关于不可压Navier-Stokes方程和相关问题的一些研究
|
摘要
本文由四部分组成:
第一部分研究带有阻尼项α|u|~(β-1)u(α＞0)的不可压Navier-Stokes方程解的存在性和唯一性。利用Galerkin方法,我们得到了带有阻尼项不可压Navier-Stokes方程Cauchy问题当β≥1时,存在全局的弱解;当β≥7/2时,存在全局的强解,并且当7/2≤β≤5时强解是唯一的。
第二部分研究的是带有阻尼项α|u|~(β-1)u(α＞0)的不可压Navier-Stokes方程Cauchy问题弱解的大时间性态。利用Fourier分解方法,我们得到了解的衰减率为-3/2,这与经典的不可压Navier-Stokes方程的衰减率是一样的。
第三部分对三维有界及无界区域上的非齐次不可压Navier-Stokes方程解的全局存在性和全局稳定性进行了讨论。利用细致的能量估计,在解满足适当的条件下,对初始数据加上更正则的条件和相容性条件,如果给初值一个小扰动,我们证明了解是整体存在的(0,∞),而且此解是全局稳定的。
第四部分对二维Boussinesq方程进行了一些讨论。利用Schauder不动点定理,我们得到了当初始旋度光滑时,二维Boussinesq方程存在唯一的古典解。但当初始旋度属于L~1(R~2)或者是Radon测度空间时,是否存在整体的解还未证明,进一步的研究正在进行中。
This thesis is composed of four parts.
In the first part,the existence and uniqueness of the incompressible Navier-Stokes equations with damping are studied.By the Galerkin method,we show that the Cauchy problem of the Navier-Stokes equations with dampingα｜u|~(β-1)u (α＞0) has global weak solutions for anyβ≥1,global strong solutions for anyβ≥7/2 and that the strong solution is unique for any 7/2≤β≤5.
In the second part,we investigate the large time behavior of weak solutions for the Cauchy problem of the Navier-Stokes equations with dampingα｜u｜~(β-1)u(α＞0,β≥3).By the Fourier splitting method,we show that the decay rate of the solutions is -3/2,which is the same as the classical incompressible Navier-Stokes equations.
In the third part,we mainly study the global existence and stability of the solutions for the nonhomogeneous incompressible Navier-Stokes equations in three-dimensional bounded or unbounded domains.By delicate energy estimates,under some more regular conditions and some compatibility condition on the initial data,we obtain that if the initial data are small perturbation on those of a known strong solution,then there exists a global solution,which defined on(0,∞) for the nonhomogeneous incompressible Navier-Stokes equations and is a perturbation of the known one.The strong solution of the nonhomogeneous incompressible Navier-Stokes equations is stable.
In the fourth part,we give some remarks on the two dimensional Boussinesq equations.By the Schauder fixed points,we get that when the initial vorticity is smooth,there exists a unique classical solutions for the Cauchy problem of the two dimensional Boussinesq equations.When the initial vorticity belongs to
L~1(R~2)(or the Radon measure space),whether there exists global existence of the solutions is still open.It will be investigated in future.
第一部分研究带有阻尼项α|u|~(β-1)u(α＞0)的不可压Navier-Stokes方程解的存在性和唯一性。利用Galerkin方法,我们得到了带有阻尼项不可压Navier-Stokes方程Cauchy问题当β≥1时,存在全局的弱解;当β≥7/2时,存在全局的强解,并且当7/2≤β≤5时强解是唯一的。
第二部分研究的是带有阻尼项α|u|~(β-1)u(α＞0)的不可压Navier-Stokes方程Cauchy问题弱解的大时间性态。利用Fourier分解方法,我们得到了解的衰减率为-3/2,这与经典的不可压Navier-Stokes方程的衰减率是一样的。
第三部分对三维有界及无界区域上的非齐次不可压Navier-Stokes方程解的全局存在性和全局稳定性进行了讨论。利用细致的能量估计,在解满足适当的条件下,对初始数据加上更正则的条件和相容性条件,如果给初值一个小扰动,我们证明了解是整体存在的(0,∞),而且此解是全局稳定的。
第四部分对二维Boussinesq方程进行了一些讨论。利用Schauder不动点定理,我们得到了当初始旋度光滑时,二维Boussinesq方程存在唯一的古典解。但当初始旋度属于L~1(R~2)或者是Radon测度空间时,是否存在整体的解还未证明,进一步的研究正在进行中。
This thesis is composed of four parts.
