The Global Solution To A Class Of Compressible Navier-Stokes Equations With Non-Newtonian Potential

Fluid dynamics mainly study the flowing and stable state, and theinteractional of the fluid and the boundary of solid.The problem of the fluid dynamics, especially the nonlinear problemabsorbed many mathematicians, engineers and physical scientists's, includingEuler and Bernoulli. They have built the mathematic models in their works.The models indicate that the subject was built. From then on, the study ofthe fluid dynamics began to use differential equations. After that, in orderto solve engineer problems, especially the effects of the viscousness, Navierbuilt the viscous fluid dynamics, in 1822; Stokes gave the equation based onmore logical law, in 1845. The equations are still used today, which is namedNavier-Stokes equations.Corresponding to Newton's law and the conservation law, the fluiddynamics give the equations of continuity, momentum and thermal energyin mathematics. Furthermore, there are some other equations, such as stateequation. We call it together the fluid dynamics.The fluid dynamics study different models, which are the ideal fluiddynamics, the viscous fluid dynamics, the incompressible fluid dynamics, thecompressible fluid dynamics and the non-Newton fluid dynamics.In this paper, we delicate to the model of the fluid dynamics, thatis, the compressible Navier-Stokes equation with non-Newtonian potential.That is,·equation of continuity: ·momentum equation:·non-Newtonian potential:·the boundary conditions:·the initial conditions:whereΩ(?) R³ is a bounded domain of the class C^2,θ, 0 <θ< 1.(?) is the outernormal vector, (?) = (?)(x, t) the density, (?) = (?)(x, t) =[u₁(x,t), u₂(x, t), u₃(x, t)]the velocity,φ=φ(x,t) the non-Newton gravitational potential, (?)▽φtheunit body force caused by the gravity,γ> 1 the adiabatic constant,μ> 0andλthe viscosity constants satisfyingλ+ 2/3μ≥0, a = e^S the constant determinedby the entropy S, and g > 0 the gravitational constant. Physically,this system describes the motion of compressible viscous isentropic gas flowunder the gravitational force.The difficult of this type model is mainly that the equations are coupledwith elliptic, parabolic and hyperbolic. And when the density (?) = 0 and(?)=∞, the vacuum and concentrate of density cause much trouble. Also thedegenerate of the elliptic equation, and so on.We get the main result as following.Theorem 1 Letγ> 3/2. Then, given (?)₀∈L^γ(Ω),|q₀ⁱ|²/(?)₀∈L¹(Ω), (?)₀≥0;q₀ⁱ(x)=0, as (?)₀ = 0. Then, given T > 0 arbitrary, there exists a finiteenergy weak solution (?), (?),φof the problem (1)-(3) satisfying the boundaryconditions (4) and the initial conditions (5). That is, ·(1), (2), (3) hold in D'(Ω×(0, T)); moreover, (1) holds in D'(R³Ã—(0, T))provided (?) and (?) were prolonged to zero on R³\Ω;·The first equation (1) is satisfied in the sense of the renormalized solution,i.e.Before we give the proof of theorem 1, we first consider the followingmodified problem:with the boundary conditions:and, we modify the initial conditions: Now, we introduce the reason why we modified the problems.The extra termε△(?) in (1) represents a "vanishing viscosity" with nospecific physical meaning. From the mathematical viewpoint, however, itconverts the hyperbolic equation (1) into a parabolic one. As a result, onecan expect better regularity properties of the " density" g constructed at thislevel of approximation. At the same time, we give the Neumann boundarycondition (10) in order to keep the conservation of the mass. And we cangive the better regularity of (?) for the convergence.The quantityδ(?)^βadded to the momentum equation (2) can be consideredas an " artificial pressure", which was introduced to make the pressureestimates compatible with the vanishing viscosity regularization of (1). ForNavier-Stokes equation, the better integrality means better regularity.In the same spirit, the new quantityε▽(?)·▽_? was introduced in (8)in order to eliminate the extra terms arising in the energy inequality to savethe a priori estimate.Finally, there is some change about (3) for eliminating the degenerateof the equation, and more easy to give the existence and the estimate of theequation.We use the method of Faedo-Galerkin to give the global smooth solution.By the uniform estimates of (?)_n, (?)_n andφ_n we get the followingtheorem.Theorem 2 For givenβ> max{4,γ}. Then for any T > 0, the problem(7)-(14) has the weak solution on [0,T]. That is, ((?),(?),φ) such that Our next goal is to letε→0 in the modified continuity equation(7). To this end, let us denote by (?)_ε,(?)_ε,φ_εthe corresponding solutionof the approximate problems. At this stage of the proof of Theorem 1, wedefinitely loose boundedness ofâ–½_?εand, consequently, strong compactnessof the sequence of {(?)_ε} in L¹(Ω×(0,T)) becomes a central issue. So we usethe method of Lions', which depends on the "effective viscous flux", that is,(a(?)^γ-(λ+ 2μ)div(?))(?). Specifically, asε→0, we havewhere (?) and (?) are the limits of them in the sense of suitable topology.In order to give the limits of the solutions asε→0, we need the moreregularity of density (?), using the test functionwhere i = 1,2,3,, and we get the following estimateBy this estimate, we getThen, we use the "Minty"' method to get the compactness of (?)_ε.For the compactness of▽φ_ε, we use the following inequalityThus, we get the following result. Theorem 3 LetΩ(?) R³ be a bounded domain of class C^2,θ and letβ>Then, given initial datas (?)₀,(?)₀ as in (13), (14), there exists a finiteenergy weak solution (?), (?),φof the problemMoreover, (?)∈L^β+1(Ω×(0, T)) and the equation (20) holds in the senseof renormalized solutions on D(R³Ã—(0,T)) provided (?), (?) were prolonged tobe zero on R³ \Ω.Finally, we have the following estimates:Our ultimate goal in the proof of Theorem 1 is to carry out the limitprocess when the parameterδtends to zero. By the modified initial conditions,we get the uniform estimates independent onδ; in order to give moreregularity of (?)_δ, we use the following test functionWe also use the cut functions. Specifically, we consider a class of functions: where T∈C^∞(R) chose as followingWe get the following estimate:where C is independent on k. (?)_δis the approximate solution, and (?) means thelimit in the suitable topology. So we get the proof of renormalized solution,that is, (?), (?) were prolonged to be zero outsideΩ, we havefor any b∈C(R), b'(z) = 0 as z is large enough.Then, we use the following function to get the strong convergence of(?)_δ.Using this function, we get (?)= (?)ln (?). So we have the strong convergenceof (?)_δin the suitable topology.Thus, we get the proof of Theorem 1.