非正则奇异性偏微分方程形式解的Borel可和性研究
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摘要
本文主要研究如下奇异偏微分方程的可解性和唯一性问题：这里m是正整数,j,α满足j+|α|≤m和j 奇异微分方程的研究可追溯到1856年Briot-Bouquet对于Briot-Bouquet型奇异常微分方程的研究和1913年Gevrey对于前进—后退扩散方程的研究,而这些问题的进一步研究最终都可以转化为研究如上的奇异偏微分方程(1).同时,对于经典的Cauchy-Kowalewski定理,如果考虑其在超曲面带有奇异性质的解,最后也归结到求解奇异偏微分方程(1).因此,这类方程的研究具有十分重要的意义.
对于m=1和n=1的情形,其研究已经比较完备.特别地,对于全特征型方程的解,Chen-Tahara,Chen-Luo-Tahara和Chen-Luo-Zhang研究了其Gevrey发散性及Borel可和性.
在本文中,我们主要把他们的工作推广到m=1和n=2的情形.即,我们研究以下方程的可解性问题t(?)u=F(t,x,y,u,(?)xu,(?)yu),(t,x,y)∈C3.(2)更具体的说来,本文研究了以下内容：
在第一章中,我们首先介绍了研究奇异方程(1.1)的一些历史背景.接着对于m=1和n=1的情况,即对于方程(1.8),我们详细叙述了其最近的一些发展.
在第二章中,我们回顾了Gevrey渐近展开和Gevrey形式幂级数的Borel可和性等基本知识,并证明了一些后面将会用到的基本引理.
在第三章中,我们研究的是二维空间变量方程t(?)tu:F(t,x,y,u,(?)xu,(?)yu),u(0,x,y)三0,(3)在合适的条件下,证明了上述方程的唯一形式幂级数解u(t,x,y)∈(?){t,y}[[x]]k,得到的奇点分布规律从[25]中的一条直线上演变成平面区域上网格状的散点.
在第四章中,我们研究了更弱条件下的奇异方程：利用Nagurno范数证明了上述方程的唯一形式幂级数解u(t,x,y)在充分小的多圆盘DR1×DR2={(t,y)∈(?)2：|t| 在第五章中,我们研究了[25]中的单个空间变量方程：在张角较小(θ<π/k)的扇形区域G(d,θ)内,得到它有无穷多个解析解.
In this dissertation, we mainly study the solvability and uniqueness problems of the following singular partial differential equation:(t(?)t)mu=F(t,x,(t(?)j(?)xαu),(t,x)∈C×Cxn. Here m is a positive integer, j,α satisfy j+|α| The study of the singular differential equations can date back to the year1856when Briot-Bouquet carried out a study of Briot-Bouquet type differential equations, and the year1913when Gevrey initiated the study of the forward-backward diffusion equations. Further studies on these problems will at last come to the study of the above singular differential equations. Meanwhile, for the classical Cauchy-Kowalewski Theorem, when we consider the singular solutions along a hypersurface, will also contribute to the study of the singular differential equation above. Hence the study of this kind of singular equations is quite interesting.
For the case m=1and n=1, the study is somehow very complete. In particular, for the solutions of totally characteristic equations, Chen-Tahara, Chen-Luo-Tahara and Chen-Luo-Zhang studied the summability and Gevrey divergency.
In this dissertation, we will generalize their work to the case m=1and n=2. Namely, we will study the solvability of the equation:t(?)tu=F(t,x,y,u,(?)xu,(?)yu),(t,x, y)∈C3. More precisely, we will study the following content:
In the first chapter, we first introduced some background of the study of the singular equations. Next, we stated some recent process for the case m=1and n=1.
In the second chapter, we recalled some basic notations and properties of the Gevrey asymptotic expansion and Borel summarability of Gevrey-type formal power series. We also proved some basic lemmas that will be used later.
In the third chapter, we consider the summability of the following equation: t(?)tu=F(t,x,y,u,(?)xu,(?)yu), u(0,x,y)=0,(4) Under some suitable conditions, we proved that the unique solution u(t,x,y)∈C{t,y}[[x]]k, whose singular points are discretely distributed on some lattice in the plane region.
In the fourth chapter, we consider the summability of the following equation: By making use of the Nagumo norm, we proved that the equation above has a unique formal power series solution u(t,x,y), which is Borel summarable with respect to the variable x in a small polydiscs DR1×DR2={(t,y)∈C2:|t| In the fifth chapter,we consider the following equation with one space variable: We proved that in the sector domain G(d,θ) with a small aperture angle(θ<π/k),we have infinity many real analytic solutions.
