發(fā)布時間：2018-05-29 23:45

  本文選題：可壓縮歐拉方程組 + 阻尼　； 參考：《南京大學(xué)》2016年博士論文

【摘要】：在流體力學(xué)中,以萊昂哈德·歐拉命名的歐拉方程組,是控制理想流體運動的一組擬線性雙曲方程。這些方程分別代表質(zhì)量、動量和能量守恒,也可以看做是零粘性和零熱導(dǎo)率的Navier-Stokes方程的特殊情形。歐拉方程組是描述無粘流體運動最重要和最基本的方程。這些方程在海洋學(xué)、氣象學(xué)、空氣動力學(xué)等領(lǐng)域有著廣泛的應(yīng)用�？蓧嚎s歐拉方程組的光滑解一般情況下會在有限時間內(nèi)爆破,可能會伴隨激波、疏散波等的形成。激波的形成是流體最重要的現(xiàn)象之一,其研究歷史可以參見文獻[8]及其中的參考文獻。在高維情形,對于一類特殊初值,Sideris [55]已經(jīng)證明三維的光滑解會在有限時間內(nèi)產(chǎn)生奇性,Rammaha在[53]中證明了一個二維的爆破結(jié)果。關(guān)于爆破結(jié)果和爆破機制的更廣泛的文獻見[1-4,6,8-10,17,33,40,56,57,59,68,69]及其中的參考文獻�？蓧嚎s流體通過多孔介質(zhì)的運動可以由如下具有摩擦阻尼的可壓縮歐拉方程組來描述其中摩擦系數(shù)v0是常數(shù)。當初始值是平衡狀態(tài)的小擾動時系統(tǒng)(0.0.1)存在全局光滑解,而且柯西問題的解預(yù)計會趨于由達西定律控制的擴散波。在某種意義下,阻尼可以阻止小振幅光滑解的奇性的產(chǎn)生�？挛鲉栴}或初邊值問題解的全局存在性以及解的大時間行為在文獻[7,15,24,26,27,29,32,34,35,38,41-49,51,58,60,61,63-65,70]中已經(jīng)建立。關(guān)于非光滑解的結(jié)果也可以見文獻[16,25,28,36,37,50]。本文中,我們將考慮下述具有退化阻尼的可壓縮歐拉方程組經(jīng)典解的全局存在性或爆破其中x∈Rd(或R+d),摩擦系數(shù)a(t)=μ/(1+t)λ中的μ0和λ≥0是常數(shù),振幅ε0充分小。由于本文只考慮經(jīng)典解,我們可以假設(shè)初始值(ρ0,u0)是足夠光滑且具有緊支集的。這里我們還需要指出,當我們研究半空間R+d中的初邊值問題時,系統(tǒng)(0.0.2)應(yīng)當提供滑移邊界條件。首先,我們將λ≥0,μ0分成如下四種情況：情況一：0≤λ1,μ0當d=2,3；情況二：λ=1,μ3-d當d=2,3；情況三：λ=1,μ≤3-d當d=2；情況四：入1,μ0當d=2,3。本文的主要結(jié)果可以簡要概括為：·在情況一中,全空間Rd中的柯西問題或半空間R+d中的初邊值問題存在全局光滑解。·在情況二中,當初始值滿足curlu0≡0時,柯西問題(0.0.2)的光滑解是全局存在的�！ぴ谇闆r三和情況四中,柯西問題(0.0.2)的光滑解會在有限時間內(nèi)爆破。在第二章中,我們研究三維無旋流的柯西問題。在第三章中,我們考慮全空間Rd中高維可壓縮歐拉方程組的柯西問題。在第四章中,我們致力于半空間R+d中高維可壓縮歐拉方程組的初邊值問題。
[Abstract]:In fluid mechanics, the Euler equations named by Leonhard Euler are a set of quasilinear hyperbolic equations which control the motion of ideal fluid. These equations represent conservation of mass momentum and energy respectively and can also be regarded as special cases of Navier-Stokes equations with zero viscosity and zero thermal conductivity. Euler equations are the most important and basic equations for describing the motion of non-viscous fluids. These equations are widely used in oceanography, meteorology and aerodynamics. The smooth solution of compressible Eulerian equations will blow up in finite time and may be accompanied by shock wave and evacuation wave. The formation of shock waves is one of the most important phenomena in fluid. In the case of high dimension, for a special initial value, Sideris [55], it has been proved that the smooth solution of 3D can produce singularity in finite time. In [53], we prove a two-dimensional blow-up result. A more extensive literature on blasting results and blasting mechanism can be found in [1-4J 68-1010 / 1734040577N59C 6869] and its references. The motion of compressible fluid through porous media can be described by the following compressible Euler equations with friction damping where the friction coefficient v _ 0 is a constant. When the initial value is a small disturbance in the equilibrium state, the system has a global smooth solution, and the solution of the Cauchy problem is expected to tend to the diffusion wave controlled by Darcy's law. In a sense, damping can prevent the singularity of smooth solutions with small amplitude. The global existence of solutions to Cauchy problems or initial-boundary value problems and the large time behavior of solutions have been established in the literature [7 / 15 / 24 / 26 / 2729 / 32 / 32 / 3 / 35 / 35 / 35 / 41 / 491 / 51 / 6061-63-670]. The results of nonsmooth solutions can also be found in reference [16 / 25 / 2836 / 3750]. In this paper, we will consider the global existence or blow-up of the classical solutions of the degenerate damped compressible Euler equations where x 鈭，

本文編號：1952978