Some Results Of The Compressible Navier-Stokes Equations

Navier-Stokes equations of fluid mechanics are the most important equations, and have an extremely wide application background in physics. In mathematics, the search for its exact solutions is very difficult.In this paper, under appropriate conditions in the Navier-Stokes equations, we study the existence and uniqueness of weak solutions, the existence of self-similar solutions, as well as the blowup solutions of such issues. Mainly divided into the following areas:●Local solutions of the multidimensional Navier-Stokes equations for isentropic compressible flow are constructed with spherically symmetric initial data between a solid core and a free boundary connected to a surrounding vacuum state. It is also proved that no vacuum develops between the solid core and the free boundary, and the free boundary expands with finite speed.●Considering the free boundary problem of the symmetric compressible isentropic Navier-Stokes equations in Rⁿ(n≥1), with density-dependent viscosity coefficients, we obtain the global existence, uniqueness and continuous dependence on initial data of a weak solution, with a Lebesgue initial velocity u₀∈L^rm, Am > n and .●Considering the one-dimensional compressible Navier-Stokes equations, we prove that there exists neither forward nor backward self-similar solutions with finite kinetic energy.●We study the pressureless Navier-Stokes equations for non-Newtonian fluid. The analytical solutions with arbitrary time blowup, in radial symmetry, are constructed.These studies help us to understand some properties of the Navier-Stokes equations. Under certain conditions, a more detailed description of the local and global state of the solution is given. At the same time, it reveals some of the more interesting physical phenomena.