退化阻尼对高维可压缩欧拉方程组经典解的影响

发布时间：2018-05-29 23:45

  本文选题：可压缩欧拉方程组 + 阻尼　； 参考：《南京大学》2016年博士论文

【摘要】：在流体力学中,以莱昂哈德·欧拉命名的欧拉方程组,是控制理想流体运动的一组拟线性双曲方程。这些方程分别代表质量、动量和能量守恒,也可以看做是零粘性和零热导率的Navier-Stokes方程的特殊情形。欧拉方程组是描述无粘流体运动最重要和最基本的方程。这些方程在海洋学、气象学、空气动力学等领域有着广泛的应用。可压缩欧拉方程组的光滑解一般情况下会在有限时间内爆破,可能会伴随激波、疏散波等的形成。激波的形成是流体最重要的现象之一,其研究历史可以参见文献[8]及其中的参考文献。在高维情形,对于一类特殊初值,Sideris [55]已经证明三维的光滑解会在有限时间内产生奇性,Rammaha在[53]中证明了一个二维的爆破结果。关于爆破结果和爆破机制的更广泛的文献见[1-4,6,8-10,17,33,40,56,57,59,68,69]及其中的参考文献。可压缩流体通过多孔介质的运动可以由如下具有摩擦阻尼的可压缩欧拉方程组来描述其中摩擦系数v0是常数。当初始值是平衡状态的小扰动时系统(0.0.1)存在全局光滑解,而且柯西问题的解预计会趋于由达西定律控制的扩散波。在某种意义下,阻尼可以阻止小振幅光滑解的奇性的产生。柯西问题或初边值问题解的全局存在性以及解的大时间行为在文献[7,15,24,26,27,29,32,34,35,38,41-49,51,58,60,61,63-65,70]中已经建立。关于非光滑解的结果也可以见文献[16,25,28,36,37,50]。本文中,我们将考虑下述具有退化阻尼的可压缩欧拉方程组经典解的全局存在性或爆破其中x∈Rd(或R+d),摩擦系数a(t)=μ/(1+t)λ中的μ0和λ≥0是常数,振幅ε0充分小。由于本文只考虑经典解,我们可以假设初始值(ρ0,u0)是足够光滑且具有紧支集的。这里我们还需要指出,当我们研究半空间R+d中的初边值问题时,系统(0.0.2)应当提供滑移边界条件。首先,我们将λ≥0,μ0分成如下四种情况：情况一：0≤λ1,μ0当d=2,3；情况二：λ=1,μ3-d当d=2,3；情况三：λ=1,μ≤3-d当d=2；情况四：入1,μ0当d=2,3。本文的主要结果可以简要概括为：·在情况一中,全空间Rd中的柯西问题或半空间R+d中的初边值问题存在全局光滑解。·在情况二中,当初始值满足curlu0≡0时,柯西问题(0.0.2)的光滑解是全局存在的。·在情况三和情况四中,柯西问题(0.0.2)的光滑解会在有限时间内爆破。在第二章中,我们研究三维无旋流的柯西问题。在第三章中,我们考虑全空间Rd中高维可压缩欧拉方程组的柯西问题。在第四章中,我们致力于半空间R+d中高维可压缩欧拉方程组的初边值问题。
[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，

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