几类薛定谔方程的整体适定性和爆破分析
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摘要
薛定谔方程一直是偏微方程研究的热点之一。尤其是70年代以后，随着调和分析方法的引入，该方面的研究获得了长足发展。著名数学家，如J. Bourgain（菲尔兹奖得主）、T. Tao（菲尔兹奖得主）、C. Kenig（国际数学家大会一小时报告人）、F. Merle等都在这方面做了许多重要工作，产生了深远影响，引起了很多后继研究。
解的整体适定性和爆破分析是薛定谔方程研究的两个基本问题。粗略地讲，整体适定性研究解的长时间存在性，常用手段有连续性方法、守恒律方法和“I方法”等；爆破分析研究有限时间爆破解在爆破时刻的行为，常用手段有集中紧方法、谱分析方法和“profile分解”方法等。薛定谔方程的差异主要体现在非线性项上，进而在解的整体适定性和爆破行为上表现不同。当非线性项为power项时，这两方面都已有很多深刻的研究结果；但当非线性项为其他形式时，结果还不很丰富。
本文研究了三种薛定谔方程解的整体适定性和爆破，即：二维广义的Gross-Pitaevskii方程解的整体适定性、带有Dipolar项的三维Gross-Pitaevskii方程解的整体适定性和爆破门槛，以及高维的质量临界四阶薛定谔方程爆破解的质量集中现象。针对这三个方程各自的特点，本文分别采用了不同的分析技巧，得到了一些新的结果，丰富了当前的研究成果。本文的创新点主要包括：
利用经典的压缩映射不动点定理和守恒律方法，证明了二维广义的Gross-Pitaevskii方程在一个自然匹配的能量空间E内是整体适定的。空间E严格包含H1(R2)，从而改进了熟知的研究结果；
对于带有Dipolar项的三维Gross-Pitaevskii方程，通过构造合适的变分问题，建立新的不变集合，获得了新的整体适定性和爆破门槛，部分地回答了数学家R. Carles、P. Markowich和C. Sparber提出的问题；
利用集中紧的分析技巧、T. Tao等人提出的“I方法”和四阶薛定谔方程的强色散效应，证明了高维质量临界四阶薛定谔方程，其Hs(Rd)(s0(d)Nonlinear Schr(o|¨)dinger equation (NLS) has received much attention. Since1970s,great progress has been made after introducing harmonic analysis technique. Sevaralfamous mathematicians, such as J. Bourgain and T. Tao (both are Fields medalists), C.Kenig (Plenary speaker of International Congress of Mathematicians), F. Merle, havedone a lot of important research and made great contributions.
Global wellposedness and blow-up analysis are two basic problems of NLS.Roughly speaking, global wellposedness is related to the long time existence of the solu-tions. The usual approaches include continuity arguments, conservation laws,“I method”etc.. Blow-up analysis aims to investigate the behavior of the finite time blow-up solutionsnear the blow-up time. The usual approaches include concentration compactness argu-ment, spectral analysis,“profile decomposition”, etc.. NLS with diferent type nonlinearterms, will behave diferently in global wellposedness and blow-up. When the nonlinearterm is power type, many deep results have been proved; while the nonlinear terms are ofother types, research and results are still limited.
This thesis will study the following three Schr(o|¨)dinger equations: the two dimen-sional generalized Gross-Pitaevskii equation, the Gross-Pitaevskii equation with trappedDipolar gases and the mass critical fourth-order Schr(o|¨)dinger equation in higher dimen-sions. According to the speciality of each equation, suitable techniques are applied andnew results are obtained. The contributions of this thesis include:
Based on Banach fixed point theorem and conservation laws, the two dimensionalgeneralized Gross-Pitaevskii equation is proved to enjoy global wellposedness in aspace bigger than H1(R2), which extends the well-known result in literature;
Through constructing proper variational problem, two novel invariant sets of theGross-Pitaevskii with trapped Dipolar gases are obtained. Hence, new sharp thresh-old of global wellposedness and blow-up is proposed;
The finite time blow-up solutions for the mass critical fourth-order Schr(o|¨)dingerequation in Hs(Rd)(s0(d)
解的整体适定性和爆破分析是薛定谔方程研究的两个基本问题。粗略地讲，整体适定性研究解的长时间存在性，常用手段有连续性方法、守恒律方法和“I方法”等；爆破分析研究有限时间爆破解在爆破时刻的行为，常用手段有集中紧方法、谱分析方法和“profile分解”方法等。薛定谔方程的差异主要体现在非线性项上，进而在解的整体适定性和爆破行为上表现不同。当非线性项为power项时，这两方面都已有很多深刻的研究结果；但当非线性项为其他形式时，结果还不很丰富。
本文研究了三种薛定谔方程解的整体适定性和爆破，即：二维广义的Gross-Pitaevskii方程解的整体适定性、带有Dipolar项的三维Gross-Pitaevskii方程解的整体适定性和爆破门槛，以及高维的质量临界四阶薛定谔方程爆破解的质量集中现象。针对这三个方程各自的特点，本文分别采用了不同的分析技巧，得到了一些新的结果，丰富了当前的研究成果。本文的创新点主要包括：
利用经典的压缩映射不动点定理和守恒律方法，证明了二维广义的Gross-Pitaevskii方程在一个自然匹配的能量空间E内是整体适定的。空间E严格包含H1(R2)，从而改进了熟知的研究结果；
对于带有Dipolar项的三维Gross-Pitaevskii方程，通过构造合适的变分问题，建立新的不变集合，获得了新的整体适定性和爆破门槛，部分地回答了数学家R. Carles、P. Markowich和C. Sparber提出的问题；
利用集中紧的分析技巧、T. Tao等人提出的“I方法”和四阶薛定谔方程的强色散效应，证明了高维质量临界四阶薛定谔方程，其Hs(Rd)(s0(d)
Global wellposedness and blow-up analysis are two basic problems of NLS.Roughly speaking, global wellposedness is related to the long time existence of the solu-tions. The usual approaches include continuity arguments, conservation laws,“I method”etc.. Blow-up analysis aims to investigate the behavior of the finite time blow-up solutionsnear the blow-up time. The usual approaches include concentration compactness argu-ment, spectral analysis,“profile decomposition”, etc.. NLS with diferent type nonlinearterms, will behave diferently in global wellposedness and blow-up. When the nonlinearterm is power type, many deep results have been proved; while the nonlinear terms are ofother types, research and results are still limited.
