On Classical Solutions Of The Incompressible Navier-Stokes Equations With Vacuum

In nature, all things obey the invariable order of nature, for instance, conservation laws (mass, momentum and energy), universal gravitation etc. We are opening out, conquering and transforming the nature by these foundational laws. Of course, it is out of question that the flowing gas or liquid (fluid) obeys the order of nature. For example, the fluid obeys conversation laws of mass and momentum. If we want to know the state of fluids, We need to study equations that are constructed according to these conversation laws.In this paper, we study, in 3-dimension, local classical existence of a unique solution to a class of incompressible Navier-Stokes equations with initial and boundary conditions. The motion of general incompressible viscous fluid in a bounded domain is descriked by the following equations with initial and boundary condition:whereρ, u, P denote the unknown density, velocity and pressure of the fluid, respectively. Andρ₀,u₀,f satisfy the following regularity conditions:Our main result is the following theorem.Theorem 1.1 Assumeρ₀,u₀,f satisfy (1.1.5),and the compatibility conditionfor some (P₀,g)∈H³(Ω)×H²(Ω). Then there existαtime T_*∈(0,T),and a unique strong solution (ρ,u,P) to the problem (1.1.1)-(1.1.4), such that Remark Let (ρ,u,P)beαsolution of (1.1.1)-(1.1.4) with the regularity (1.1.7). Then it is easy to show that (ρ, u,P) is indeed a classical solution of (1.1.1)-(1.1.4) in (0,T_*]×Ω.Firstly, in view of the sobolev embedding results ,we havewhich implies that (ρ,u) satisfies (1.1.1), (1.1.3), (1.1.4) in a classical sense .Then using the standard embedding resultsandfor any 2≤q < 6,we deduce from (1.1.5)and(1.1.7)thatOn the other hand ,by virtue of the continuity equation (1.1.1),we can rewrite the momentum equation (1.1.2) as which implies that for each t∈[0,T_*],u∈H₀¹(Ω)∩H³(Ω),P∈H²(Ω) satisfy the equtions of the form,Note that tF∈C([0, T_*]; W^1,4(Ω)),hence follows from the result in [15] ,we haveTherefore,in view of the sobolev embedding results,we conclude thatand (ρ,u,P)is a classical solution of (1.1.1)-(1.1.4)in (0, T]×Ω.To proof the theorem,we firstly study the following linearized problemwith initial boundary value conditionsBy using standard semi-discrete Galerkin method,we could prove the existence and uniqueness of the classical solution of'(1.1.8)-(1.1.10).Since vacuum may appear ,which would bring more difficulties,we consider the problem in two steps .Firstly,we consider the case of non-vacuum .Then we study the original case with vacuum. For the case of non-vacuum,we consider the follwing initial boundary problemwhereρ₀≥δ> 0.we construct the following approximating problem :Let u⁰ = 0,and forThen we obtain that (ρ^k,u^k,P^k) is classical solution of approximating problem and we get a uniform prior estimates to the approximating solutions, Taking limit about k,we obtain the following theorem :Theorem 1.2 Assumeρ₀,u₀,f satisfies (1.1.5) and (1.1.6),ρ₀≥δ> 0.Then there exist a time T_*∈(0,T),and a, unique strong solution (ρ,u,P) to the problem, (1.1.1)-(1.1.4),such thatAccording to the Theorem 1.1, we may prove the existence and uniqueness of the solution for the initial and boundary value problem with vacuum. To this end, we need to regularize the initial value of the original problem, such that they satisfy the conditions of Theorem 1.1.Sinceρ₀ is smooth, then for smallδ, 0 <δ＜＜1, letρ₀^δ=ρ₀ +δ, be the unique solution of the following zero boundary value problem For the uniform estimate of u₀^δ,P₀^δ,we could takeδ→0 ,and obtain that the limit (ρ₀,u₀,P₀) be a classical solution of following problemSo we obtain that (ρ^δ, u^δ) is the unique solution of the following problemand the following uniform estimate holdsAccording to. the uniform estimate(1.1.17),by taking limit aboutδ,then the existence of Theorem 1.1 is proved.. Next,we will prove the uniqueness .By equatins (1.1.1)-(1.1.4),we getThen we obtainSo,this complete the proof of Theorem 1.1.