对于m=1和n=1的情形,其研究已经比较完备.特别地,对于全特征型方程的解,Chen-Tahara,Chen-Luo-Tahara和Chen-Luo-Zhang研究了其Gevrey发散性及Borel可和性.
在本文中,我们主要把他们的工作推广到m=1和n=2的情形.即,我们研究以下方程的可解性问题t(?)u=F(t,x,y,u,(?)xu,(?)yu),(t,x,y)∈C3.(2)更具体的说来,本文研究了以下内容：
在第一章中,我们首先介绍了研究奇异方程(1.1)的一些历史背景.接着对于m=1和n=1的情况,即对于方程(1.8),我们详细叙述了其最近的一些发展.
在第二章中,我们回顾了Gevrey渐近展开和Gevrey形式幂级数的Borel可和性等基本知识,并证明了一些后面将会用到的基本引理.
在第三章中,我们研究的是二维空间变量方程t(?)tu:F(t,x,y,u,(?)xu,(?)yu),u(0,x,y)三0,(3)在合适的条件下,证明了上述方程的唯一形式幂级数解u(t,x,y)∈(?){t,y}[[x]]k,得到的奇点分布规律从[25]中的一条直线上演变成平面区域上网格状的散点.
在第四章中,我们研究了更弱条件下的奇异方程：利用Nagurno范数证明了上述方程的唯一形式幂级数解u(t,x,y)在充分小的多圆盘DR1×DR2={(t,y)∈(?)2：|t|
In this dissertation, we mainly study the solvability and uniqueness problems of the following singular partial differential equation:(t(?)t)mu=F(t,x,(t(?)j(?)xαu),(t,x)∈C×Cxn. Here m is a positive integer, j,α satisfy j+|α|
For the case m=1and n=1, the study is somehow very complete. In particular, for the solutions of totally characteristic equations, Chen-Tahara, Chen-Luo-Tahara and Chen-Luo-Zhang studied the summability and Gevrey divergency.
In this dissertation, we will generalize their work to the case m=1and n=2. Namely, we will study the solvability of the equation:t(?)tu=F(t,x,y,u,(?)xu,(?)yu),(t,x, y)∈C3. More precisely, we will study the following content:
In the first chapter, we first introduced some background of the study of the singular equations. Next, we stated some recent process for the case m=1and n=1.
In the second chapter, we recalled some basic notations and properties of the Gevrey asymptotic expansion and Borel summarability of Gevrey-type formal power series. We also proved some basic lemmas that will be used later.
In the third chapter, we consider the summability of the following equation: t(?)tu=F(t,x,y,u,(?)xu,(?)yu), u(0,x,y)=0,(4) Under some suitable conditions, we proved that the unique solution u(t,x,y)∈C{t,y}[[x]]k, whose singular points are discretely distributed on some lattice in the plane region.
In the fourth chapter, we consider the summability of the following equation: By making use of the Nagumo norm, we proved that the equation above has a unique formal power series solution u(t,x,y), which is Borel summarable with respect to the variable x in a small polydiscs DR1×DR2={(t,y)∈C2:|t|
引文
[1]W. Balser, B.L.J. Braaksma, J.P. Ramis and Y. Sibuya, Multisummability of formal power series solutions of linear ordinary differential equations,Asymptotic Anal.5(1991),27-45.
[2]W. Balser, Formal power series and linear systems of meromorphic ordinary dif-ferential equations, Universitext XVIII, Springer-Verlag, New York,2000.
[3]W. Balser, Multisummability of formal power series solutions of partial differential equations with constant coefficients, J. Differential Equations,201 (2004),63-74.
[4]B. L. J. Braaksma, Multisummability of formal power series solutions of nonlinear meromorphic differential equations, Ann. Inst. Fourier,42 (1992), p.517-540
[5]M. Canalis-Durand, J.-P. Ramis, R. Schafke and Y. Sibuya, Gevrey solutions of singularly perturbed differential equations, J. Reine Angew. Math.,518 (2000), 95-129.
[6]H. Chen, Z. Luo and H. Tahara, Formal solution of nonlinear first order totally characteristic type PDE with irregular singularity, Ann. Inst. Fourier,51 (2001), 1599-1620.
[7]H. Chen and H. Tahara, On totally characteristic type non-linear differential equa-tions in the Complex Domain, Publ. RIMS, Kyoto Univ.,35 (1999),621-636.
[8]H. Chen and H. Tahara, On the holomorphic solution of non-linear totally charac-teristic equations, Mathematische Nachrichten,219 (2000),85-96.
