Well-posedness And Singularity Formation Of The Compressible Isentropic Navier-Stokes Equations

The theory on well-posedness and singularity formation is an important branch of the field of study on hydrodynamics equations. In this thesis, a series of research has been done for the Cauchy problem to the compressible isentropic Navier-Stokes equations with density-dependent viscosities in a power law and vacuum. Due to the high degeneracy of this system in the vacuum domain, the construction of the multi-dimensional classical solutions with arbitrarily large data and vacuum has been a long-standing open problem. We have given valuable contributions in this thesis toward this direction.The first chapter is an introduction. We describe the related mathematical models and the physical background. We also recall the related known results and analyze the possible mathematical difficulties for the problems under consideration, and show the new ingredients for solving our problems. In addition, we present some necessary preliminary knowledge. The main results of this thesis can be shown as follows:1. Local existence of classical solutions with vacuum.We analyze the mathematical structure of this system in the vacuum domain, and reasonably give the evolution mechanism of the fluid velocity along with the time in the vacuum domain. Based on this, via making full use of the symmetrical structure of the hyperbolic operator and the weak smoothing effect of the elliptic operator, and introducing some artificial viscosity, we obtain a series of a priori estimates independent of the lower bound of the initial mass density, and success-fully establish the local existence theory of classical solutions with arbitrarily large data and vacuum. Here, we particularly point out that, in the vacuum domain, different powers of the density appearing in the form of viscosity corresponds to different mathematical structures which control the behavior of the fluid velocity.2. Vanishing viscosity limit.We establish the vanishing viscosity limit of the Navier-Stokes equations to the Euler equations for three-dimensional compressible isentropic flow with vacu-um appearing in some open set or at the far field, and also give the convergence rates in Sobolev space Hs for any constant s’∈ [2,3). Actually, we show that the life span of the solution to compressible isentropic Navier-Stokes equations is uniformly positive with respect to the viscosity coefficients. It is worth paying spe-cial attention that we could also establish a uniform energy-type estimate in H3 space with respect to the viscosity coefficients for our solutions, which leads the convergence of the regular solution of the Navier-Stokes equations to that of the Euler equations for compressible isentropic flow.3. Singularity formation of solutions.Based on the analysis on the evolution mechanism of the fluid velocity along with time in the vacuum domain, we propose two classes of initial data which lead our smooth solution to blow up in finite time. In truth, we show that for some initial data, the classical solution that we obtained will break down if vacuum appears in some local domain. This makes it impossible to get a general theory of global existence with vacuum, even when the initial data are sufficiently small and smooth in any sense. At the same time, we also point out that because the solution that we obtained satisfy the conservation law of momentum in the sense of integration, then we could not get such global solution whose L∞ norm of the velocity decays to zero as time goes to infinity.4. Beale-Kato-Majda blow-up criterion.Corresponding to the non-global existence theory shown in the above chapter, when viscosity coefficients depend on density linearly in R2, we show the blow-up mechanism or singularity structure via establishing a Beale-Kato-Majda type blow-up criterion. More precisely, we show that the L∞ norms of ±▽ρ/ρ and D(u)= (▽u+▽nT)/2 control the possible blow-up for our classical solutions, which means that if a solution is initially smooth and loses its regularity at some later time, then the formation of singularity must be caused by losing the bound of the L∞ norm of D(u) or ▽ρ/ρ as the critical time approaches.