On The Compressible Navier-Stokes Equations With Large External Potential Forces

The Navier-Stokes equations describe the motion of viscous Newtonian fluids and have a very wide range of applications in fluid mechanics,astrophysics,geophysics and so on.It has been a hot topic in both theoretical and numerical studies of partial differential equations.Since the pioneering work of the incompressible Navier-Stokes equations due to Leray in 1934,there has been a lot of literature on the mathematical theory of compressible/incompressible Navier-Stokes equations.However,due to the complicated coupling and strong nonlinearity of the mixed type equations,many mathematically fundamental and physically important problems are still open,for example,the regularity and uniqueness of the global weak solution of the three dimensional incompressible Navier-Stokes equation with large data,the global existence of weak solution of the three dimensional compressible isentropic Navier-Stokes equation with adiabatic exponent 1<γ<3/2,and so on.In reality,the fluid motion is often driven by the external forces,which will significantly affect the dynamic stability of flows when the external forces are large.Moreover,the(possible)presence of mass concentration and vacuum will also cause some serious difficulties in the mathematical analysis of compressible Navier-Stokes equations.In this thesis,we aim to study the existence and large-time behavior of the global solution with vacuum for the initial(-boundary)value problem of three dimensional compressible Navier Stokes equations subject to large external potential forces.It is divided into three chapters.In Chapter 1,we briefly review the closely related research results of multidimensional compressible Navier-Stokes equations,and state the main theorems of this paper.We also recall some known results and elementary inequalities,which will be frequently used throughout this thesis.In Chapter 2,we focus our interest on the Cauchy problem of the three dimensional viscous heat-conductive compressible Navier-Stokes equations subject to large external potential force on the whole space R3.Due to the strong coupling between velocity field and temperature and the strong nonlinearities(e.g.the quadratic term|▽u|2),the theoretical study of the heat-conductive flows is more complicated than the isentropic case.For discontinuous data with small energy and vacuum,we prove the global existence of "intermediate weak" solutions with large oscillations and large external potential forces,provided the unique steady state is strictly away from vacuum.The proof is based on the classical local existence result and the global a priori estimates.To derive the global-in-time(weighted)estimates,we make full use of the mathematical structure of the steady solutions and introduce the so-called "modified"effective viscous flux and vorticity.The asymptotic behavior of the solutions is also obtained.It is worth pointing out that the "modified" effective viscous flux and vorticity,depending strongly on the stationary solutions,are different from the ones defined in[8,13,16,20]for the Navier-Stokes equations without external potential force,and play important roles in proving the LP estimates of the gradient of velocity.The solutions obtained are called "intermediate weak solutions",which were firstly introduced by Hoff(cf.[11-13])and have better regularity than the usual finiteenergy/variational weak solutions(see[8,20]).However,the question that whether the uniqueness of such weak solutions holds or not is still unclear.In the second part of Chapter 2,we aim to discuss the regularity and uniqueness of the weak solutions.Under the assumptions that the initial density is strictly positive from below and the viscosity coefficients satisfy the additional and non-physical condition 7μ>λ,we prove the solutions,belonging to a larger class of functions,are unique.Compared with the existing results where the regularity ▽∈u H1 was required,to prove the uniqueness,it suffices to show that Vu ∈ L3.In Chapter 3,we study an initial and boundary value problem of the compressible viscous isentropic Navier-Stokes equations with large external potential forces in 3D half space.For technical reasons,we only consider the case of Navier’s boundary condition.Analogously to that in Chapter 2,the modified effective viscous flux and vorticity(depending on the stationary solutions)are crucial for the entire analysis.However,compared with the initial value problem,some additional boundary integrals arise from the integration by parts.Moreover,due to the lack of the boundary conditions for the effective viscous flux,it seems difficult to apply the standard theory of elliptic system to deduce some estimates of the effective viscous flux.To overcome these difficulties,we make full use of the Navier’s boundary condition to infer that the equation of vorticity is equipped with suitable boundary conditions,so that,the regularity theory of elliptic system yields the estimates of vorticity.This,combined with the standard Lp-estimates and the momentum equations,also gives rise to the desired estimates of the modified"effective viscous flux" and the gradient of velocity.With the global a priori estimates at hand,we can then establish the global well-posedness of strong solutions with vacuum and large oscillations,provided the initial energy is suitably small.Compared with the previous literature where an initial compatibility condition was technically needed,we assume only that u0=m0/ρ0 ∈H1 is well defined,where m0 and ρ0 are the initial momentum and the initial density,respectively.