This thesis will study the following three Schr(o|¨)dinger equations: the two dimen-sional generalized Gross-Pitaevskii equation, the Gross-Pitaevskii equation with trappedDipolar gases and the mass critical fourth-order Schr(o|¨)dinger equation in higher dimen-sions. According to the speciality of each equation, suitable techniques are applied andnew results are obtained. The contributions of this thesis include:
Based on Banach fixed point theorem and conservation laws, the two dimensionalgeneralized Gross-Pitaevskii equation is proved to enjoy global wellposedness in aspace bigger than H1(R2), which extends the well-known result in literature;
Through constructing proper variational problem, two novel invariant sets of theGross-Pitaevskii with trapped Dipolar gases are obtained. Hence, new sharp thresh-old of global wellposedness and blow-up is proposed;
The finite time blow-up solutions for the mass critical fourth-order Schr(o|¨)dingerequation in Hs(Rd)(s0(d)
引文
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[2] Kuzin I, Pohozaev S. Entire solutions of semilinear elliptic equations. Basel Boston Berlin:Birkha¨user,1997.
[3] Tao T, Visan M, Zhang X. The nonlinear Schro¨inger equation with combined power-typenonlinearities. Comm. Partial Diferential Equations,2007,32(8):1281–1343.
[4] Tao T. Nonlinear Dispersive Equations: Local and Global Analysis. Providence, Rhode Island:American Mathematical Society,2007.
[5] Cazenave T. Semilinear Schro¨dinger Equations. New York: Courant Institute of MathematicalSciences,2003.
[6] Glassey R T. On the blowing up of solutions to the Cauchy problem for nonlinear Schro¨dingeroperators. J. Math. Phys.,1977,8(9):1794–1797.
[7] Ogawa T, Tsutsumi Y. Blow-up of H1solutions for the nonlinear Schro¨dinger equation. J.Diferential Equations,1991,92(2):317–330.
[8] Weinstein M. Nonlinear Schro¨dinger equations and sharp interpolation estimates. Comm.Math. Phys.,1982,87(4):567–576.
[9] Weinstein M. On the structure and formation of singularities in solutions to the nonlineardispersive evolution equations. Comm. Partial Diferential Equations,1986,11(5):545–565.
[10] Ogawa T, Tsutsumi Y. On blow-up for the pseudo-conformally invariant nonlinear Schro¨dingerequation. Funkcialaj Ekvacioj,1989,32:417–428.
[11] Merle F, Tsutsumi Y. L2concentration of blow-up solutions for the nonlinear Schro¨dingerequation with critical power nonlinearity. J. Diferential Equations,1990,84(2):205–214.
[12] Hmidi T, Keraani S. Blowup theory for the critical nonlinear Schro¨dinger equations revisited.Int. Math. Res. Not.,2005,2005(46):2815–2828.
[13] Colliander J, Raynor S, Sulem C, et al. Ground state mass concentration in the L2-criticalnonlinear Schro¨dinger equation below H1. Math. Res. Lett.,2005,12(2-3):357–375.
[14] Visan M, Zhang X. On the blowup for the L2critical focusing nonlinear Schro¨dinger equationin higher dimensions below the energy class. SIAM J. Math. Anal.,2007,39(1):34–56.
[15] Bourgain J. Refinements of Strichartz’s inequality and applications to2D-NLS with criticalnonlinearity. Intern. Mat. Res. Not.,1998,1998(5):253–283.
[16] Tao T, Visan M, Zhang X. Global well-posedness and scattering for the mass-critical nonlinearSchro¨dinger equation for radial data in high dimensions. Duke Math. J.,2007,140(1):165–202.
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