Introduction Of Research Advance In Fluid Dynamics

This paper is a review of the article, a brief introduction to the nearly 10 years of fluid dynamic equations of progress. We can select the main line with viscous compressible flow, through pressure does not depend on temperature and pressure-dependent temperature divided into two categories for the study of a weak solution, mainly to discuss the two-dimensional and three-dimensional space and the global weak solution oriented.Generally, a compressible and heat conducting fluid governed by the Navier-Stokes equations satisfies the following system:In the equation involved in the meaning of symbols can be found in the second part of the body, there is described in detail.In the third part, we introduce pressure-independent case,namely compressibleisentropic flow. The form of equation can be written as:We give initial condition and boundary condition as follow: initial condition:ρ|t=0=ρ₀,ρu|t=0=m₀Dirichlet boundary condition (no-slip boundary condition):u|(?)Ω=0We take into account problem in the domain (0, T)×Ω, where (0, T) is time interval,Ω, is a space region.We divide two section to discuss in the basis of pressure depends on the density, the ideal viscous compressible flow and general viscous compressible flow. In the case of ideal viscous compressible flow, namely:p= ap^γ, a > 0,γ> 1.We introduce a classic result of the 1990s by Lions, under the assume:μ> 0,μ+λ> 0, a > 0,γ∈(1, +∞).he obtained the results in the following cases:(1)The whole space case:Ω= R^N, R^N×(0, T), T∈(0, +∞);(2)Dirichlet boundary condition (4);(3)Periodic case: the problem (2) hold in R^N×(0,T), we require all the data are periodic in each x_iand period T_i∈(0, T)(1≤i≤N).Theorem 1. We assume:there exists a weak solution (ρ, u) of (2) satisfying the initial conditions (3), and in the the whole space case and periodic case whereρ∈L^p(Ω×(0, T)); in the case of Dirichlet boundary conditions whereρ∈L^p(K×(0, T)) for any compact set K (?)Ω, where p =γ+ 2/Nγ- 1. In addition, (ρ,u) satisfies the following energy inequality for almost all t∈[0, T]:Then in the introduction of the definition of renormalized solution and finite energy weak solution, We present another classic results, It is received in 2001 by Feireisl.Theorem 2. AssumeΩ(?)R³ is a bounded domain of the class C^2+v, v > 0,Let the dataρ₀, m₀ = (m₀¹,m₀²,m₀³) are supposed to comply with compatibility conditions of the form:ρ₀∈L^γ(Ω),ρ(0)≥0, m₀ⁱ(x) = 0, wheneverρ₀(x) = 0, |m₀ⁱ|²/ρ₀∈L¹(Ω),i=1,2,3.and letγ>3/2.Then given T > 0 arbitrary, there exists a finite energy weak solution (ρ, u) of the problem (2)satisfying the initial conditions (4).This theorem is also set up in the two-dimensional. That indicate the existenceof global finite energy weak solution for the problem (2), (4)whenγ> n/2.In addition, we have also introduced weak time-period solution driven by a time-period external force, global weak solutions subject to large external potentialforces, global weak solution of the long time stability, global weak solutions with discontinuous and small initial conditions, and so on. Text can be found in details.We also mention a very remarkable case studied in two space dimension about period solution, see [5], it is obtained under the following unnatural hypothesis:μ= C_st,λ= bρ^β, b>0,β> 0. Theorem 3. Letβ> 3, andγ≥1, T² = R²/Z², Letρ₀(x), m₀ such that: (ρ₀, m₀(x))∈L^∞(T²)×(W^1,2(T²))².Then, there exists a weak solution of the problem(2), (3). The solution exists for all time t>0.In the case of general viscous compressible flow, p = p(ρ) is a common function on p. Function on the pressure in certain assumptions circumstances, we have some results, Text can be found in details.In the fourth part, that is pressure dependent on temperature circumstances, duing to the energy conservation equation, and with the temperature, so the more difficult to solve. Therefore, the present results are obtained in the presence of many of the restrictive conditions and come under the assumption that, for example, in the rules area (plane, spherical, etc.) to consider, on the viscosity, thermal conductivity, equation of state, as well as the certain special assumption, and so on.We introduce a result of global weak solution, it is from Lions; A result from Feireisl, it is obtained in the sense of variational solution.System (1) is supplemented with initial conditions:ρ|t=0=ρ₀,ρu|t=0=m₀,ρE|t=0=G₀+|m₀|²/2ρ₀. (5)For the stability of the global weak solution, we have:Theorem 4. LetΩbe either the periodic box T³ or the whole space R³. Viscosity,thermal conduction and equation of state, initial condition satisfy special assumptions, and have:H(0)=∫_Ω(G₀ +|m₀|²/2ρ₀)dx<+∞.that the initial density posatisfy for some positive constantρ∞,ρ₀-ρ_∞∈L¹(Ω),ρ₀logρ_∞/ρ₀∈L¹(Ω), ρ₀e_c(ρ₀)∈L¹(Ω),▽μ(ρ₀)/(?)∈L²(Ω)^d.and that the initial entropy densitys₀=C_vlog(θ₀/ρ₀^г),satisfy:ρ₀s₀∈L¹(Ω).Then, there exists a global in time weak solution to (1), (5).In the fifth part, we have the whole issue of a summary and outlook. Publicly on the unresolved issues are introduced, as follows:The question of global regularity and uniqueness of solutions without restrictionson the size of the data is still open in dimension d=3 both in incompressible and in the compressible case.In dimension d = 2, global regularity and uniqueness are well known for incompressibleflows. Concerning the compressible case, some answers are provided in [5] with particular assumptions on the constitutive relations namely constantμand density dependentλ. General result is still open because of the lack of estimates in regions where the density vanishes.The problem(2)-(4) with initial condition and no-slip boundary condition, we assume that the equations of state are of ideal polytropic gas type:P = aρ^γ,γ> 1, a > 0.when 1≤γ≤n/2, n = 3, the existence of the global large solutions is still not clear, this is a very difficult open problem.In some of the weak solution under the regularity assumption, Hoff recently studied the uniqueness of the weak solution, but with finite energy weak solution is still not clear, This is a very challenging issue.