一类不定常热耦合Stokes问题
摘要
本文针对较为典型的工程问题及流体力学中常见的一类不可压拟牛顿流不定常热耦合Stokes问题进行研究，在假设初始温度θ_0∈L~2(Ω)，耦合函数κ∈L~∞(Ω×(0，T))，考虑齐次边界条件并假设耦合函数μ，κ∈C(IR×(0，T))是有界的条件下，主要研究了这类问题方程组弱解的存在性，惟一性和解的爆破的一些结果，模型如下：
全文主要包括三部分的内容：
第一部分，在引入变分问题的基础上，利用Faedo-Galerkin方法证明了此类不可压拟牛顿流不定常热耦合Stokes问题弱解的存在性。
第二部分，针对此类问题，利用嵌入定理的相关知识以及Meyers估计和Schauder不动点定理证明了弱解的存在性。
第三部分，通过建立弱解对初边值的估计式，利用相关技巧得到弱解的惟一性。同时在一定的假设条件下得到了解的爆破。
各部分中均将微分方程组转化为变分问题，通过研究相应变分问题的不动点来研究原微分方程组解的存在性。在文章的证明过程中，嵌入定理，Meyers估计，Schauder不动点定理和Faedo-Galerkin方法发挥了重要作用。
Several typical kinds of PDE models called coupled nonlinear equations have been considered in this thesis and the existence , uniqueness and blowup results of the respective elliptic equations are obtained.The model in this thesis is:
This thesis is mainly composed of three parts of contents:
In the first part, under the condition of the variational formulation, existence of the unsteady thermally coupled Stokes problem which describing the unsteady flow of a quasi-Newtonian fluid with temperature-dependent viscosity and with a viscous heating is proved under Faedo-Galerkin method.
In the second part, by Meyers estimate, Schauder fixed point theorem together we study the boundary value problem for the thermally coupled Stokes problem and the coexistence of existence is obtained.
The third part studies the estimate of the weak solution which depends on the initial and boundary conditions is established to prove the uniqueness. Results on blowup of the weak solution is also studied.
Differential equations are transformed to variational formulation all the above parts. By the existence of the fixed point for the operaor equation, we obtain the existence for the differential equations, in which the Meyers estimate, Faedo-Galerkin method and Schauder fixed point theorem play an important role.
全文主要包括三部分的内容：
第一部分，在引入变分问题的基础上，利用Faedo-Galerkin方法证明了此类不可压拟牛顿流不定常热耦合Stokes问题弱解的存在性。
第二部分，针对此类问题，利用嵌入定理的相关知识以及Meyers估计和Schauder不动点定理证明了弱解的存在性。
第三部分，通过建立弱解对初边值的估计式，利用相关技巧得到弱解的惟一性。同时在一定的假设条件下得到了解的爆破。
各部分中均将微分方程组转化为变分问题，通过研究相应变分问题的不动点来研究原微分方程组解的存在性。在文章的证明过程中，嵌入定理，Meyers估计，Schauder不动点定理和Faedo-Galerkin方法发挥了重要作用。
Several typical kinds of PDE models called coupled nonlinear equations have been considered in this thesis and the existence , uniqueness and blowup results of the respective elliptic equations are obtained.The model in this thesis is:
This thesis is mainly composed of three parts of contents:
In the first part, under the condition of the variational formulation, existence of the unsteady thermally coupled Stokes problem which describing the unsteady flow of a quasi-Newtonian fluid with temperature-dependent viscosity and with a viscous heating is proved under Faedo-Galerkin method.
In the second part, by Meyers estimate, Schauder fixed point theorem together we study the boundary value problem for the thermally coupled Stokes problem and the coexistence of existence is obtained.
The third part studies the estimate of the weak solution which depends on the initial and boundary conditions is established to prove the uniqueness. Results on blowup of the weak solution is also studied.
Differential equations are transformed to variational formulation all the above parts. By the existence of the fixed point for the operaor equation, we obtain the existence for the differential equations, in which the Meyers estimate, Faedo-Galerkin method and Schauder fixed point theorem play an important role.
引文
[1] Zhu J, Loula A F D, Karam F J,Guerreiro J N C,Finite element analysis of a coupled thermally dependent viscosity flow problem[J]. Comput Appl Math, 2007,26(1): 45-66.
[2] Baranger J, Mikelic A, Statinary solutions to a quasi-newtonian flow with visous heating[J].M3AS: Math Model Meth Appl Sci, 1995, 5(6): 725-738.
