The Initial-Boundary Value Problem To A Class Of The Regularization For The Vlasov Equations With Non-Newtonian Potential

The kinetic 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.In the macroscopic scales where the gas and fluid are regarded as a continuum, their motion is described by the macroscopic quantities such as macroscopic mass density, bulk velocity, temperature, pressure, stresses, heat flux and so on. The Euler and Navier-Stokes equations, compressible or incompressible, are the most famous equations among governing equations proposed so far in fluid dynamics.The extreme contrary is the microscopic scale where the gas, fluid, and hence any matter, are looked at as a many-body system of microscopic particles. Thus, the motion of the system is governed by the coupled Newton equations, within the framework of the classical mechanics. Although the Newton equation is the first principle of the classical mechanics, it is not of practical use because the number of the equations is so enormous.From a cognitive perspective, we also hope that a way can be reflected macroscopic scales and microscopic scale at the same time. The Boltzmann Equation is able to embody the characteristics. So we can say that the Boltzmann equation, which is the subject of these notes, is the most classical but fundamental equation in the mesoscopic kinetic theory. In fact, the Boltzmann equation is a convergence between the Euler and Navier-Stokes equations.The first existence theorem of the solutions to the Boltzmann equation goes back to 1932 when Carleman proved the existence of global (in time) solutions to the Cauchy problem for the spatially homogeneous case. It should be stressed that this is two years before the incompressible Navier-Stokes equation was solved by Leray on the existence of global weak solutions. On the other hand, the research on the spatially inhomogeneous Boltzmann equation started much later. It is only in 1963 when Grad constructed the first local solutions near the Maxwellian, and it is in 1974 when the first author of these notes constructed global solutions that are also near the Maxwellian, extending Grad's mathematical framework.The Vlasov equation of this paper is the Boltzmann equation in a special case of collision-free items.In this paper, we delicate to the model of the fluid dynamics, that is, the initial-boundary value problem to a class of the regularization for the Vlasov equations with non-Newtonian potential. That is, with initial and boundary condition:where f is the mass density function of gas particles having position x and velocity v at time t.Φ=Φ(x, t) the non-Newton gravitational potential,ε> 0. (x, v)∈(?)= I×(-L, L), I = (0,1), (?), T>0.x∈I,v∈(-L,L). Physically, this system describes the motion of compressible viscous isentropic gas flow under the gravitational force. The difficult of this type model is mainly that the equations are coupled with elliptic and parabolic. Also the degenerate of the elliptic equation, and so on. For the case of the parameter p∈(1,2), we consider its solution defined by:Definition 1 The (f,φ) is called a solution to the initial boundary value problem(1)-(2), if the following conditions are satisfied:(i)(ii) For all (?), for a.e.t∈(0,T), we have: (iii) For allφ∈L^∞(0,T;H₀¹(I)), for a.e.t∈(0,T), we have:We get the main result as following.定理1 Assume f₀, f₀≥0, f₀∈L^∞(?), for any fix T > 0. Then there exist aweak solution of the problem (1)-(2) in (?) [0, T], such that:定理2 Assume f₀, f₀≥0, (?), for any fix T > 0. 1 < p < 2.Assume(?) is a weak solution of the problem (1)-(2) in (?) [0, T], and satisfies (?).If (f,φ) is a weak solution of the problem (1)-(2)in (?)[0, T], then (?)=(f,φ) .In this chapter, we use the method of iterative to construct the approximate solution, and get the uniform estimate, then get the Theorem 1 and Theorem 2.So we first consider the following: (?), (3) with initial boundary value:(?), (4)whereδ> 0,ε> 0, 1 < p < 2, (?)(0,T), T > 0, (?),I=(0,1), x∈I, v∈(-L,L).For the singularity, we need to regularize it: Letφ^δ,0 = 0(?), (5) (?), (6)(?), (7)wherewe get the uniform estimates of solution for (5)-(7).ess For the uniform estimate, we can take limits with respect to k,δand get the Theorem 1. Finally, we obtain the proving of the Theorem 2.For the case of the parameter p∈(2, +∞), we consider the following initial boundary problem:(?). (8)with(?), (9)where f is the mass density function of gas particles having position x and velocity v at time t.Φ=Φ(x, t) the non-Newton gravitational potential,μ> 0 and (?) =we consider its solution defined by:Definition 2 The (f,φ) is called a solution to the initial boundary value problem(8)-(9), if the following conditions are satisfied:(i)(ii) For all (?), for a.e. t∈(0, T), we have: (iii) For allφ∈L^∞(0,T; H₀¹(I)), for a.e.t∈(0,T), we have:The main result as following:定理3 Assume f₀, f₀≥0, f₀∈L^∞(?), for any fix T > 0.p > 2,μ> 0. Then there exist a weak solution of the problem (8)-(9) in (?)[0, T], such that:定理4 Assume f₀, f₀≥0, (?), for any fix T > 0. p > 2,μ> 0. Assume(?) is a weak solution of the problem (8)-(9) in (?) [0. T], and satisfies (?). If (f,φ) is a weak solution of the problem (8)-(9) in (?)[0, T], then (?) =(f,φ).In this chapter, the proof by the same technique as in the case of the p∈(1.2). We use the method of iterative to construct the approximate solution, and get the uniform estimate, then get the Theorem 3 and Theorem 4.