In the first part,the existence and uniqueness of the incompressible Navier-Stokes equations with damping are studied.By the Galerkin method,we show that the Cauchy problem of the Navier-Stokes equations with dampingα｜u|~(β-1)u (α＞0) has global weak solutions for anyβ≥1,global strong solutions for anyβ≥7/2 and that the strong solution is unique for any 7/2≤β≤5.
In the second part,we investigate the large time behavior of weak solutions for the Cauchy problem of the Navier-Stokes equations with dampingα｜u｜~(β-1)u(α＞0,β≥3).By the Fourier splitting method,we show that the decay rate of the solutions is -3/2,which is the same as the classical incompressible Navier-Stokes equations.
In the third part,we mainly study the global existence and stability of the solutions for the nonhomogeneous incompressible Navier-Stokes equations in three-dimensional bounded or unbounded domains.By delicate energy estimates,under some more regular conditions and some compatibility condition on the initial data,we obtain that if the initial data are small perturbation on those of a known strong solution,then there exists a global solution,which defined on(0,∞) for the nonhomogeneous incompressible Navier-Stokes equations and is a perturbation of the known one.The strong solution of the nonhomogeneous incompressible Navier-Stokes equations is stable.
In the fourth part,we give some remarks on the two dimensional Boussinesq equations.By the Schauder fixed points,we get that when the initial vorticity is smooth,there exists a unique classical solutions for the Cauchy problem of the two dimensional Boussinesq equations.When the initial vorticity belongs to
L~1(R~2)(or the Radon measure space),whether there exists global existence of the solutions is still open.It will be investigated in future.
引文
[1] S.A.Antontsev, A.V.Kazhikov, V.N. Monakhov, Boundary Value Problem in Mechanics of Nonhomogeneous Fluids, Norht-Holland, Amsterdam, 1990.
[2] G.K.Batchelor, An introduction to fluid dynamics. Cambridge University Press, Cambridge, 1967.
[3] H.O.Bae, H. J.Choe, Decay rate for the incompressible flows in half spaces. Math Z, 238 (2001), 799-816.
[4] H.Beirao da Veiga, A new regularity class for the Navier-Stokes equations in R~n, Chin. Ann. of Math., 16 B: 4 (1995), 407-412.
[5] M.Ben-Artzi, Planar Navier-Stokes equations vorticity approach.
[6] M.Ben-Artzi, Global solutions of two-dimensional Navier-Stokes and Euler equations, Arch. Rat. Mech. Anal. 128 (1994), 329-358.
[7] D.Bresch and B.Desjardins, Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model, Comm. Math. Phys., 238 (1-2) (2003), 211-223.
[8] D.Bresch, B.Desjardins, Chi-Kun Lin, On some compressible fluid models: Korteweg, lubrication, and shallow water systems, Comm. Partial Differential Equations, 28 (3-4) (2003), 843-868.
[9] W.Borchers, T.Miyakawa, L~2 decay rate for the Navier-Stokes flow in half spaces. Math Ann,282 (1988), 139-155.
[10] L.Caffarelli, R.Kohn, L.Nirenberg, Partial regularity of suitably weak solutions of the Navier-Stokes equations. Comm on Pure and Applied Math, 25 (1982), 771-831.
