一类带有非牛顿位势的正则化非线性Vlasov方程初边值问题
摘要
本文主要研究一类带有非牛顿位势的正则化非线性的Vlasov方程(?)atf+v(?)xf+(?)xΦ(?)vf-εdiv((|▽f|2+μ0)q-2/2▽f)=-(|(?)xΦ|p-2(?)xΦ)x=∫-LLf(x,v,t)dv,探讨了这类方程满足下列条件的初边值问题其中f是在时刻t,在(x,v)处单位体积及单位速度变化范围中的粒子分布函数.φ为非牛顿重力位势.q>2,ε>0,μ0>0,(?)=(?)×(0,T),T>0,(x,v)∈(?)=I×(-L,L),I=(0,1),x∈I,v∈(-L,L).本文重点研究上述问题弱解的存在性.
The kinetic theory of gas dynamics is also called as compressible fluid dynamics. It is a fluid mechanics that mainly studies the variability of fluid density or fluid compressibility which plays a significant role.
In macroscopic scales,fluid is considered as a whole,and its motion can be described by some macroscopic quantities such as density,velocity,temperature,pressure,etc.In this re-spect,the Euler and Navier-Stokes equations,compressible or incompressible,are very famous.
In microscopic scales,fluid is considered as a many-body system of microscopic parti-cles.So the motion of the system is governed by the coupled Newton equations within the framework of the classieal mechanies. Though the Newton equation is the first principle of the classieal mehanies.it is not of practically used because the number of the equations is too big.
From the cognitive perspective,we wish that be a way which could show the macroseopic scales and microscopic scales at the same time. And Boltzmann equation can reflect this feature. Thus,we could say Boltzmann equation is between the macroseopic scales and mi-croscopic scales. Actually,the Boltzmann equation is a convergence between the Euler and Navier-Stokes equations.
The first proof of the existence of Boltzmann equation can be traced back to the year 1932 when Carleman proved the existence of global solutions to the Cauchy problem for the spatially homogeneous case. And it's 2 years earlier 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 Boltzman equation began much later.It was only in the year 1963 when Grad constructed the first local solutions near the Maxwellian,and it was in the year 1974 when the first author of these notes constructed global solutions which were also near the Maxwellian, extending Grad's mathematieal 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,｛(?)tf+υ(?)xf+(?)xΦ(?)υf-εdiv((|▽f|2+-(|(?)xΦ|p-2(?)xΦ)x=∫-LLf(x,υ,t)dυwith initial and boundary condition: where f is the mass density funcation of gas particles having position x and velocity v at time t.Φ=Φ(x, t) the non-Newton gravitational potential,ε> 0,μ0>0, q>2,(?)=(?)×(0, T), T> 0, (x, v)∈(?)=I×(-L,L),I=(0,1), x∈I,υ∈(-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∈(3/2,2),q∈(2,+∞),we consider its solution defined by:
Definition 0.1 The(f,Φ)is called a solution to the initial boundary valude problem (0.1)一(0.2),if the following conditions are satisfied:
(i) f∈Lq(0,T;W01,q((?)))∩L∞(0,T;W01,q((?))), ft∈L2(0,T;L2((?))),Φ∈L∞(0,T;H2(I)∩H14(I)).
(ⅱ)For allφ∈Lq(0,T;W01,q((?))),q>2,μ0>0,φt∈L2(0,T；L2((?))),for a.e.t∈(0,T), we have:∫(?)f(x,υ,t)φ(x,υ,t)dxdυ-∫0t∫(?)(fφt+υfφx+fΦxφυ-ε(|▽f|2+μ0)q-2/2▽f▽φ)dxdvds∫(?)f0φ(x,υ,0)dxdυ
(ⅲ)For allψ∈L∞(0,T;H01(I)),for a.e.t∈(0,T),we have:∫01|Φx|p-2Φxψxdx=∫(?) We get the main result as following:
Theorem 0.1 Assume f0≥0,f0∈L∞(0,T;W1,q((?)))for any fix T>0.3/2 2,μ>0,μ0>0,Then there exist a weak solution of the problem(0.1)一(0.2)in (?)×[0,T], such that: f∈Lq(0,T;W01,q((?)))∩L∞(0,T;W01,q((?))), diυ((|▽f|2+μ0)q-2/2▽f)∈L2(0,T;L2((?))),Φ∈L∞(0,T;H2(I)∩H01(I)).
