Some Studies On The Mathematical Theory Of Compressible Isentropic Navier-Stokes Equations

The basic series of equations in theoretical fluid mechanics is Navier-Stokes equations, and it is used to describe some fluid substance such as liquid and gas. It shows the dynamic balance among all kinds of force which acting on any given domain in fluid, and it’s the basic model of Partial Differential Equation. In this paper, we study the existence of weak solutions with Cauchy problem and free boundary for Vaigant-Kazhikhov model in Navier-Stokes equation, as well as the existence of analytical solutions and the asymptotic behavior. The main result are as follows:● Global solution to 3D spherically symmetric compressible Navier-Stokes equations with large data. In this chapter, the global classical solution is obtained away from the symmetry center r=0 with arbitrarily large and non-vacuum data. In particular, it is shown that the solution will not develop the vacuum states in any finite time away from the symmetry center if the initial density does not contain vacuum states. Then the global weak solutions with the symmetry center r= 0 are obtained as the limit of the classical solutions.● Global existence of spherically symmetric weak solutions to the free boundary value problem of 3D isentropic compressible Navier-Stokes equa-tions(CNS). we prove the global existence of spherically symmetric weak so-lutions to the free boundary problem for Vaigant-Kazhikhov model with vac-uum and free boundary separating fluids and vacuum. In addition, the free boundary is shown to expand outward at an algebraic rate in time.● Analytical solutions to the 3-D compressible Navier-Stokes equations with free boundaries. By constructing a class of radial symmetric and self-similar analytical solutions in RN (N≥2), we derive that the free boundary expand outward in the radial direction at an algebraic rate in time and also have the asymptotic behavior.