[11] X.J.Cai, Q.S.Jiu, Weak and strong solutions for the incompressible Navier-Stokes equations with damping term. J. math. Anal. Appl, In press, Available online 24 January, 2008.
[12] D.Chae, H.-S.Nam, Local existence and blow-up criterion for the Boussinesq equations, , Vol 155 (1999), 55-80.
[13] D.Chae, S.Kim and H.-S. Nam, Local existence and blow-up criterion of Holder continuous solutions of the Boussinesq equations, Nogaga Math. J., Vol 155 (1999), 55-80, .
[14] H.J.Choe, H. Kim, Strong Solutions of the Navier-Stokes Equations for Nonhomogeneous Incompressible Fluids, Comm. Partial Differentiao Equations 28 (2003),1183-1201.
[15] Y.Cho, H.Kim, Unique solvability for the density-dependent Navier-Stokes equations, Nonlinear Analysis 59 (2004),465-489.
[16] L.Escauriaza, G,Seregin, V.Sverak, On L_(3,∞)-solutions to the Navier-Stokes equations and backward uniqueness, Russian Math. Surveys, 58 (2003), 211-250.
[17] C.Foias, Une remarque sur l'unicite des solutions des equations de Navier-Stokes en dimension n, Bull. Soc. Math. France, 89 (1961), 1-8.
[18] C.Foias, C.Guillope, and R. Temam, New a priori estimates for Navier-Stokes equations in dimension 3, Comm. Partial Diff Eqs. 6 (1981), 329-359.
[19] G.P.Galdi, An introduction to the Navier-Stokes initial-boundary value problem, in " Fundemental directions in Mathematical Fluid Mechanics" editen by G.P. Galdi, John G. Heywood and Rolf Rannacher. Birkauser Verlag, Basel. Boston, Berlin.
[20] G.P.Galdi, An introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. Linearized Steady Problems. Springer-Verlag, New York, Berlin etc. 1994.
[21] Y.Giga, Solutions for semilinear parabolic equations in L~p and regularity of weak solutions of the Navier-Stokes system, J. Diff. Eqs., 61 (1986), 186-212.
[22] C.He, T.Miyakawa, On weighted norm estimates for solutions to nonsta-tionary Navier-Stokes equations in an exterior domain, preprint.
[23] E.Hopf, Uber die Anfangswertaufgabe fur die hydrody namischen Grund-gleichungen, Math. Nachr. 4(1951), 213-231.
[24] C.He, The Cauchy problem for the Navier-Stokes equations. J Math Anal App, 209 (1997), 228-242.
[25] C.He, L.Z.Wang, Weighted L~p-estimates for Stokes flow in R_+~n(n ≥ 2), with applications to the Non-stationary Navier-Stokes flow, preprint.
[26] J.G.Heywood, The Navier-Stokes equations: On the existence, regularity and decay of solutions. Indiana University Math. J., 29 (1980), 641-681.
[27] L.Hsiao, Quasilinear hyperbolic systems and dissipative mechanisms. World Scientific 1997.
[28] F.M.Huang, R.H.Pan, Convergence rate for compressible Euler equations with damping and vacuum, Arch. Rational Mech. Anal. 166 (2003), 359-376.
[29] Y.Kagei, On weak solutions of nonstationary Boussinesq equations, Diff and Integral Equ, 9, (1993), 587-611.
[30] R.Kajikiya, T.Miyakawa, On L~2 decay of weak solutions of the Navier-Stokes equations in R~n. Math. Z., 192 (1986), 136-148.
[31] T.Kato, Strong L~p solutions of the Navier-Stokes equations in R~n, with application to weak solutions, Math. Z. 187(1984), 471-480.
[32] A.V.Kazhikov, Resolution of boundary Value problems for nonhomogeneous viscous fluids, Dokl. Akarl. Nauh.,216 (1974), 1008-1010(in Russian).