Theorem 0.2 Assume f0≥0,f0∈L∞(0,T;W1,q((?))),for any fix T>0.3/2 2. Assume((?))is a weak solution of the problem(0.1)一(0.2)in (?)×[0,T],and satisfies (?)∈L∞(0,T;H2((?))).If(f,Φ)is a weak s.lution of the problem(0.1)一(0.2)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 0.1 and Theorem 0.2.
So we first consider the following: with initial boundary value: whereδ>0,μ0>0,ε>0,3/22,(?)=(?)×(0,T),T>0,(x,υ)∈(?) I×(-L,L),I=(0,1),x∈I,υ∈(-L,L).
For the singularity,we need to regularize it:LetΦδ,0=0, ftδ,k+υfxδ,k+[Jδ*(ξδΦδ,k,k-1)]xfυδ,k-εdiυ((|▽fδ,k|2+μ0)q-2/2▽fδ,k)=0,(0.5) Lpδδ,k=∫-LLfδ,k(x,υ,t)dυ, (0.6)-[(δ(Φxδ,k2+1)/(Φxδ,k)2+δ)2-p/2Φxδ,k]x we get the uniform estimates of the solution for(0.5)一(0.7). sup0≤t≤T(∥Φδ(t)∥H2(I)+∥fδ(t)∥W1,q(?))≤C For the uniform estimate,we can take limits with respect toκ,δand get the Theorem 0.1. Finally,we obtain the proving of the Theorem 0.2.
For the case of the parameter p∈(2,+∞),q∈(2,+∞),we c.nsider the following initial boundary problem: with where f is the mass density function of gas particles having position x and velosityυat time t.Φ=Φ(x,t)the non-Newton gravitational potential,μ0>0,p>0,ε>0,p>2,q>2,(?)= (?)×(0,T),T>0,(x,υ)∈(?)=I×(L,L),I=(0,1),x∈I,υ∈(一L,L).
We consider its solution defined by:
Definition 0.2 The(f,Φ)is called a solution to the initial boundary valude problem (0.8)一(0.9),if the following conditions are satisfied:
(i) f∈Lq(0,T;W01,q((?)))∩L∞(0,T;W01,q((?))), ft∈L2(0,T;L2((?))),Φ∈L∞(0,T;H2(I)∩H01(I)).
(ii)For allφ∈Lq(0,T;W01,q((?))),q>2,μ0>0,φt∈L2(0,T;L2((?))),for a.e.t∈(0,T), we have:∫(?)f(x,υ,t)φ(x,υ,t)dxdυ-∫0t∫(?)(fφt+υfφx+fΦxφυ-ε(|▽f|2+μ0)q-2/2▽f▽φ)dxdυds
∫(?)f0φ(d,υ,0)dxdυ
(iii)For allψ∈L∞(0,T;H01(I)),for a.e.t∈(0,T),we have:∫01(|Φx|2+μ)p-2/2Φxψxdx=∫(?)fψdxdv
The main result as following:
Theorem 0.3 Assume f0≥0,f0∈L∞(0,T;W1,q((?)))for any fix T>0.p>2,q> 2,μ>0,μ0>0.Then there exist a weak solution of the problem(0.8)-(0.9)in(?)×[0,T], such that: f∈Lq(0,T;W01,q((?)))∩L∞(0,T;W01,q((?))), diυ((|▽f|2+μ0)q-2/2▽f)∈L2(0,T;L2((?))),Φ∈L∞(0,T;H2(I)∩H01(I)).
