依赖密度的Boussinesq方程组的局部强解的存在唯一性
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摘要
本文考虑非齐次不可压Boussinesq方程组的如下初边值问题．
其中Ω(?)R~3为边界光滑的有界开区域，未知向量函数u=u(x，t)表示流体速度，未知函数ρ=ρ(x, t),p = p(x, t),θ=θ(x,t)分别表示密度，压力与温度函数，f=f(x,t)为已知外力向量函数，ρ_0u_0=ρ_0u_0(x)，ρ_0 =ρ_0(x)分别表示初始动量与初始密度，系数μ:=μ(ρ.θ),κ:=κ(ρ,θ), C_v:= C_v(ρ,θ)分别为流体粘性系数，热传导系数和比热容．它们都是关于密度ρ和温度θ的正的函数．
本文主要研究问题(*)强解的局部存在唯一性，内容分为如下两部分：
1．考虑(*)的一个线性问题，证明了线性问题强解的存在唯一性，并得到了强解的一致估计，且这些估计不依赖于初始密度ρ_0的下界．
2．在线性问题强解的存在唯一性和一致估计的基础上，根据经典的迭代讨论，从而证明了问题(*)强解的局部存在唯一性．
In this paper ,We consider the following initial-boundary value problem of homogeneous incompressible Boussinesq equations.whereΩis an open bounded subset of R~3 with a smooth boundary.The unknown functions are u = u(x, t),ρ=ρ(x, t),p =ρ(x, t) andθ=θ(x,t).which represent the velocity field, the density, the pressure and the temperature of the flow, respectively. f= f(x,t) is the known external potential.ρ_0u_0=ρ_0u_0(x) is the initial momentum andρ_0 =ρ_0(x) is the initial density.μ:=μ(ρ.θ),κ:=κ(ρ,θ), C_v:= C_v(ρ,θ) is the viscosity coefficient. heat conductivity coefficient and specific heat at constant volume of the flow, respectively. They are all positive functions ofρandθ.
In this paper,we mainly study the local existence and uniqueness of the strong solutions of the problem (*). The contents of the paper include two parts:
First , we consider a linearized problem of (*). It is proved that the strong solution of the linearized problem exists uniquely. And it is obtained that the strong solutions satisfy some uniform estimates, which is independent of the lower bound of the initial density.
Second, we prove the existence and uniqueness of strong solutions of the problem (*) by applying the classical iteration argument , based on the above uniform estimates.
其中Ω(?)R~3为边界光滑的有界开区域，未知向量函数u=u(x，t)表示流体速度，未知函数ρ=ρ(x, t),p = p(x, t),θ=θ(x,t)分别表示密度，压力与温度函数，f=f(x,t)为已知外力向量函数，ρ_0u_0=ρ_0u_0(x)，ρ_0 =ρ_0(x)分别表示初始动量与初始密度，系数μ:=μ(ρ.θ),κ:=κ(ρ,θ), C_v:= C_v(ρ,θ)分别为流体粘性系数，热传导系数和比热容．它们都是关于密度ρ和温度θ的正的函数．
本文主要研究问题(*)强解的局部存在唯一性，内容分为如下两部分：
1．考虑(*)的一个线性问题，证明了线性问题强解的存在唯一性，并得到了强解的一致估计，且这些估计不依赖于初始密度ρ_0的下界．
2．在线性问题强解的存在唯一性和一致估计的基础上，根据经典的迭代讨论，从而证明了问题(*)强解的局部存在唯一性．
In this paper ,We consider the following initial-boundary value problem of homogeneous incompressible Boussinesq equations.whereΩis an open bounded subset of R~3 with a smooth boundary.The unknown functions are u = u(x, t),ρ=ρ(x, t),p =ρ(x, t) andθ=θ(x,t).which represent the velocity field, the density, the pressure and the temperature of the flow, respectively. f= f(x,t) is the known external potential.ρ_0u_0=ρ_0u_0(x) is the initial momentum andρ_0 =ρ_0(x) is the initial density.μ:=μ(ρ.θ),κ:=κ(ρ,θ), C_v:= C_v(ρ,θ) is the viscosity coefficient. heat conductivity coefficient and specific heat at constant volume of the flow, respectively. They are all positive functions ofρandθ.
In this paper,we mainly study the local existence and uniqueness of the strong solutions of the problem (*). The contents of the paper include two parts:
First , we consider a linearized problem of (*). It is proved that the strong solution of the linearized problem exists uniquely. And it is obtained that the strong solutions satisfy some uniform estimates, which is independent of the lower bound of the initial density.
Second, we prove the existence and uniqueness of strong solutions of the problem (*) by applying the classical iteration argument , based on the above uniform estimates.
引文
[1] R.A.Adams,Sobolev Spaces.Academic Press, 1978.
[2] D.Chae,H.-S.Nam, Local existence and blow-up criterion of Holder continuous solutions of the Boussinesq equations.Nogaya Math.J..Vol 155(1999),55-80.
[3] D.Chae,H.-S.Nam, Local existence and blow-up criterion for the Boussinesq equations,Proceedings of the Royal Society of Ediburgh,127A(1997),935-946.
[4]陈亚浙，吴兰成，《二阶椭圆型方程与椭圆型方程组》，科学出版社， 1997．
[5] P.Constantin, C.Foias,Navier-Stokes equations, The University of Chicago Press, 1988.