[33] O.A.Ladyzhenskaya, The mathematical theory of viscous incompressible flow. Gordon and Breach, New York, 1969.
[34] O.Ladyzhenskaya, V.A.Solonnikov, Unique Solvability of an initial and boundary value problem foe viscous incompressible nonhomogeneous fluids, J. Soviet. Math. 9 (1978), 697-749.
[35] L.Laudau and E.Lifschitz, Mecanique des fluides. 2nd edition, Editions MIR, Moscow, 1989.
[36] J.Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math. 63 (1934), 193-248.
[37] X.J.Li, Q.S.Jiu, Global L~2 stability of large solutions to the 3D Boussinesq Equations, preprint (in Chinese).
[38] P.L.Lions, Mathematical topics in fluid mechanics, Vol.1: incompressible models, in Oxford Lecture Series in Mathematics and Its Applications, Vol. 10, 1996.
[39] P.Maremonti, On the asymptotic behavior of the L~2-norm of suitable weak solutions to the Navier-Stokes equations in three-dimensional exterior domains. Math. Phys., 118 (1988), 405-420.
[40] K.Masuda, L~2 decay of solutions of the Navier-Stokes equations in the exterior domains, in " Proceedings of Symposia in Pure Mathematics". 45 (1986), Part 2, 179-182. American Mathematical Society, 1986.
[41] K.Masuda, Weak solutions of the Navier-Stokes equations, Tohoku Math. J., 36 (1984), 623-646.
[42] F.J.McGrath, Nonstationary plane flow of viscous and ideal fluids, Arch. Rat. Mech. Anal. 27 (1968), 328-348.
[43] C.L.Navier. Memoire sur les lois du mouvement des fluids. Mem. Acad. Sci. de France., 6 (1827), 389-440.
[44] M.Padula, An existence theorem for nonhomogeneous in compressible fluids, Rend. Circ. Mat, Palemo, 31. 4 (1982).
[45] G.Ponce, R.Rache, T.C.Sideris, E.S.Titi, Global stability of large solutions to the 3D Navier-Stokes equations, Comm.Math. Phys., 159 (1994), 329-341.
[46] M.E.Schonbek, L~2 decay for weak solutions of the Navier-Stokes equations. Arch. Rational. Mech. Anal., 88 (1985), 209-222.
[47] M.E.Schonbek, Large time behavior of solutions to the Navier-Stokes equations. Comm. Partial Diff. equations., 11 (1986), 733-763.
[48] M.E.Schonbek, Lower bounds of rates of decay for solutions to the Navier-Stokes equations. J. Amer. Math. Soc, 4 (1991), 423-449.
[49] M.E.Schonbek, Asymptotic behavior of solutions to the three-dimensional Navier-Stokes equations. Indiana Univ. Math. J., 41(3) (1992), 809-823.
[50] T.Nagasawa, A new energy inequality and partial regularity foe weak solutions of Navier-Stokes equations, J.Math. Fluid Mech., 3 (2001), 40-56.
[51] R.Salvi, The equations of viscous incompressible nonhomogeneous fluids: on the existence and regularity, J. Austral. Math. Soc. Ser. B 33 (1991).
[52] J.Serrin, The ininial value problem for the Navier-Stokes equations, Univ. Wisconsin Press, Nonlinear Problem, Ed. R. langer, 1963. J.Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal. 9 (1962), 187-195.
[54] J.Simon, On the existence fo the pressure for solutions of variational Navier-Stokes equations, J.Math. Fluid Mech., 1 (1999), 225-234.
[55] J.Simon, Ecoulementd'un fluide nonhomogene avec une densite initiale s'annulant. CR. Acad. Sci. Paris, 15 (1978), 1009-1012.
[56] J.Simon, Nonhomogeneous viscous incompressible fluids: existence of velocity, density and pressure, SIAM J. Math. Anal., 21(5) (1990), 1093-1117.