Theorem 0.4 Assume f0≥0,f0∈L∞(0,T;W1,q((?))),for any fix T>0.p>2,q> 2,μ>0,μ0>0.Assume(f,Φ)is a weak solution of the problem(0.8)一(0.9)in(?)×[0,T], and satisfies f∈L∞(0,T;H2((?))).If(f,Φ)is a weak solution of the problem(0.8)一(0.9)in (?)×[0,T],then(f,Φ)=(f,Φ).
In this chapter,the proof by the same technique as in the case of the p∈(3/2,2),q∈(2,+∞).We use the method of iterative to construct the approximate solution,and get the uniform estimate,then get the Theorem 0.3 and Theorem 0.4.
The kinetic theory of gas dynamics is also called as compressible fluid dynamics. It is a fluid mechanics that mainly studies the variability of fluid density or fluid compressibility which plays a significant role.
In macroscopic scales,fluid is considered as a whole,and its motion can be described by some macroscopic quantities such as density,velocity,temperature,pressure,etc.In this re-spect,the Euler and Navier-Stokes equations,compressible or incompressible,are very famous.
In microscopic scales,fluid is considered as a many-body system of microscopic parti-cles.So the motion of the system is governed by the coupled Newton equations within the framework of the classieal mechanies. Though the Newton equation is the first principle of the classieal mehanies.it is not of practically used because the number of the equations is too big.
From the cognitive perspective,we wish that be a way which could show the macroseopic scales and microscopic scales at the same time. And Boltzmann equation can reflect this feature. Thus,we could say Boltzmann equation is between the macroseopic scales and mi-croscopic scales. Actually,the Boltzmann equation is a convergence between the Euler and Navier-Stokes equations.
The first proof of the existence of Boltzmann equation can be traced back to the year 1932 when Carleman proved the existence of global solutions to the Cauchy problem for the spatially homogeneous case. And it's 2 years earlier 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 Boltzman equation began much later.It was only in the year 1963 when Grad constructed the first local solutions near the Maxwellian,and it was in the year 1974 when the first author of these notes constructed global solutions which were also near the Maxwellian, extending Grad's mathematieal 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,｛(?)tf+υ(?)xf+(?)xΦ(?)υf-εdiv((|▽f|2+-(|(?)xΦ|p-2(?)xΦ)x=∫-LLf(x,υ,t)dυwith initial and boundary condition: where f is the mass density funcation of gas particles having position x and velocity v at time t.Φ=Φ(x, t) the non-Newton gravitational potential,ε> 0,μ0>0, q>2,(?)=(?)×(0, T), T> 0, (x, v)∈(?)=I×(-L,L),I=(0,1), x∈I,υ∈(-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∈(3/2,2),q∈(2,+∞),we consider its solution defined by:
Definition 0.1 The(f,Φ)is called a solution to the initial boundary valude problem (0.1)一(0.2),if the following conditions are satisfied:
(i) f∈Lq(0,T;W01,q((?)))∩L∞(0,T;W01,q((?))), ft∈L2(0,T;L2((?))),Φ∈L∞(0,T;H2(I)∩H14(I)).
(ⅱ)For allφ∈Lq(0,T;W01,q((?))),q>2,μ0>0,φt∈L2(0,T；L2((?))),for a.e.t∈(0,T), we have:∫(?)f(x,υ,t)φ(x,υ,t)dxdυ-∫0t∫(?)(fφt+υfφx+fΦxφυ-ε(|▽f|2+μ0)q-2/2▽f▽φ)dxdvds∫(?)f0φ(x,υ,0)dxdυ
(ⅲ)For allψ∈L∞(0,T;H01(I)),for a.e.t∈(0,T),we have:∫01|Φx|p-2Φxψxdx=∫(?) We get the main result as following:
Theorem 0.1 Assume f0≥0,f0∈L∞(0,T;W1,q((?)))for any fix T>0.3/2
Theorem 0.2 Assume f0≥0,f0∈L∞(0,T;W1,q((?))),for any fix T>0.3/2
In this chapter,we use the method of iterative to construct the approximate Solution, and get the uniform estimate,then get the Theorem 0.1 and Theorem 0.2.
