| In this paper, we consider the existence and the uniqueness of radially symmetric strong solutions to the Navier-Stokes-Poisson equations, and the existence of the strong solutions to the Navier-Stokes-Poisson equations and the full Navier-Stokes equations.The Navier-Stokes-Poisson equations are derived from physical problems, which mainly describe the motion of gasous stars with viscosity and self-gravitation. However, the Euler-Poisson equations depict the motion of celestial bodies without viscosity, and the Navier-Stokes equations may reflect the motion of the fluids too, which do not consider the infection of the self-gravition to fluids. The Navier-Stokes-Poisson equations are the more rigorous and better models that reflect the motion of gasous stars than the Navier-Stokes equations and the Euler-Poisson equations.In the 19th century, as to the Euler-Poisson equations, many mathematicians have had lots of research and gotten various results, including existence, uniqueness, stability, the existence for stationary solutions, and the existence of the symmetric solutions with solid core and without solid core. Moreover, presently, the Navier-Stokes equations are the very hot direction in fluids. Many excellent mathematicians devote to this field, and have gotten a great deal of results, including the global existence of weak solutions in 3-dimension, the global existence of the symmetric weak and strong solutions with solid core in 3-dimension, the global existence of the strong solutions in 1-dimension, the local existence of the strong solutions in 3-dimension, the large time behavior of weak or strong solutions, some stabilities relying on initial condition or viscous terms, the large time blow-up behavior for the smooth solutions, and so on. For details, please refer the references. But, as to the the Navier-Stokes-Poisson equations, there are a few results, mainly including (1) in [66], the global existence of the limited energy weak solutions in 3-dimension; (2) in [81-82], the global existence of the symmetric weak solution to the free-boundary systems; (3) in [95], when the viscosity of the Navier-Stokes-Poisson equations disappears, the Navier-Stokes-Poisson equations would converge to the Euler-Poisson equations. Compared to the Navier-Stokes and Euler-Poisson equations, the Navier-Stokes-Poisson equations have a lack of results. Since the Navier-Stokes-Poisson equations can more properly describe the motions of gasous stars, naturally, it is very important to consider the well posedness of the weak and strong solutions to the Navier-Stokes-Poisson equations. Hence we do some research and have gotten many significant results.As to methods, usually, compact method can handle the existence of the weak or strong solutions to the Navier-Stokes and Euler-Poisson equations. Obviously, it can be used to deal with the Navier-Stokes-Poisson equations. Moreover, using compact method, we concluded the following results: (1) the global existence and uniqueness of the radially symmetric strong solution to the Navier-Stokes-Poisson equations in three dimension; (2) the global existence, uniqueness of strong solution to the Navier-Stokes-Poisson equations in one dimension; (3) the local existence, uniqueness, stability of the strong solution to the Navier-Stokes-Poisson equations in three dimension. The main ideas of compact method are that: (i) we get the uniform estimates of approximate system; (ii) using the compactness for the spaces of approximate solutions, we can obtain that a subsequence of approximate solution converges to a limit function; (iii) we are required to prove that the solution of original problem is this limitation. The mainly difficult problem of compact method is step (iii). To avoid and overcome fussy discussion to compactness, we innovate methods to deal with the existence of strong solutions. We introduce the iterative method. And using our iterative method, we have gotten the following results: (1) the local existence of the strong solutions to the Navier-Stokes-Poisson equations and a blow-up criterion in three dimensions; (2) the local existence of strong solutions to the full Navier-Stokes-Poisson equations in three dimensions;(3) the local existence of strong solutions to the density-dependent full Navier-Stokes equations in three dimensions. The main ideas of iterative method are that: (i) prove the existence of the strong solution to linearized systems; (ii) from the existence of the strong solution to linearized systems, construct iterative approximate systems, and estimate uniformly the strong solution of iterative approximate system; (iii) deal with the convergence to the sequence of the iterative approximate strong solutions, precisely, for example, as to the full Navier-Stokes-Poisson equations, if we suppose that {pk}, {uk}, {ek}, {Φk} are the sequences of the strong solutions to the iterative approximate systems, using the uniform estimates in step (ii), we deal with the difference systems of the iterative approximate systems,thenThrough this estimates, naturally, we can get the strong convergence of the iterative approximate solutions. The virtue of iterative method is that we use the estimates of the iterative approximate systems to deal with the convergence of iterative approximate solution,and then avoid the fussy discussion of compact methods. Furthermore, in this paper, we get the local existence of strong solutions to the Navier-Stokes-Poisson equations with heat conductivity, and some more meaningful and novel results.This paper is organized as follows:Chapterâ… :Introduction. It includes physical backgrounds and researched history.Chapterâ…¡:Basic knowledge. It includes basic notions and basic Lemmas.Chapterâ…¢:The uniqueness and existence of the radially symmetric strong solutions to the Navier-Stokes-Poisson equations in three dimension. This result was dealed with by compact method. For detail, the main results are: if 0≤p0∈H1,u0∈H01∩H2 and satisfy the compatibility conditionwe get the existence and the uniqueness of the symmetric strong solutions to Navier-Stokes-Poisson equations in three dimensions.Chapterâ…£:The global existence and uniqueness of the strong solutions to the Navier-Stokes-Poisson equations in one dimension. This result was gotten by compact method. For detail, the main results are: supposed that p0∈H1(0,1),u0∈H01(0,1),we get the global strong solutions to the Navier-Stokes-Poisson equations in one dimension. Furthermore, if p0∈H1,u0∈H01∩H2 and the compatibility conditionthere exists the global strong solution with higher regularity for 1-D NFavier-Stokes-Poisson equations. It is important that through Lagrangian techniques(we translate the initial system into Lagrangian fluids, then combine the character of Lagrangian fluids), we get the global uniform boundary of density. Many excellent estimates can be concluded by this boundary;Chapterâ…¤:The well-posedness problems of the strong solutions to the Navier-Stokes-Poisson equations in three dimensions. This result can be proved by compact method. For detail, the main results are: if p0∈W1,6,u0∈H01∩H2 and satisfy the compatibility conditionwe get the exitence, uniqueness, stability of local strong solutions to the Navier-Stokes-Poisson equations in three dimensions;Chapterâ…¥:The local existence and uniqueness of the strong solutions to the Navier-Stokes-Poisson equations and a blow-up criterion in three dimensions. This result was dealed with by iterative method. For detail, the main results are: if p0∈W1,q,u0∈H01∩H2(30=4Ï€g(p0-(?)),and satisfy the compatibility conditionwe get the local existence and uniqueness of the strong solutions to the Navier-Stokes-Poisson equations and the blow-up criterion with large time, describe the relation between the existence of the strong solutions and the compatibility conditions;Chapterâ…¦:The uniqueness and existence of the strong solutions to the full Navier-Stokes-Poisson equations in three dimensions. This result was gotten by iterative method. For detail, the main results are: if p0>0,p0∈W1,q,(e0,u0)∈H01∩H2 and satisfy the compatibility condition there exists the local strong solutions to the full Navier-Stokes-Poisson equations in threedimensions;Chapterâ…§:The uniqueness and existence of the strong solutions to the full Navier-Stokesequations with density-dependent in three dimensions. This result was gotten by iterativemethod. For detail, the main results are: if 0≤p0∈W1,q,(e0,u0)∈H01∩H2 and satisfythe compatibility condition,the existence and uniqueness of local strong solution to the density-dependent full Navier-Stokes equations in three dimensions can be concluded.
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