[9]H. Chen, Z. Luo, On the holomorphic solution of non-linear totally characteristic equations with several space variables, Acta Mathematica Scientia,22B (2002), 393-403.
[10]H. Chen and Z. Luo, Formal solutions for higher order nonlinear totally char-acteristic pdes with irregular singularities, Contemporary Mathematics Vol.368: Geometric Analysis of PDE and Several Complex Variables, (2005),121-131.
[11]O. Costin and S. Tanveer, Existence and uniqueness for a class of nonlinear higher-order partial differential equations in the complex plane, Comm. Pure and Appl. Math., LⅢ (2000),0001-0026.
[12]L. Di Vizio, An ultrametric version of the Maillet-Malgrange theorem for non linear q-difference equations, Proc. Amer. Math. Soc.,136 (2008),2803-2814.
[13]P.C. Fife and J.B. McLeod, The Approach of solutions of nonlinear diffusion equa-tions to travelling front, solutions, Arch. Ration. Mech. Anal.,65 (1977),335-361.
[14]R. Gerard and H. Tahara, Solutions holomorphes et singulieres d'equations aux derivees partielles singulieres non lineaires, Publ. Res. Inst. Math. Sci.,29 (1993), 121-151.
[15]R. Gerard and H. Tahara, Holomorphic and singular solutions of non-linear singu-lar partial differential equations, Ⅱ, In:Structure of differential equations, Kata-ta/Tyoto,1995, (eds. M. Morimoto and T. Kawai),1996, World Scientific,135-150.
[16]R. Gerard and H. Tahara, Singular nonlinear partial differential equations, Aspects of Mathematics, E 28, Vieweg,1996.
[17]M. Gevrey, Sur les equations aux derivees partielles du type parabolique, J. de Mathematique,9(1913),305-476.
[18]P. S. Hagan and J. R. Ockendon, Half-range analysis of a counter-current separator, J. Math. Anal. Appl.,160 (1991),358-378.
[19]M. Hibino, Borel summability of divergent solutions for singular first order lin-ear partial differential equations with polynomial coefficients. J. Math. Sci. Univ. Tokyo,10 (2003),279-309.
[20]M. Hibino, Borel summability of divergence solutions for singular first-order partial differential equations with variable coefficients. I. J. Differential Equations,227 (2006),499-533.
[21]M. Hibino, Borel summability of divergent solutions for singular first-order partial differential equations with variable coefficients. II. J. Differential Equations,227 (2006),534-563.
[22]L. Hormander, An introduction to complex analysis in several variables, Third edition, North-Holland Mathematical Library,7. North-Holland Publishing Co., Amsterdam,1990.
[23]T. Kobayashi, Singular solutions and prolongation of holomorphic solutions to nonlinear differential equations, Publ. RIMS. Kyoto Univ.,34 (1998),43-63.
[24]Z. Luo, H. Chen and C. Zhang, On the summability of the formal solutions for some PDEs with irregular singularity, C.R. Acad. Sci. Paris, Ser. I,336 (2003), 219-224.
[25]Z. Luo, H. Chen and C. Zhang, On the summability of formal solutions for a class of nonlinear singular PDEs with irregular singularity, Contemporary of Mathematics, 400 (2006),53-64.
[26]Z. Luo, H. Chen and C. Zhang, Exponential-type Nagumo norms and summability of formal solutions of singular partial differential equations, Annales De L'institut Fourier,62 (2012),571-618.
[27]Z. Luo and C. Zhang, On the Borel summability of divergent power series respective to two variables, Preprint.
[28]D. A. Lutz, M. Miyake and R. Schafke, On the Borel summability of divergent solutions of the heat equation, Nagoya Math. J.,154 (1999),1-29.
[29]B. Malgrange, Sur le theoreme de Maillet, Asymptot. Anal,2 (1989),1-4.
[30]J. Martinet and J.-P. Ramis, Problemes de modules pour des equations dif-ferentielles non lineaires du premier ordre, Publ. Math., Inst. Hautes Etudes Sci., 55 (1982),63-164.
[31]J. Martinet and J.-P. Ramis, Elementary acceleration and multisummability I, Annales de l'.H.P. Physique theorique,54 (1991),331-401.
[32]M. Nagumo, Uber das Anfangswertproblem partieller Differentialgleichungen, Jap. J. Math.,18 (1942),41-47.
[33]S. Ouchi, Multisummability of formal solutions of some linear partial differential equations, J. Differential Equations,185 (2002),513-549.
[34]S. Ouchi, Borel summability of formal solutions of some first order singular partial differential equations and normal forms of vector fields, J. Math. Soc. Japan,57 (2005),415-460.