So we first consider the following: with initial boundary value: whereδ>0,μ0>0,ε>0,3/2
For the singularity,we need to regularize it:LetΦδ,0=0, ftδ,k+υfxδ,k+[Jδ*(ξδΦδ,k,k-1)]xfυδ,k-εdiυ((|▽fδ,k|2+μ0)q-2/2▽fδ,k)=0,(0.5) Lpδδ,k=∫-LLfδ,k(x,υ,t)dυ, (0.6)-[(δ(Φxδ,k2+1)/(Φxδ,k)2+δ)2-p/2Φxδ,k]x we get the uniform estimates of the solution for(0.5)一(0.7). sup0≤t≤T(∥Φδ(t)∥H2(I)+∥fδ(t)∥W1,q(?))≤C For the uniform estimate,we can take limits with respect toκ,δand get the Theorem 0.1. Finally,we obtain the proving of the Theorem 0.2.
For the case of the parameter p∈(2,+∞),q∈(2,+∞),we c.nsider the following initial boundary problem: with where f is the mass density function of gas particles having position x and velosityυat time t.Φ=Φ(x,t)the non-Newton gravitational potential,μ0>0,p>0,ε>0,p>2,q>2,(?)= (?)×(0,T),T>0,(x,υ)∈(?)=I×(L,L),I=(0,1),x∈I,υ∈(一L,L).
We consider its solution defined by:
Definition 0.2 The(f,Φ)is called a solution to the initial boundary valude problem (0.8)一(0.9),if the following conditions are satisfied:
(i) f∈Lq(0,T;W01,q((?)))∩L∞(0,T;W01,q((?))), ft∈L2(0,T;L2((?))),Φ∈L∞(0,T;H2(I)∩H01(I)).
(ii)For allφ∈Lq(0,T;W01,q((?))),q>2,μ0>0,φt∈L2(0,T;L2((?))),for a.e.t∈(0,T), we have:∫(?)f(x,υ,t)φ(x,υ,t)dxdυ-∫0t∫(?)(fφt+υfφx+fΦxφυ-ε(|▽f|2+μ0)q-2/2▽f▽φ)dxdυds
∫(?)f0φ(d,υ,0)dxdυ
(iii)For allψ∈L∞(0,T;H01(I)),for a.e.t∈(0,T),we have:∫01(|Φx|2+μ)p-2/2Φxψxdx=∫(?)fψdxdv
The main result as following:
Theorem 0.3 Assume f0≥0,f0∈L∞(0,T;W1,q((?)))for any fix T>0.p>2,q> 2,μ>0,μ0>0.Then there exist a weak solution of the problem(0.8)-(0.9)in(?)×[0,T], such that: f∈Lq(0,T;W01,q((?)))∩L∞(0,T;W01,q((?))), diυ((|▽f|2+μ0)q-2/2▽f)∈L2(0,T;L2((?))),Φ∈L∞(0,T;H2(I)∩H01(I)).
Theorem 0.4 Assume f0≥0,f0∈L∞(0,T;W1,q((?))),for any fix T>0.p>2,q> 2,μ>0,μ0>0.Assume(f,Φ)is a weak solution of the problem(0.8)一(0.9)in(?)×[0,T], and satisfies f∈L∞(0,T;H2((?))).If(f,Φ)is a weak solution of the problem(0.8)一(0.9)in (?)×[0,T],then(f,Φ)=(f,Φ).
In this chapter,the proof by the same technique as in the case of the p∈(3/2,2),q∈(2,+∞).We use the method of iterative to construct the approximate solution,and get the uniform estimate,then get the Theorem 0.3 and Theorem 0.4.
引文
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