Properties Of Solutions For One Dimensional Compressible Navier-Stokes Equations

In this thesis, we are concerned with some properties of solutions for one dimensional compressible Navier-Stokes equations.In Chapter one, we study the global stability of strong rarefaction waves for compressible Navier-Stokes equationswith initial data(v(t,x),u(t,x),s(t,x))|t=0=(v₀(x),u₀(x),s₀(x))→(v_±,u_±,s_±) as x→±∞. (0.4) Here the unknowns v＞0,u,θ＞0, p＞0, e, and s represent the specific volume, the velocity, the absolute temperature, the pressure, the internal energy, and the entropy of the gas respectively. The coefficients of viscosity and heat-conductivity,μandκ, are assumed to be positive constants, and v_±＞0, u_±, s_±are constants. Our main trick is to introduce a positive parameter t₀ in the construction of smooth approximation of the rarefaction wave solutions for the compressible Euler equations and then use the quantity e=t₀/δ(whereδdenotes the strength of the rarefaction waves). which is chosen to besufficiently large, to control the possible growth induced by the nonlinearity of the compressible Navier-Stokes equations and/or the interactions of waves from different families, and then based on the fundamental energy method together with the continuation argument, we can indeed obtain three types of global stability results.In Chapter two, we study vacuum problem for weak solutions of the compressible Navier-Stokes equations. Note that such a problem was studied by D. Hoff and J. A. Smoller in [18], but their proof is incomplete. We give a refinement of Lemma 2.2 in [18] which measures the time evolution of the vacuum regions and complete the proof of non-formation of vacuum states for one-dimensional compressible Navier-Stokes equation.