[57] J.Simon, Sur les fluids visqueux incompressibles et non homogenes. CR. Acad. Sci. Paris, 309 (1989), 447-452.
[58] H.Sohr, The Navier-Stokes Equations: an elementary functional analytic approach, Birkhauser Verlag, 2001.
[59] G.G.Stokes. On the Theories of the Internal Friction of Fluids in Motion. Trans. Cambridge Phil. Soc, 8 (1845), 287-319.
[60] M.Struwe, On partial regularity results for the Navier-Stokes equa-tions,Comm. Pure. Apll. Math., 41 (1988), 437-458.
[61] R.Temam, Navier-Stokes equations theory and numerical Analysis, North-Holland-Amsterdam, New York, 1984.
[62] S.Ukai, A solution formular for the Navier-Stokes equations in R~n. Comm. Pure. Appl. Math., XL (1987), 611-621.
[63] M.Weigner, Decay results for weak solutions of the Navier-Stokes equations in R~n. J. London. Math. Soc, 35 (1987), 303-313.
[64] C.S.Zhao, K.T.Li,Global L~2 stability of large solutions to the 3D MHD Equations, Acta. Math. Sinica, Vol.44, No.6, (2001), 961-976 (in Chinese).
[2] G.K.Batchelor, An introduction to fluid dynamics. Cambridge University Press, Cambridge, 1967.
[3] H.O.Bae, H. J.Choe, Decay rate for the incompressible flows in half spaces. Math Z, 238 (2001), 799-816.
[4] H.Beirao da Veiga, A new regularity class for the Navier-Stokes equations in R~n, Chin. Ann. of Math., 16 B: 4 (1995), 407-412.
[5] M.Ben-Artzi, Planar Navier-Stokes equations vorticity approach.
[6] M.Ben-Artzi, Global solutions of two-dimensional Navier-Stokes and Euler equations, Arch. Rat. Mech. Anal. 128 (1994), 329-358.
[7] D.Bresch and B.Desjardins, Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model, Comm. Math. Phys., 238 (1-2) (2003), 211-223.
[8] D.Bresch, B.Desjardins, Chi-Kun Lin, On some compressible fluid models: Korteweg, lubrication, and shallow water systems, Comm. Partial Differential Equations, 28 (3-4) (2003), 843-868.
[9] W.Borchers, T.Miyakawa, L~2 decay rate for the Navier-Stokes flow in half spaces. Math Ann,282 (1988), 139-155.
[10] L.Caffarelli, R.Kohn, L.Nirenberg, Partial regularity of suitably weak solutions of the Navier-Stokes equations. Comm on Pure and Applied Math, 25 (1982), 771-831.
[11] X.J.Cai, Q.S.Jiu, Weak and strong solutions for the incompressible Navier-Stokes equations with damping term. J. math. Anal. Appl, In press, Available online 24 January, 2008.
[12] D.Chae, H.-S.Nam, Local existence and blow-up criterion for the Boussinesq equations, , Vol 155 (1999), 55-80.
[13] D.Chae, S.Kim and H.-S. Nam, Local existence and blow-up criterion of Holder continuous solutions of the Boussinesq equations, Nogaga Math. J., Vol 155 (1999), 55-80, .
[14] H.J.Choe, H. Kim, Strong Solutions of the Navier-Stokes Equations for Nonhomogeneous Incompressible Fluids, Comm. Partial Differentiao Equations 28 (2003),1183-1201.
[15] Y.Cho, H.Kim, Unique solvability for the density-dependent Navier-Stokes equations, Nonlinear Analysis 59 (2004),465-489.
[16] L.Escauriaza, G,Seregin, V.Sverak, On L_(3,∞)-solutions to the Navier-Stokes equations and backward uniqueness, Russian Math. Surveys, 58 (2003), 211-250.
[17] C.Foias, Une remarque sur l'unicite des solutions des equations de Navier-Stokes en dimension n, Bull. Soc. Math. France, 89 (1961), 1-8.