[3] Antontsev S N, Chipot M,The thermistor problem: existence, smoothness, uniqueness, blowup[J]. SIAM J Math ANAL, 1994,25(4): 1128-1156.
[4]管平，李刚，一类热流问题解的存在性[J]．南京气象学院学报，1997，20(2)，199-204．
[5] Wardi S, A convergence result for an interative method for the equations of a stationaryquasi-newtonian flow with temperature dependent viscosity[J].M2AN: Model Math Anal Numer,1998, 32(1): 391-404.
[6] Loula A F D, Zhu J, Finite element analysis of a coupled nonlinear system, Comput ApplMath[J]. 2001,20(1): 321-339.
[7] Elliott C M, Larsson S, A Finite element model for the time-dependent joule heating problem[J].Comput Appl Math, 1995, 64(212): 1433-1453.
[8] Loula A F D, Zhu J, Mixed finite element analysis of a thermally nonlinear coupled problem[J].Numer Methods PDE, 2006, 22(1): 180-196.
[9]黄艳，李刚，黄瑜，一类非线性抛物型方程熵解的存在性[J]．南京气象学院院报，2006， 29(4)：544-548．
[10]王平，李刚，一类退化抛物型方程全局解的存在性和爆破[J]．南京气象学院院报，2006， 29(4)：569-575．
[11] Maria T G M, Francisco O G,On the existence of solutions for a strongly degenerate system[J]. CR. Acad. Sci. Paris, 2006, 343(1): 119-123.
[12] W. Allegretto and H. Xie. Solutions for microsensor thermistor equations in the small bias case[J].Proc. Roy.Soc. Edingburgh Ser. A, 1993 ,123(1), 987-999.
[13] G. Cimatti. A bound for the temperature in the thermistor problem, [J]. Appl. Math. 1988,40(1),15-22.
[14] G. Cimatti and G. Prodi. Existence results for a nonlinear elliptic system modelling a temperaturedependent electrical resistor[J], Ann. Mat. Pura Appl. 1989 152 (1), 227-236.
[15] A. Friedman. Partial Differential Equations of Parabolic Type, Englewood Cliffs, N. J.,Prentice-Hall, 1964.
[16] T. GallouAet and R. Herbin. Existence of a solution to a coupled elliptic system, Appl. Math.Lett. 7, 1994,49-55.
[17] P. Guan. Existence, boundedness and uniqueness of weak solution for the??thermistor problem with mixed boundary conditions[J], Acta Mathematica Scientia, 1998, 18,326-332.
[18] H. D. Howison and J. F. Rodrigues, M. Shillor. Stationary solutions to the thermistor problem[J].Math.Anal. Appl. 1993, 174, 573-588.
[19] A. Quarteroni and A. Valli. Numerical Approximation of Partial Differential Equations, SpringerSeries in Computational Mathematics, Springer-Verlag, Vol.23,1997.
[20] P. Shi, M. Shillor and X. Xu. Existence of a solution to the Stefan problem with Joule'sheating[ J].Differential Eq. 1993, 105,239-263.
[21] H. M. Yin. The semiconductor systems with temperature effect[ J] .Math. Anal. Appl. 1995,196,135-152.
[22] G. Yuan and Z. Liu. Existence and uniqueness of the Cff-solution for the thermistor problem withmixed boundary values[J]. SIAM J. Math. Anal. 1994,25,1157-1166.
[23] J. Bass. Thermoelasticity. McGraw-Hill Encyclopedia of Physics, Parker SP (ed).McGraw-Hill,New York, 1982.
[24] J. Karam F. Thermally Coupled Flow of Non-Newtonian Fluids in Conjugated Problems.Doctoral Thesis, COPPE, Federal University of Rio de Janeiro, 1996 (in Portuguese).
[25] G. Cimatti. Remarks on existence and uniqueness for the thermistor problem under mixedboundary conditions[J]. Quart. Appl. Math. 1989; 47: 117 - 121.
[26] V. Girault, P.A. Raviart. Finite Element Methods for Navier - Stokes Equations.Springer-Verlag: Berlin, 1986.
[27] G.P. Galdi, C.G. Simader, H. Sohr. On the Stokes problem in Lipschitz domains[J].Ann. Mat.pura appl. 1994; CLXVII: 147 - 163.
[28] K. Groger. A W~(1,p) - estimate for solutions to mixed boundary value problems for second orderelliptic differential equations[J]. Math. Ann. 1989; 283: 679 - 687.