[35]S. Ouchi, Multisummability of formal power series solutions of nonlinear partial differential equations in complex domains, Asymptot. Anal.,47 (2006),187-225.
[36]C. D. Pagani and G. Talenti, On a forward-backward parabolic equation, Ann. Mat. Pura. Appl.,90 (1971),1-57.
[37]J.-P. Ramis, Les series k-sommables et leurs applications,Complex Analysis, Mi-crolocal Calculus and Relativistic Quantum Theory, Lecture Notes in Physics,126 (1980),178-199.
[38]H. Tahara, Uniqueness of the solution of non-linear singular partial differential equations, J. Math. Soc. Japan,48 (1996),729-744.
[39]H. Tahara, On the uniqueness theorem for nonlinear singular partial differential equations, J. Math. Sci. Univ. Tokyo,5 (1998),477-506.
[40]H. Tahara and H. Yamazawa, Structure of Solutions of Nonlinear Partial Differ-ential Equations of Gerard-Tahara Type, Publ. Res. Inst. Math. Sci.,41 (2005), 339-373.
[41]J.-C. Tougeron, Sur les ensembles semi-analytiques avec conditions Gevrey au bord, Ann. Sci. Ecolc Norm. Sup.,27 (1994),173-208.
[42]Y. Tsuno, On the prolongation of local holomorphic solutions of nonlinear partial differential equations. J. Math. Soc. Japan,27,454-466 (1975).
[43]C. Zhang, Sur un theoreme du type de Maillet-Malgrange pour les equations q-differences-differentielles, Asymptot. Anal.,17 (1998),309-314.
[2]W. Balser, Formal power series and linear systems of meromorphic ordinary dif-ferential equations, Universitext XVIII, Springer-Verlag, New York,2000.
[3]W. Balser, Multisummability of formal power series solutions of partial differential equations with constant coefficients, J. Differential Equations,201 (2004),63-74.
[4]B. L. J. Braaksma, Multisummability of formal power series solutions of nonlinear meromorphic differential equations, Ann. Inst. Fourier,42 (1992), p.517-540
[5]M. Canalis-Durand, J.-P. Ramis, R. Schafke and Y. Sibuya, Gevrey solutions of singularly perturbed differential equations, J. Reine Angew. Math.,518 (2000), 95-129.
[6]H. Chen, Z. Luo and H. Tahara, Formal solution of nonlinear first order totally characteristic type PDE with irregular singularity, Ann. Inst. Fourier,51 (2001), 1599-1620.
[7]H. Chen and H. Tahara, On totally characteristic type non-linear differential equa-tions in the Complex Domain, Publ. RIMS, Kyoto Univ.,35 (1999),621-636.
[8]H. Chen and H. Tahara, On the holomorphic solution of non-linear totally charac-teristic equations, Mathematische Nachrichten,219 (2000),85-96.
[9]H. Chen, Z. Luo, On the holomorphic solution of non-linear totally characteristic equations with several space variables, Acta Mathematica Scientia,22B (2002), 393-403.
[10]H. Chen and Z. Luo, Formal solutions for higher order nonlinear totally char-acteristic pdes with irregular singularities, Contemporary Mathematics Vol.368: Geometric Analysis of PDE and Several Complex Variables, (2005),121-131.
[11]O. Costin and S. Tanveer, Existence and uniqueness for a class of nonlinear higher-order partial differential equations in the complex plane, Comm. Pure and Appl. Math., LⅢ (2000),0001-0026.
[12]L. Di Vizio, An ultrametric version of the Maillet-Malgrange theorem for non linear q-difference equations, Proc. Amer. Math. Soc.,136 (2008),2803-2814.
[13]P.C. Fife and J.B. McLeod, The Approach of solutions of nonlinear diffusion equa-tions to travelling front, solutions, Arch. Ration. Mech. Anal.,65 (1977),335-361.
[14]R. Gerard and H. Tahara, Solutions holomorphes et singulieres d'equations aux derivees partielles singulieres non lineaires, Publ. Res. Inst. Math. Sci.,29 (1993), 121-151.
[15]R. Gerard and H. Tahara, Holomorphic and singular solutions of non-linear singu-lar partial differential equations, Ⅱ, In:Structure of differential equations, Kata-ta/Tyoto,1995, (eds. M. Morimoto and T. Kawai),1996, World Scientific,135-150.
[16]R. Gerard and H. Tahara, Singular nonlinear partial differential equations, Aspects of Mathematics, E 28, Vieweg,1996.