[18] C.Foias, C.Guillope, and R. Temam, New a priori estimates for Navier-Stokes equations in dimension 3, Comm. Partial Diff Eqs. 6 (1981), 329-359.
[19] G.P.Galdi, An introduction to the Navier-Stokes initial-boundary value problem, in " Fundemental directions in Mathematical Fluid Mechanics" editen by G.P. Galdi, John G. Heywood and Rolf Rannacher. Birkauser Verlag, Basel. Boston, Berlin.
[20] G.P.Galdi, An introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. Linearized Steady Problems. Springer-Verlag, New York, Berlin etc. 1994.
[21] Y.Giga, Solutions for semilinear parabolic equations in L~p and regularity of weak solutions of the Navier-Stokes system, J. Diff. Eqs., 61 (1986), 186-212.
[22] C.He, T.Miyakawa, On weighted norm estimates for solutions to nonsta-tionary Navier-Stokes equations in an exterior domain, preprint.
[23] E.Hopf, Uber die Anfangswertaufgabe fur die hydrody namischen Grund-gleichungen, Math. Nachr. 4(1951), 213-231.
[24] C.He, The Cauchy problem for the Navier-Stokes equations. J Math Anal App, 209 (1997), 228-242.
[25] C.He, L.Z.Wang, Weighted L~p-estimates for Stokes flow in R_+~n(n ≥ 2), with applications to the Non-stationary Navier-Stokes flow, preprint.
[26] J.G.Heywood, The Navier-Stokes equations: On the existence, regularity and decay of solutions. Indiana University Math. J., 29 (1980), 641-681.
[27] L.Hsiao, Quasilinear hyperbolic systems and dissipative mechanisms. World Scientific 1997.
[28] F.M.Huang, R.H.Pan, Convergence rate for compressible Euler equations with damping and vacuum, Arch. Rational Mech. Anal. 166 (2003), 359-376.
[29] Y.Kagei, On weak solutions of nonstationary Boussinesq equations, Diff and Integral Equ, 9, (1993), 587-611.
[30] R.Kajikiya, T.Miyakawa, On L~2 decay of weak solutions of the Navier-Stokes equations in R~n. Math. Z., 192 (1986), 136-148.
[31] T.Kato, Strong L~p solutions of the Navier-Stokes equations in R~n, with application to weak solutions, Math. Z. 187(1984), 471-480.
[32] A.V.Kazhikov, Resolution of boundary Value problems for nonhomogeneous viscous fluids, Dokl. Akarl. Nauh.,216 (1974), 1008-1010(in Russian).
[33] O.A.Ladyzhenskaya, The mathematical theory of viscous incompressible flow. Gordon and Breach, New York, 1969.
[34] O.Ladyzhenskaya, V.A.Solonnikov, Unique Solvability of an initial and boundary value problem foe viscous incompressible nonhomogeneous fluids, J. Soviet. Math. 9 (1978), 697-749.
[35] L.Laudau and E.Lifschitz, Mecanique des fluides. 2nd edition, Editions MIR, Moscow, 1989.
[36] J.Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math. 63 (1934), 193-248.
[37] X.J.Li, Q.S.Jiu, Global L~2 stability of large solutions to the 3D Boussinesq Equations, preprint (in Chinese).
[38] P.L.Lions, Mathematical topics in fluid mechanics, Vol.1: incompressible models, in Oxford Lecture Series in Mathematics and Its Applications, Vol. 10, 1996.
[39] P.Maremonti, On the asymptotic behavior of the L~2-norm of suitable weak solutions to the Navier-Stokes equations in three-dimensional exterior domains. Math. Phys., 118 (1988), 405-420.
[40] K.Masuda, L~2 decay of solutions of the Navier-Stokes equations in the exterior domains, in " Proceedings of Symposia in Pure Mathematics". 45 (1986), Part 2, 179-182. American Mathematical Society, 1986.
