Study Of Two Mathematical Problems In Fluid Dynamics

In fluid dynamics, there are two fundamental equations, i.e. Euler equations and Navier-Stokes equations. The Euler equations are a set of quasilinear hyperbolic equa-tions governing adiabatic and inviscid flow while the Navier-Stokes equations describe the motion of viscous fluid substances. The research on solutions of the two equations is not only interesting and amazing in mathematics but also meaningful and useful in our real life. It admits many strong physical backgrounds. In this thesis, we will study two mathematical problems related to these two equations.First, we will study a problem related to the incompressible Navier-Stokes equa-tions in 3 dimensions. The incompressible Navier-Stokes equations in cartesian coor-dinates are given by a_tv+（v·▽）v+▽p=△v, ▽· v= 0, （0.0.4） where v is the velocity field and p is the pressure. We consider the axisymmetric solution of the equations. That means, in cylindrical coordinates γ,θ,z with x=（x₁,x₂,x₃）= （rcosθ,rsinθ,z）, the solution is of this form v= v_re_r+v_θe_θ+v_ze_z, where the basis vectors e_r, e_θ, e_z are and the components v_r,v_θ,v_z do not depend on 9. We denote b= v_re_r+v_ze_z.Global in-time regularity of the solution to the axisymmetric Navier-Stokes e-quations is still open. Under the no swirl assumption, v_θ= 0, Ladyzhenskaya and Ukhovskii-Iudovich independently proved that weak solutions are regular for all time. When the swirl v_θ is non-trivial, recently, some efforts and progress have been made on the regularity of the axisymmetric solutions. The main result is that if the solution satisfies r|b|< C_*<∞, then the solution is regular.The natural scaling of Navier-Stokes equations is:if （v,p） is a solution of equa-tions （0.0.4）, then for any λ> 0, the scaled pair v^λ（x,t）=λv（λx, λ²t）, pλ（x,t） λ²p（λx, λ²t） is also a solution.It seems that the above assumption on b is critical in the following scaled sense:Denote b^λ= v_r^λe_r+v_z^λe_z and let x₀= （r₀cosθ₀, r₀sinθ₀,z₀） be a fixed space point. For the scaled solution, the assumption r₀|b（x₀,t）|< C_* becomes When λ→ 0, the bound C* is invariant, independent of λ.We study the regularity problem under a slightly supercritical assumption on the term b. To be precise, we consider b such that where a is a small positive number. Our assumption （0.0.5） is supercritical which means for the scaled solution, the assumption r0|b（x₀,t）|< C_* becomes When A λ→ 0, the bound goes to infinity.It shows that when one zooms in at a point, the bound on the term b becomes worse, so the regularity of our solution must be handled more carefully.In Chapter 2, we will be dedicated to proving the regularity of the solution to the axi-symmetric Navier-Stokes equations under the assumption （0.0.5）.Second, we will study the isentropic Euler equations with time-dependent damp-ing.l.e. where ρ（t,x）,u（t,x）and p（t,x）represent the density,fluid velocity and pressure re-spectively and λ,μ>0 are two positive numbers to describe the decay rate of the damping with respect to time.As is well known,when the damping vanishes（μ=0）,shock will form and there is no global smooth solutions to system（0.0.6）.While the solution of the Euler equations with constant damping（λ=0）and small data is globally existed.It is natural to ask whether there is a pair of non-negative critical exponent（λ,μ）to separate the global existence and the finite-time blow up of solutions with small data.It is believed and proved that there is a pair of（λc（n）,μc（n））,depending on the space dimension n,such thatwhen 0≤λ<λ_c（n）,0<μ or λ=λ_c（n）, μ>μ_c（n）,（0.0.6）have global existence of small-data solutions:while when λ=λ_c（n）,μ≤μ_c（n） and λ>λ_c（n）,0≤μ,the smooth solution of（0.0.6）will blow up in finite time.The first work on this topic comes from[50]where the authors proved that in 3 dimensions,（λ_c（3）,μ_c（3））=（1,0）and later they also show that（λ.（2）,μ_c（2））=（1,1） in[51].The second part of the main result of this paper is to show that in 1 dimension, we have（λ_c（1）,μ_c（1））=（1,2）.All the clues indieate that the critical exponent can be stated as follow. （λ_c（n）,μ（n））=（1,3-n） n≤3；In Chapter 3,we will be devoted to proving the fact（λ_c（1）,μ_c（1））=（1,2）.