一类流体力学方程组的初值问题的爆破与非存在性

发布时间：2019-02-17 12:11

【摘要】：本文研究了带电场的Euler-Poisson方程组的初值问题的经典解的爆破以及可压缩Navier-Stokes方程组的初值问题在Sobolev空间中的非存在性问题,共包含两部分内容。第一部分考虑带电场的完全的Euler-Poisson方程组以及等熵的Euler-Poisson方程组的初值问题。我们证明了在初值满足一定的条件下,该初值问题的经典解在有限时间内会爆破。由于Euler-Poisson方程组是由Euler方程组耦合上重力场或电场而得到,而重力场或电场均是非局部项,其破坏了密度的紧支集的有限传播,这使得Sideris[39]证明Euler方程组的爆破的方法在处理Euler-Poisson方程组的爆破时失效。Sideris的方法包含两个关键的元素,一个是:密度的紧支集的有限传播,另一个是:动量的径向分量的估计。其中动量的径向分量的估计可以用内能的上界估计代替。因此,我们需要找到一个新的元素来替代密度的紧支集的有限传播。对于本问题,我们需要克服两个困难,一个是估计由电场引出的非局部项,另一个是密度不具有紧支集的情况下如何证明爆破。这两个困难是结合在一起的。我们的做法是:应用Hardy-Littlewood-Sobolev不等式来处理由电场引出的非局部项进而来估计内能的上界,应用Chemin不等式给出内能的下界估计。最后通过比较内能的上下界的系数,确定出初值在满足一定的条件下,带电场的Euler-Poisson方程组的初值问题的光滑解会在有限时间内爆破。第二部分研究可压缩任意维数的完全Navier-Stokes方程组和一维等熵的Navier-Stokes方程组的初值问题在Sobolev空间中的非存在性问题。辛周平[45],Cho和Jin[61]证明了:在初始密度具有紧支集的前提下,可压缩任意维数的完全Navier-Stokes方程组和一维等熵的Navier-Stokes方程组的初值问题的解在某类Sobolev空间中会爆破。我们的工作证明了在初始密度具有紧支集的前提下,有热交换的可压缩任意维数的完全Navier-Stokes方程组在此类Sobolev空间中无解以及一维等熵的Navier-Stokes方程组在初值满足一定的条件下在此类Sobolev空间中无解。我们的办法是首先通过一个观察将初值问题化成了一个有界区域上的超定的积分微分方程组的初边值问题,然后定义一个适当的沿时间导数退化的抛物或积分微分算子并建立相应的Hopf引理和强极值原理,最终用反证法完成证明。
[Abstract]:In this paper, we study the blow-up of the classical solution of the initial value problem of the Euler-Poisson equations with electric field and the nonexistence of the initial value problem of the compressible Navier-Stokes equations in the Sobolev space, which contains two parts. In the first part, we consider the initial value problem of complete Euler-Poisson equations with electric field and Euler-Poisson equations with equal entropy. We prove that the classical solution of the initial value problem will burst in finite time if the initial value satisfies certain conditions. Because the Euler-Poisson equations are derived from the coupling of the Euler equations with the gravity field or electric field, the gravity field or electric field is a nonlocal term, which destroys the finite propagation of the compact support set of the density. This makes Sideris [39] prove that the method of blow-up of Euler equations is invalid when dealing with the blow-up of Euler-Poisson equations. Sideris's method contains two key elements, one is the finite propagation of compact support set of density, The other is the estimation of the radial component of momentum. The estimation of radial component of momentum can be replaced by the upper bound estimation of internal energy. Therefore, we need to find a new element to replace the finite propagation of compact set of density. For this problem, we need to overcome two difficulties, one is to estimate the nonlocal term derived from the electric field, and the other is how to prove the explosion when the density does not have a compact support set. The two difficulties are combined. Our approach is to use the Hardy-Littlewood-Sobolev inequality to deal with the nonlocal term derived from the electric field and then to estimate the upper bound of the internal energy, and to use the Chemin inequality to give the lower bound estimate of the internal energy. Finally, by comparing the coefficients of the upper and lower bounds of the internal energy, it is determined that the smooth solution of the initial value problem of the Euler-Poisson equations with electric field will burst in a finite time under certain conditions. In the second part, the nonexistence of the initial value problem for the complete Navier-Stokes equations with compressible arbitrary dimension and the one-dimensional isentropic Navier-Stokes equations in Sobolev space is studied. Xin Zhouping [45], Cho and Jin [61] prove that if the initial density has compact support, The solution of the initial value problem for the complete Navier-Stokes equations with compressible arbitrary dimension and the one-dimensional isentropic Navier-Stokes equations will burst in a certain Sobolev space. Our work proves that under the condition that the initial density has compact support set, The complete Navier-Stokes equations with compressible arbitrary dimension with heat exchange have no solution in this kind of Sobolev space and the one-dimensional isentropic Navier-Stokes equations have no solution in this kind of Sobolev space under the condition that the initial value satisfies certain conditions. Our approach is to first turn the initial value problem into an initial-boundary value problem for an overdetermined system of integro-differential equations on a bounded domain. Then a proper parabolic or integro-differential operator degenerate along the time derivative is defined and the corresponding Hopf Lemma and strong extremum principle are established.
【学位授予单位】：清华大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175.8

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