[41] K.Masuda, Weak solutions of the Navier-Stokes equations, Tohoku Math. J., 36 (1984), 623-646.
[42] F.J.McGrath, Nonstationary plane flow of viscous and ideal fluids, Arch. Rat. Mech. Anal. 27 (1968), 328-348.
[43] C.L.Navier. Memoire sur les lois du mouvement des fluids. Mem. Acad. Sci. de France., 6 (1827), 389-440.
[44] M.Padula, An existence theorem for nonhomogeneous in compressible fluids, Rend. Circ. Mat, Palemo, 31. 4 (1982).
[45] G.Ponce, R.Rache, T.C.Sideris, E.S.Titi, Global stability of large solutions to the 3D Navier-Stokes equations, Comm.Math. Phys., 159 (1994), 329-341.
[46] M.E.Schonbek, L~2 decay for weak solutions of the Navier-Stokes equations. Arch. Rational. Mech. Anal., 88 (1985), 209-222.
[47] M.E.Schonbek, Large time behavior of solutions to the Navier-Stokes equations. Comm. Partial Diff. equations., 11 (1986), 733-763.
[48] M.E.Schonbek, Lower bounds of rates of decay for solutions to the Navier-Stokes equations. J. Amer. Math. Soc, 4 (1991), 423-449.
[49] M.E.Schonbek, Asymptotic behavior of solutions to the three-dimensional Navier-Stokes equations. Indiana Univ. Math. J., 41(3) (1992), 809-823.
[50] T.Nagasawa, A new energy inequality and partial regularity foe weak solutions of Navier-Stokes equations, J.Math. Fluid Mech., 3 (2001), 40-56.
[51] R.Salvi, The equations of viscous incompressible nonhomogeneous fluids: on the existence and regularity, J. Austral. Math. Soc. Ser. B 33 (1991).
[52] J.Serrin, The ininial value problem for the Navier-Stokes equations, Univ. Wisconsin Press, Nonlinear Problem, Ed. R. langer, 1963. J.Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal. 9 (1962), 187-195.
[54] J.Simon, On the existence fo the pressure for solutions of variational Navier-Stokes equations, J.Math. Fluid Mech., 1 (1999), 225-234.
[55] J.Simon, Ecoulementd'un fluide nonhomogene avec une densite initiale s'annulant. CR. Acad. Sci. Paris, 15 (1978), 1009-1012.
[56] J.Simon, Nonhomogeneous viscous incompressible fluids: existence of velocity, density and pressure, SIAM J. Math. Anal., 21(5) (1990), 1093-1117.
[57] J.Simon, Sur les fluids visqueux incompressibles et non homogenes. CR. Acad. Sci. Paris, 309 (1989), 447-452.
[58] H.Sohr, The Navier-Stokes Equations: an elementary functional analytic approach, Birkhauser Verlag, 2001.
[59] G.G.Stokes. On the Theories of the Internal Friction of Fluids in Motion. Trans. Cambridge Phil. Soc, 8 (1845), 287-319.
[60] M.Struwe, On partial regularity results for the Navier-Stokes equa-tions,Comm. Pure. Apll. Math., 41 (1988), 437-458.
[61] R.Temam, Navier-Stokes equations theory and numerical Analysis, North-Holland-Amsterdam, New York, 1984.
[62] S.Ukai, A solution formular for the Navier-Stokes equations in R~n. Comm. Pure. Appl. Math., XL (1987), 611-621.
[63] M.Weigner, Decay results for weak solutions of the Navier-Stokes equations in R~n. J. London. Math. Soc, 35 (1987), 303-313.
[64] C.S.Zhao, K.T.Li,Global L~2 stability of large solutions to the 3D MHD Equations, Acta. Math. Sinica, Vol.44, No.6, (2001), 961-976 (in Chinese).