[17]M. Gevrey, Sur les equations aux derivees partielles du type parabolique, J. de Mathematique,9(1913),305-476.
[18]P. S. Hagan and J. R. Ockendon, Half-range analysis of a counter-current separator, J. Math. Anal. Appl.,160 (1991),358-378.
[19]M. Hibino, Borel summability of divergent solutions for singular first order lin-ear partial differential equations with polynomial coefficients. J. Math. Sci. Univ. Tokyo,10 (2003),279-309.
[20]M. Hibino, Borel summability of divergence solutions for singular first-order partial differential equations with variable coefficients. I. J. Differential Equations,227 (2006),499-533.
[21]M. Hibino, Borel summability of divergent solutions for singular first-order partial differential equations with variable coefficients. II. J. Differential Equations,227 (2006),534-563.
[22]L. Hormander, An introduction to complex analysis in several variables, Third edition, North-Holland Mathematical Library,7. North-Holland Publishing Co., Amsterdam,1990.
[23]T. Kobayashi, Singular solutions and prolongation of holomorphic solutions to nonlinear differential equations, Publ. RIMS. Kyoto Univ.,34 (1998),43-63.
[24]Z. Luo, H. Chen and C. Zhang, On the summability of the formal solutions for some PDEs with irregular singularity, C.R. Acad. Sci. Paris, Ser. I,336 (2003), 219-224.
[25]Z. Luo, H. Chen and C. Zhang, On the summability of formal solutions for a class of nonlinear singular PDEs with irregular singularity, Contemporary of Mathematics, 400 (2006),53-64.
[26]Z. Luo, H. Chen and C. Zhang, Exponential-type Nagumo norms and summability of formal solutions of singular partial differential equations, Annales De L'institut Fourier,62 (2012),571-618.
[27]Z. Luo and C. Zhang, On the Borel summability of divergent power series respective to two variables, Preprint.
[28]D. A. Lutz, M. Miyake and R. Schafke, On the Borel summability of divergent solutions of the heat equation, Nagoya Math. J.,154 (1999),1-29.
[29]B. Malgrange, Sur le theoreme de Maillet, Asymptot. Anal,2 (1989),1-4.
[30]J. Martinet and J.-P. Ramis, Problemes de modules pour des equations dif-ferentielles non lineaires du premier ordre, Publ. Math., Inst. Hautes Etudes Sci., 55 (1982),63-164.
[31]J. Martinet and J.-P. Ramis, Elementary acceleration and multisummability I, Annales de l'.H.P. Physique theorique,54 (1991),331-401.
[32]M. Nagumo, Uber das Anfangswertproblem partieller Differentialgleichungen, Jap. J. Math.,18 (1942),41-47.
[33]S. Ouchi, Multisummability of formal solutions of some linear partial differential equations, J. Differential Equations,185 (2002),513-549.
[34]S. Ouchi, Borel summability of formal solutions of some first order singular partial differential equations and normal forms of vector fields, J. Math. Soc. Japan,57 (2005),415-460.
[35]S. Ouchi, Multisummability of formal power series solutions of nonlinear partial differential equations in complex domains, Asymptot. Anal.,47 (2006),187-225.
[36]C. D. Pagani and G. Talenti, On a forward-backward parabolic equation, Ann. Mat. Pura. Appl.,90 (1971),1-57.
[37]J.-P. Ramis, Les series k-sommables et leurs applications,Complex Analysis, Mi-crolocal Calculus and Relativistic Quantum Theory, Lecture Notes in Physics,126 (1980),178-199.
[38]H. Tahara, Uniqueness of the solution of non-linear singular partial differential equations, J. Math. Soc. Japan,48 (1996),729-744.
[39]H. Tahara, On the uniqueness theorem for nonlinear singular partial differential equations, J. Math. Sci. Univ. Tokyo,5 (1998),477-506.
[40]H. Tahara and H. Yamazawa, Structure of Solutions of Nonlinear Partial Differ-ential Equations of Gerard-Tahara Type, Publ. Res. Inst. Math. Sci.,41 (2005), 339-373.
[41]J.-C. Tougeron, Sur les ensembles semi-analytiques avec conditions Gevrey au bord, Ann. Sci. Ecolc Norm. Sup.,27 (1994),173-208.
[42]Y. Tsuno, On the prolongation of local holomorphic solutions of nonlinear partial differential equations. J. Math. Soc. Japan,27,454-466 (1975).
[43]C. Zhang, Sur un theoreme du type de Maillet-Malgrange pour les equations q-differences-differentielles, Asymptot. Anal.,17 (1998),309-314.