[6] Y.Cho and H.Kim ,Existence result for heat-conducting viscous incompressiblefluids with vacuum, Preprint 2005.
[7] G.Duvaut. J.L.Lions, Inequalities in Mechanics and Physics. Springer-Verlag, Berlin-Heidelberg-New York,1976.
[8] E.Feiresl, Dynamics of Compressible Fluids. Oxford University. 2004.
[9] Jishan Fan and Wanghui Yu ,Strong solution to the incompressible magnetohydrodynamicequations with vacuum,2007.
[10] N. Ishimura, H. Morimoto, Remarks on the blow-up criterion for the 3-D Boussinesqequations,mathematical Model's and Methods in Applied Sciences,Vol9,No9(1999), 1323-1332.
[11] Antontsev,Kazhikov,Monakhov.Boundarv Value Problems in Mechanics of NonhomogeneousFluids,North Holland,Amsterdam. 1990.
[12] U.J.Kim,Weak solutions of an initial boundary value problem for an incompressibleviscous fluid with nonnegative density.SIAM J.,Math.Anal..18 (1987).pp.89-96.
[13] Y.Kagei, On weak solutions of nonstationary Boussinesq equations,Differential and Integral Equations, 6(1993),587-611.
[14]李大潜，陈韵梅，非线性发展方程，科学出版社，1997．
[15] J.L.Lions,On some problems connected with Navier-Stokes equations in nonliner evolution equations, ,Academic Press.1978.
[16] P.-L.Lions, Mathematical topic in fluid dynamics, vol2, compressible models, OxfordScience Publication, Oxford, 1998.
[17]李明杰；Boussinesq方程组解的正则性问题研究,首都师范大学；2006年
[18] J.Simon,Compact sets in the spaceL~p(0,T;B),Ann.Mat.pura Appl.(4),146(1987), pp.65-96.
[19] J.Simon,Nonhomogeneous viscous incompressible fluids:existence of velocity,density, and pressure,SIAM J.Math.Anal, 21(5),(1990),pp.1093-1117.
[20] R . Temam , Navier-Stokes equations theory and numerical Analysis , North-Holland-Amsterdam New York Oxford,1984.
[21] C.-X.Miao, Harmomc Analysis and Its Applications in Partial Differential Equations,Science Press, 2nd ed, 2004.
[2] D.Chae,H.-S.Nam, Local existence and blow-up criterion of Holder continuous solutions of the Boussinesq equations.Nogaya Math.J..Vol 155(1999),55-80.
[3] D.Chae,H.-S.Nam, Local existence and blow-up criterion for the Boussinesq equations,Proceedings of the Royal Society of Ediburgh,127A(1997),935-946.
[4]陈亚浙，吴兰成，《二阶椭圆型方程与椭圆型方程组》，科学出版社， 1997．
[5] P.Constantin, C.Foias,Navier-Stokes equations, The University of Chicago Press, 1988.
[6] Y.Cho and H.Kim ,Existence result for heat-conducting viscous incompressiblefluids with vacuum, Preprint 2005.
[7] G.Duvaut. J.L.Lions, Inequalities in Mechanics and Physics. Springer-Verlag, Berlin-Heidelberg-New York,1976.
[8] E.Feiresl, Dynamics of Compressible Fluids. Oxford University. 2004.
[9] Jishan Fan and Wanghui Yu ,Strong solution to the incompressible magnetohydrodynamicequations with vacuum,2007.
[10] N. Ishimura, H. Morimoto, Remarks on the blow-up criterion for the 3-D Boussinesqequations,mathematical Model's and Methods in Applied Sciences,Vol9,No9(1999), 1323-1332.
[11] Antontsev,Kazhikov,Monakhov.Boundarv Value Problems in Mechanics of NonhomogeneousFluids,North Holland,Amsterdam. 1990.
[12] U.J.Kim,Weak solutions of an initial boundary value problem for an incompressibleviscous fluid with nonnegative density.SIAM J.,Math.Anal..18 (1987).pp.89-96.
[13] Y.Kagei, On weak solutions of nonstationary Boussinesq equations,Differential and Integral Equations, 6(1993),587-611.
[14]李大潜，陈韵梅，非线性发展方程，科学出版社，1997．
[15] J.L.Lions,On some problems connected with Navier-Stokes equations in nonliner evolution equations, ,Academic Press.1978.
[16] P.-L.Lions, Mathematical topic in fluid dynamics, vol2, compressible models, OxfordScience Publication, Oxford, 1998.
[17]李明杰；Boussinesq方程组解的正则性问题研究,首都师范大学；2006年
[18] J.Simon,Compact sets in the spaceL~p(0,T;B),Ann.Mat.pura Appl.(4),146(1987), pp.65-96.
[19] J.Simon,Nonhomogeneous viscous incompressible fluids:existence of velocity,density, and pressure,SIAM J.Math.Anal, 21(5),(1990),pp.1093-1117.
[20] R . Temam , Navier-Stokes equations theory and numerical Analysis , North-Holland-Amsterdam New York Oxford,1984.
[21] C.-X.Miao, Harmomc Analysis and Its Applications in Partial Differential Equations,Science Press, 2nd ed, 2004.