Study On Compressible Fluid Mechanics Mathematical Theory With Energy Conservation Equation

The nonlinear partial differential equations in the fluid dynamics as the basic model are very important, such as Navier-Stokes equations, magnetohydrodynamic (MHD) equations, electric-magnetohydrodynamic equations, Rayleigh-Benard convection, κ-ε model equations for turbulent flows, Navier-Stokes-Korteweg eqautions and so on. They are used to describe fluid flow and many important mathematical model of physical phenomena. Navier-Stokes equations as a typical system lie in the hot research areas of mathematics and physics at home and abroad, especially, the study of theirs definite solution problem. This system has been intensively studied due to its physical background and mathematical significance.In this paper, we talk about the nature of the solution to the full compressible fluid mechanics equations from five sides.Firstly, we show the existence and uniqueness of global strong spherically symmetric solutions to the full compressible Navier-Stokes equations in the whole space RN (N=2,3) with large initial data and the stress free boundary condition. The main difficulty is to obtain the L∞L2-norm for ux due to the stress free boundary condition. To overcome this obstacle, we first estimate the L∞L2-norm for θx in terms of the L∞L2-norm for uxx, then combining this estimate with the energy estimate for9in the H2level, we can obtain the desired estimates for the first and second derivatives of u. See my paper12.Secondly, we study the Cauchy problem to the κ-ε model equations for turbulent flows about gas explosion in coal mine. The classical energy method and technical estimates are used to prove the existence and uniqueness of global smooth solution to the κ-ε model equa-tions for turbulent flows with small initial data. At the same time, the large-time behavior of the global smooth solution is also considered and the optimal convergence rates are estimated as follows||(ρ-ρ,u,h,k-k,ε)||q≤C(1+t)-σ(p,q;0),2≤q≤6,||(ρ-ρ,u,h,k,ε)||∞≤C(1+t)-σ(p,2;1),||▽(ρ-ρ,u,h,k-k,ε)||H2≤C(1+t)-σ(p,2;1),||(ρt,ut,ht,kt,εt)||2≤C(1+t)-σ(p,2;1), where σ(p, q; l) is defined as σ(p,q;l)=3/2(1/p-1/q)+l/2. See my papers2and5. Thirdly, we investigate the vanishing capillarity limit of the compressible fluid models of Korteweg type to the Navier-Stokes equations. We show that the unique smooth solution of the three dimensional Navier-Stokes-Korteweg system converges globally in time to the strong solution of the three dimensional Navier-Stokes system, as k tends to zero. The main method is that we firstly establish the uniform estimates of global smooth solution with respect to the capillary coefficient k, then by Lions-Aubin lemma, for any given positive time t, we establish the following convergence estimate where C is a positive constant independent of k and t. See my paper1.Fourthly, we focus on local well-posedness in critical Besov spaces for the non-isentropic compressible magnetohydrodynamics (CMHD) equations in RN,N≥2. We mainly apply Terence Tao’s abstract bootstrap principle to prove existence of the solution to the com-pressible MHD equations. By the Littlewood-Paley decomposition, interpolation inequality, Young inequality, Bernstein inequality, commutator estimate, product estimate, Gronwall lemma and so on, we get that for T>0small enough, the solutions (an, Hn,un,θn)n∈N are uniformly bounded in space Then by using compact principle, the convergence of the solutions is obtained. Finally, by Osgood lemma, we show the uniqueness of the solution. See my paper4.Rayleigh-Benard convection is a typical mathematical model to study the convection of fluid, which is very important in our modern life. We firstly establish the local well-posedness for the full compressible Rayleigh-Benard convection equations. Next, we are concerned with global existence and large-time behavior of solutions to the isentropic electric-magnetohydrodynamic equations in a bounded domain Ω (?) RN, N=2,3. Finally, we prove the blow-up of smooth radially symmetric solutions to the isentropic compressible MHD equations in RN,2≤N≤3. See my papers7,6and3.