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，我们得到了以下结果： 1.由于众所周知，即使初始数据光滑，等熵欧拉方程的演化问题也承认冲击波，因此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，可以方便地构造这样一个解作为抛物型近似方程解的极限。我们成功地构造了该近似方程的有界变分函数类中的时间全局解。到目前为止，我们已经有了 Nishida 和 Smoller 构建这样一个解决方案的场景。然而，他们的场景仅在密度有界时才有效。我们首先成功地证明了它的有界性，因此他们的方案在真正意义上是有效的。2.在不可压缩粘性流体的二维演化自由边界问题中，我们研究了自由边界和容器边界有接触线的情况。作为我们对具有滑移边界条件的容器中纳维-斯托克斯方程可解性的一系列研究，这里我们研究了两个问题：(1)无限扇区中的斯托克斯方程，(2)分段光滑边界域中的纳维-斯托克斯方程。然后我们证明了它们在加权Sobolev空间中的可解性。3.对于自引力粘性辐射和反应气体的一维模型方程，我们发现了唯一的整体全局方程属于 Hoelder 空间的时间解。对于这个问题，我们使用了Stefan-Boltzmann关系。4.在带有半无限裂纹的二维无限弹性或粘弹性条带中，我们研究了稳态问题的可解性并确定了裂纹的扩展。此外，我们还证明了其演化问题的弱可解性。目前正在准备以下成果： (1)围绕格斯特纳摆线波和不可压缩无粘流的3D域构造解。 (2)非线性声学中的非线性问题。较少的

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