Global Existence Of Strong Solutions Of Navier-Stokes Equations With Non-Newtonian Potential

The Knetic theory of the gas is also known as compressible fluid dynamics.It is research the fluid dynamics of variable density or fluid played a significant compression.The Euler and Navier-Stokes equations,are the most famous equations among govern-ing equations proposed so far in fluid dynamics.In this paper,we only consider thatΩis a one-dimensional bounded interval. For simplicity we only consider thatΩ=(0,1),T<+∞.The initial and boundary conditions areρ(?)t=0=ρ0(x)≥0,μ(?)t=0=μ0 (?)x∈(0,1) u(0,t)=u(1,t)=0,Φ(0,t)=Φ(1,t)=0,(?)t>0The Navier stokes equations for compressible fluids have been studied by many au-thors.Lions[53]used the weak convergence method and first showed the existence of global weak solutions for isentropic flow under the assumption thatγ≥3/2if n=2 andγ≥9/5if n=3. In a paper[2],Feireisl,Novotny and Petzeltova extended Lions' global existence result in R3 to the caseγ>3/2.Kim in[3]have showed that the global existence of strong solution for the one-dimensional Navier-Stokes equation.For Navier-Stokes Equations With Non-Newtonian Potential,in[4],Junping Yin and Zhong Tan proved the existence of strong solution to Navier-Stokes-Poisson equations.For the existence,uniqueness,or other virtues of the strong solutions of Navier-Stokes equations,we may refer to[5]-[12]. in[14][15],we know the viscosity coefficient may depends on the density. It is worth to mentioning a recent result due to Huanyao Wen and Lei Yao in [16]. They consider the initial boundary value problem ofTheir results showed the existence of global strong solution and uniqueness. But as for Navier-Stokes Equations With Non-Newtonian Potential, the results of strong solutions are few.In this paper, we mainly study four class of compressible fluid models with non-Newtonian potential.Firstly,Here the unknown functions p=p(x, r) and u=u(x, t) denote the density and velocity respectively. P= apy(a>0,γ>1) is the pressureΦ=Φ(x, t) is the non.Newtonian gravitational potential.1<p<2,0<δ<l.In this paper, we normalize a=1 in the rest of this paper. Physically, this system describes the motion of compressible viscous isentropic flow under the non-Newtonian gravitational force.In this paper, we only consider thatΩ. is a one-dimensional bounded interval. For simplicity we only consider thatΩ= (0,1), T<+∞. The initial and boundary conditions are Theorem Assume that the initial conditions satisfyThen there is a global strong solution (ρ, u,Φ), such that for all T∈(0,∞), we haveTheorem Assume that the initial conditions satisfyand compatibility conditionThen there is a unique strong solution (ρ, u,Φ), such that for all T∈(0,∞), we have ut, Gx∈L2(0, T; H1(0,1))Φt,∈L∞(0, T; H2(0,1)) where G=λux-PSecondly, Where P= aργ(a>0,γ>1),λ>0,p>2,μ0>0. In this paper, we only consider thatΩis a one-dimensional bounded interval. For simplicity we only consider thatΩ= (0,1), T<+∞. The initial and boundary conditions areAssume that the initial conditions satisfy Then there is a global strong solution (ρ,u,Φ) of (1.1)-(1.5), such that for all T∈(0,∞), we haveThirdly, We consider the strong solutions to the initial boundary value problems for the isentropic compressible Navier-Stokes equations in one dimension:Here the unknown functionsρ=ρ(x, t) and u=u(x, t) denote the density and velocity respectively. P=aργ(a>0,γ>1) is the pressure.Φ=Φ(x, t) is the non-newtonian gravitational potential.0<p<2,0<δ<1. In this paper, we normalize a=1 in the rest of this paper. Whereμ1 is a positive constant. T<+∞. The initial and boundary conditions areTheorem Assume that the viscosity coefficientμ(ρ) satisfies (1.4), the initial data Then there is a global strong solution (ρ,μ,Φ) of (1.1)-(1.6), such that for all T∈(0,∞), we haveFinally,Here the unknown functionsρ=ρ(x, t) and u=u(x, t) denote the density and velocity respectively. P= aργ(a>0,γ>1) is the pressure.Φ=Φ(x, t) is the non-newtonian gravitational potential. p≥2,μ0> 0. In this paper, we normalize a=1 in the rest of this paper. Whereμ1 is a positive constant. T<+∞. The initial and boundary conditions areTheorem Assume that the viscosity coefficientμ(ρ) satisfies (1.4), the initial data Then there is a global strong solution (ρ,μ,Φ) of (1.1)-(1.6), such that for all T∈(0,∞), we have...