[29] N.G. Meyers. An L~p -estimate for the gradient of solutions of second order elliptic divergenceequations[J]. Ann. Sc. Norm. Sup. Pisa 1963; 17: 189 - 206.
[30] A. Bossavit and J.-F. Rodrigues. On the electromagnetic induction heating problem in boundeddomains[J], Adv. Math. Sci. Appl. 1994,4, 79-92.
[31] P. G. Ciarlet. The Finite Element Method for Elliptic Problems,North-Holland, Amsterdam,1978.
[32] C. G. Simader. On Dirichlet's Boundary Value Problem, Lecture Notes in Mathematics 268,Springer-Verlag, Berlin, 1972.
[33] V. Thom ee. Galerkin Finite Element Methods for Parabolic Problems, Lecture Notes in Mathematics 1054, Springer-Verlag, Berlin, 1984.
[34] F. J. Hyde. Thermistors, Ili(?)e Books, London, 1971.
[35] Gang, Li, Jiang Zhu and Xijun, Yu,A stabilized mixed finite element method for a thermally nonlinear coupled problem,Proceedings of the Forth International Workshop on ScientificComputing and Its Applications, Press of Science, Beijing, China, 2007.
[36] S.N. ANTONTSEV, A. V. KAZHIKHOV, AND V. N. MONAKHOV, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids, Stud. Math. Appl., 1990,22, North Holland, Amsterdam.
[37] A. BENSOUSSAN ,J. L. LIONS, Applications des Indquations Variationnelles en ContrSle Stochastique, Dunod, Paris, 1978.
[38] O.A. LADYENSKAJA, V. i. SOLONIKOV, AND N. N. URAL'CEVA, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, RI, 1968.
[39] O.A. LADYENSKAJA AND i. N. URAL'CEVA, Equations aux Ddrivdes Partielles de Type Elliptique, Dunod, Paris, 1968.
[40] S. N. ANTONTSEV AND M. CHIPOT, Some results on the thermistor problem, Proceedings of the Conference Free-Boundary Problems in Continuum Mechanics, Novosibirsk,Russia, July 1991.
[2] Baranger J, Mikelic A, Statinary solutions to a quasi-newtonian flow with visous heating[J].M3AS: Math Model Meth Appl Sci, 1995, 5(6): 725-738.
[3] Antontsev S N, Chipot M,The thermistor problem: existence, smoothness, uniqueness, blowup[J]. SIAM J Math ANAL, 1994,25(4): 1128-1156.
[4]管平，李刚，一类热流问题解的存在性[J]．南京气象学院学报，1997，20(2)，199-204．
[5] Wardi S, A convergence result for an interative method for the equations of a stationaryquasi-newtonian flow with temperature dependent viscosity[J].M2AN: Model Math Anal Numer,1998, 32(1): 391-404.
[6] Loula A F D, Zhu J, Finite element analysis of a coupled nonlinear system, Comput ApplMath[J]. 2001,20(1): 321-339.
[7] Elliott C M, Larsson S, A Finite element model for the time-dependent joule heating problem[J].Comput Appl Math, 1995, 64(212): 1433-1453.
[8] Loula A F D, Zhu J, Mixed finite element analysis of a thermally nonlinear coupled problem[J].Numer Methods PDE, 2006, 22(1): 180-196.
[9]黄艳，李刚，黄瑜，一类非线性抛物型方程熵解的存在性[J]．南京气象学院院报，2006， 29(4)：544-548．
[10]王平，李刚，一类退化抛物型方程全局解的存在性和爆破[J]．南京气象学院院报，2006， 29(4)：569-575．
[11] Maria T G M, Francisco O G,On the existence of solutions for a strongly degenerate system[J]. CR. Acad. Sci. Paris, 2006, 343(1): 119-123.
[12] W. Allegretto and H. Xie. Solutions for microsensor thermistor equations in the small bias case[J].Proc. Roy.Soc. Edingburgh Ser. A, 1993 ,123(1), 987-999.
[13] G. Cimatti. A bound for the temperature in the thermistor problem, [J]. Appl. Math. 1988,40(1),15-22.
[14] G. Cimatti and G. Prodi. Existence results for a nonlinear elliptic system modelling a temperaturedependent electrical resistor[J], Ann. Mat. Pura Appl. 1989 152 (1), 227-236.
[15] A. Friedman. Partial Differential Equations of Parabolic Type, Englewood Cliffs, N. J.,Prentice-Hall, 1964.
