结构,二维各向异性Boussinesq方程的整体正则性问题也得以解决.而三维Boussinesq马解的唯一性和光滑解的整体适定性依然是一个公开问题.为了更好的理解对流项在流体运动中的影响,人们也考虑了具有特殊结构的流体(如：轴对称无旋流体).本文主要致力于研究各向异性Boussinesq方程的Cauchy问题.利用方程的耦合结构,轴对称流的性质以及非Lipschitz向量场的导数损失估计,我们建立了一些三维具有轴对称结构各向异性Boussinesq方程的整体适定性理论；利用Fourier局部化技术和速度场的增长估计,我们对粗糙初值建立了二维各向异性非线性Boussinesq方程的整体适定性.另外,我们也考虑了流体动力学方程其它一些相关模型.研究了三维不可压Navier-Stokes方程弱解的正则性准则、高维可压Navier-Stokes-Poisson方程在Lp框架下小解的整体适定性、分数阶Keller-Segel系统在临界Fourier-Herz空间中的不适定性等问题.本文具体内容如下：第二章回忆Littlewood-Paley理论的一些基本知识.同时给出了一些预备引理和交换子估计.最后,我们回顾一下轴对称流体的代数和几何性质.第三章研究三维水平耗散Boussinesq方程的Cauchy问题.在假设初始值是轴对称无旋的情形下,通过利用方程的耦合结构和水平方向上的光滑效应,我们证明了解的整体适定性.这里的一个重要因素是利用轴对称流体的结构和调和分析的技巧,我们首次建立了γ/uγ与γ/wθ的代数关系.第四章继续研究三维各向异性Boussinesq系统的Cauchy问题.在假设轴对称初值ρ0(γ,z)的支集与轴((Oz)不相交的条件下,我们证明了三维水平黏性Boussinesq系统的整体适定性.由输运方程的性质,我们首先建立了γ-ρ的增长估计.再结合水平方向上的光滑效应,我们进一步建立了估计此意味着然而,空间L2-3似乎容许了太强的奇性阻止我们获得(ρ,u)的高阶正则性.为此,利用微局部化技术和水平方向上的光滑效应,我们开发了时空估计由此,通过建立时空Log型不等式,我们就获得了解的整体适定性.第五章研究三维不可压Navier-Stokes方程弱解的正则性准则.由不可压流的性质,我们首先建立了经由一个元素Λiγuj(γ∈[0,1]且i,j∈{1,2,3})描述的Leray-Hopf弱解的正则性准则,即这里F是指标(α,β)的集合,且Λi:=√-(?)i2.这就推广和改进了有关Leray-Hopf弱解正则性准则方面的结果,包括C. Cao和E. S. Titi (Arch. Ration. Mech. Anal.202(2011)919-932)的工作.更重要的,通过利用Bony仿积分解,我们给出了如下端点情形(α=∞)的正则性准则,其中,i,j∈{1,2,3}.在第六章,我们首先来研究二维垂直耗散非线性Boussinesq方程的大解的整体适定性.接下来,我们考虑高维可压Navier-Stokes-Poisson方程在Lp框架下小解的整体适定性.再者,我们还建立了分数阶Keller-Segel模型在临界Fourier-Herz空间中的不适定性.更多还原

【Abstract】 The fluid dynamic equation (system) as the basic model to describe fluid flow, is an important nonlinear partial differential equations to understand the natural phe-nomena. It lies in the hot research areas of mathematics and physics. For instance, Boussinesq equations describe the two distinctive features in the dynamics of the ocean or of the atmosphere:rotation and stratification. This system have been intensively studied due to their physical background and mathematical significance.Great progress about Boussinesq system has been made in the past years, espe-cially the study for the two-dimensional subcritical or critical Boussinesq equations is in a satisfactory state. Recently, by establishing the growth estimate of the velocity field and using the structure of equations, the global well-posedness for the two-dimensional anisotropic Boussinesq system has been achieved. However, the uniqueness of weak so-lution or the global well-posedness of smooth solution for the tridimensional Boussinesq equations is still an open problem. To better understand the influence of the convec-tion term in the fluid, some researchers consider some fluids with special structure (for example:axisymmetric flow without swirl).This thesis is devoted to the study of Cauchy problem for the anisotropic Boussi-nesq equations. By using the coupling structure of Boussinesq equations, properties of axisymmetric flow and losing estimates for the non-Lipschitz vector field, we establish some global results on tridimensional anisotropic Boussinesq system under assumption that the initial data is axisymmetric without swirl; by using Fourier localization method and the growth estimate of the velocity, we prove the global well-posedness of the two-dimensional anisotropic non linear Boussinesq system for the rough initial data. In addition, we are concerned with some else relevant models. We investigate some prob-lems such as the regularity criterion of tridimensional incompressible Navier-Stokes equations、the global well-posedness for the high dimensional compressible Navier-Stokes-Poisson equations in the LP framework、 the ill-posedness to Keller-Segel system with fraction diffusion.The detail of this thesis is arranged as follows.In the second chapter, we recall the basic Littlewood-Paley theory. Next, we give some useful lemmas and a commutator estimate. At last we review algebraic and geometric properties of axisymmetric flow.In the third chapter, we investigate the Cauchy problem for the tridimensional Boussinesq equations with horizontal dissipation. Under the assumption that the initial data is axisymmetric without swirl, by using structure of the coupling of Boussinesq equations and the horizontal smoothing effect, we prove the global well-posedness for this system. In our proof, the main ingredient is to establish a magic relationship between yr/r and wθ/r by taking full advantage of the structure of the axisymmetric fluid without swirl and some tricks in harmonic analysis.In the fourth chapter, we continue to consider Cauchy problem of the tridimen-sional anisotropic Boussinesq equations. Under the assumption that the support of the axisymmetric initial data po(r,z) does not intersect with the axis (Oz), we prove the global well-posedness for this system with axisymmetric initial data. We first show the growth of the quantity p/r for large time by taking advantage of characteristic of trans-port equation. This growing property together with the horizontal smoothing effect enables us to establish the estimate for the quantity which implies However, space L.2admits forbidden singularity to prevent us from getting the high regularity of (ρ,u). To bridge this gap, we exploit the space-time estimate about by making good use of the horizontal smoothing effect and micro-local techniques. The global well-posedness for the large initial data is achieved by establishing a new type space-time logarithmic inequality.In the fifth chapter, we investigate the regularity criterion of the tridimensional Navier-Stokes equations via one velocity component. By deeply using properties of the incompressible fluid, we mainly establish regularity criteria of Leray-Hopf weak solutions via only one element Λriuj with γ∈[0,1] and i, j∈{1,2,3}, that is Here F is the set of index (α,β) which appears in our results and the fractional operator This extends and improves some known regularity criterions of Leray-Hopf weak solutions in term of one velocity component, including the notable works of C. Cao and E. S. Titi (Arch. Ration. Mech. Anal.202(2011)919-932). More importantly, by making full use of the Bony paraproduct decomposition, we show that Leray-Hopf weak solutions is smooth on [0, T] in term of (at the endpoint a=∞) where i,j∈{1,2,3}.In the sixth chapter, we firstly study the global well-posedness for the two dimen-sional non Boussinesq equations with vertical dissipation. Next, we consider a global well-posedness of the compressible Navier-Stokes-Possion equations in IP framework with small initial data. Finally, we prove the ill-posedness of Keller-Segel equations with fractional diffusion in critical Fourier-Herz space.更多还原