Mathematical analysis on the structure of solutions for the fundamental systems of equations in continuum mechanics

在连续介质力学的基本方程中，我们得到了如下结果：1.由于等熵Euler方程的发展问题，即使初始数据是光滑的，也允许有激波存在，所以我们通常试图找到属于有界变分函数空间的解。为了保证解的唯一性，构造这样的解作为抛物型近似方程解的极限是方便的。我们成功地构造了这个近似方程在有界变差函数类中的时间整体解。到目前为止，我们有一个方案，由于西田和Smoller构建这样一个解决方案。然而，只有当密度有界时，他们的方案才有效。我们首次成功地证明了它的有界性，从而在真实的意义上证明了它们是可行的。2.在不可压缩粘性流体的二维发展自由边界问题中，我们研究了 关于我们 其中自由边界和容器的边界具有接触线。作为我们在滑移边界条件下研究容器中Navier-Stokes方程的可解性的一系列工作，本文研究了两个问题：（1）无限扇形区域中的Stokes方程，（2）分段光滑边界区域中的Navier-Stokes方程，并证明了它们在加权Sobolev空间中的可解性。对于这个问题，我们使用Stefan-Boltzmann关系。4.在二维无限弹性或粘弹性带半无限裂纹的情况下，我们研究了定常问题的可解性，并确定了裂纹的扩展。目前，以下结果正在准备中：（1）构造绕Gerstner余摆线波和三维不可压无粘流场的解;（2）非线性声学中的非线性问题。少

Among the fundamental equations in continuum mechanics we have obtained the following results.1.Since it is well known that the evolution problems of isentropic Euler equation admit shock waves even if the initial data are smooth, we usually try to find the solution belonging to the function spaces of bounded variations. In order to guarantee the uniqueness of the solution it is convenient to construct such a solution as a limit of the solution to the approximate equation of parabolic type. We succeeded to construct the temporally global solution in the class of functions of bounded variations to this approximate equation. Up to the present time we have had a scenario due to Nishida and Smoller to construct such a solution. However, their scenario is valid only if the density is bounded. We firstly succeeded to prove its boundedness, so that in real sense their scenario works.2.Among the two-dimensional evolution free boundary problems for incompressible viscous fluid we study the case … More where the free boundary and the boundary of the container has a contact line. As a series of our study on the solvability of Navier-Stokes equations in a container with slip boundary conditions, here we investigated the two problems :(1)Stokes equations in infinite sector,(2)Navier-Stokes equations in a domain with piecewise smooth boundary.Then we proved their solvability in the weighted Sobolev spaces.3.For the one-dimensional model equations of a self-gravitating viscous radiative and reactive gas we found the unique global in time solution belonging to Hoelder spaces. For this problem we used the Stefan-Boltzmann relation.4.In a two-dimensional infinite elastic or visco-elastic strip with a semi-infinite crack we studied the solvability to the stationary problem and determined the propagation of the crack. Moreover, we proved the weak solvability of its evolution problem.Now the following results are preparing : (1)To construct the solution around the Gerstner's trochoidal wave and 3D domain for incompressible inviscid flow (2)Nonlinear problems in nonlinear acoustics. Less