[16] T. GallouAet and R. Herbin. Existence of a solution to a coupled elliptic system, Appl. Math.Lett. 7, 1994,49-55.
[17] P. Guan. Existence, boundedness and uniqueness of weak solution for the??thermistor problem with mixed boundary conditions[J], Acta Mathematica Scientia, 1998, 18,326-332.
[18] H. D. Howison and J. F. Rodrigues, M. Shillor. Stationary solutions to the thermistor problem[J].Math.Anal. Appl. 1993, 174, 573-588.
[19] A. Quarteroni and A. Valli. Numerical Approximation of Partial Differential Equations, SpringerSeries in Computational Mathematics, Springer-Verlag, Vol.23,1997.
[20] P. Shi, M. Shillor and X. Xu. Existence of a solution to the Stefan problem with Joule'sheating[ J].Differential Eq. 1993, 105,239-263.
[21] H. M. Yin. The semiconductor systems with temperature effect[ J] .Math. Anal. Appl. 1995,196,135-152.
[22] G. Yuan and Z. Liu. Existence and uniqueness of the Cff-solution for the thermistor problem withmixed boundary values[J]. SIAM J. Math. Anal. 1994,25,1157-1166.
[23] J. Bass. Thermoelasticity. McGraw-Hill Encyclopedia of Physics, Parker SP (ed).McGraw-Hill,New York, 1982.
[24] J. Karam F. Thermally Coupled Flow of Non-Newtonian Fluids in Conjugated Problems.Doctoral Thesis, COPPE, Federal University of Rio de Janeiro, 1996 (in Portuguese).
[25] G. Cimatti. Remarks on existence and uniqueness for the thermistor problem under mixedboundary conditions[J]. Quart. Appl. Math. 1989; 47: 117 - 121.
[26] V. Girault, P.A. Raviart. Finite Element Methods for Navier - Stokes Equations.Springer-Verlag: Berlin, 1986.
[27] G.P. Galdi, C.G. Simader, H. Sohr. On the Stokes problem in Lipschitz domains[J].Ann. Mat.pura appl. 1994; CLXVII: 147 - 163.
[28] K. Groger. A W~(1,p) - estimate for solutions to mixed boundary value problems for second orderelliptic differential equations[J]. Math. Ann. 1989; 283: 679 - 687.
[29] N.G. Meyers. An L~p -estimate for the gradient of solutions of second order elliptic divergenceequations[J]. Ann. Sc. Norm. Sup. Pisa 1963; 17: 189 - 206.
[30] A. Bossavit and J.-F. Rodrigues. On the electromagnetic induction heating problem in boundeddomains[J], Adv. Math. Sci. Appl. 1994,4, 79-92.
[31] P. G. Ciarlet. The Finite Element Method for Elliptic Problems,North-Holland, Amsterdam,1978.
[32] C. G. Simader. On Dirichlet's Boundary Value Problem, Lecture Notes in Mathematics 268,Springer-Verlag, Berlin, 1972.
[33] V. Thom ee. Galerkin Finite Element Methods for Parabolic Problems, Lecture Notes in Mathematics 1054, Springer-Verlag, Berlin, 1984.
[34] F. J. Hyde. Thermistors, Ili(?)e Books, London, 1971.
[35] Gang, Li, Jiang Zhu and Xijun, Yu,A stabilized mixed finite element method for a thermally nonlinear coupled problem,Proceedings of the Forth International Workshop on ScientificComputing and Its Applications, Press of Science, Beijing, China, 2007.
[36] S.N. ANTONTSEV, A. V. KAZHIKHOV, AND V. N. MONAKHOV, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids, Stud. Math. Appl., 1990,22, North Holland, Amsterdam.
[37] A. BENSOUSSAN ,J. L. LIONS, Applications des Indquations Variationnelles en ContrSle Stochastique, Dunod, Paris, 1978.
[38] O.A. LADYENSKAJA, V. i. SOLONIKOV, AND N. N. URAL'CEVA, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, RI, 1968.
[39] O.A. LADYENSKAJA AND i. N. URAL'CEVA, Equations aux Ddrivdes Partielles de Type Elliptique, Dunod, Paris, 1968.
[40] S. N. ANTONTSEV AND M. CHIPOT, Some results on the thermistor problem, Proceedings of the Conference Free-Boundary Problems in Continuum Mechanics, Novosibirsk,Russia, July 1991.