发展型方程中若干问题的研究

【作者】 章志飞；

【导师】 王斯雷； 陈杰诚；

【作者基本信息】 浙江大学 ， 基础数学， 2003， 博士

【摘要】 具有近二百年发展历史的调和分析至今仍是一个十分活跃的数学分支，它的方法几乎渗透到数学的所有领域，特别是对偏微分方程的应用，愈来愈受到人们的重视。例如第三代Calderon-Zygmund算子及T(1)，T(b)定理，对于非光滑区域上的一类椭圆边值问题，提供了用位势求解的理论基础[46]；又如以振荡积分估计及位势估计为基础，建立线性发展方程的L~p-L~q估计以及相应的时空估计，为研究发展型方程提供了新的工作空间，这方面的工作参见T．Tao的主页(http://www.math.ucla.edu/tao)中所列的参考文献；此外调和分析中的Littlewood-Paley理论，Hardy空间的实变理论，对波映照方程，Euler方程，Navier-Stokes方程的求解提供了简单而有力的工具[18，27，47，48，49，60，61]。 本文以调和分析技术为基础，对偏微分方程中的若干问题作了研究。全文共分四章：第一章研究Navier-Stokes方程弱解的正则性；第二章研究修正Navier-Stokes方程的柯西问题；第三章研究一个广义KdV方程在负指数Sobolev空间中的整体适定性；第四章研究一个二维半线性Schrōdinger方程的整体适定性。下面分别阐述各章的主要内容：第一章考虑Navier一Stokes方程的初值问题。。一△二+。·甲二十V7r二无inR几x(O.T)(NS)甲·u=0，in。(0)=a，inRnx(0，T)R几(0 .0.1)其中。二(。‘(亡，x)，…，。“(，，二))和7r二二(t，x)分别表示流体在(t，习处的流速和压力，f二(f‘(t，x)，…，了“(t，x))和a二(al(x)，…，an(x”分别为给定的外力和初始流速.(0 0.1)描述不可压缩枯性流体的运动. 当。任五2(R”)n，甲·a=o时，J.Ler盯和E.Hop叮43，331构造T(NS)方程的一个整体弱解u(t，x)任L‘(o，T;石2(Rn)”)n石2(o，T:万‘(Rn)“).众所周知，当。=2时，弱解是正则和惟一的，参见[63}.当。七3时，弱解的正则性和惟一性仍是一个公开的问题.发展(NS)方程的正则性理论有两条路线:一条是给出弱解的正则性判别;另外一条是研究弱解更好的部分正则性.关于弱解的部分正则性，目前最好的结果是由L .c盯farelli，R.Kohn，L.Nirenberg【111得到的，他们证明了某种弱解的奇异点集的一维Hausdorff测度为0.对于弱解的正则性半IJ别，.r.Serrin{52，53}首先作了研究，后来，Fabes，Jones和形viere!25}，Sohr{54}，Gigal3O〕，Struwe{56}，Takahashi!57」推广和改进了Serrin的结果，证明若弱解。。护(0，毛厂(R”)n)，苦+警三‘，。<r三二，则。是正则的.H.BeiraodaVeiga〔11进一步推广他们的结果，证明若甲。任，(0，T;Lr(俨)“)，其中号十攀二2，晋<:兰oc，则。也是正则的· 自然要问，能否找到更大的函数类，使得在这个类中的弱解是正则的.第一章主要致力于这方面的研究，得到了弱解的一个更大正则类.主要思想来源于T.Jho【60，叫行在临界Sobolev空间中波映照正则性的工作，证明的关键在于对(NS)方程的非线性项。定理·，二用一种称之为“Littlewood一Paley triehotomy”的分解.本章的主要结果是:0.0.1(先验估计) 设。(‘，x)为阴呀夕方程在旧，T)上的正则解，若。。尸(0，T;卑，co(R“)“)，号+登=1+s，0三3三1，击<:三oo，则对任意的p全2，a七O，存在与。无关的常数C>0使得 ，u oC([o，T);劣·p(几。)。)n:”(o，::户g+吞p(*n)n).(0 .0.2)且 。羚‘l·(‘’‘’”;一(一)+for 11·(下’}‘二;·; 三C!tall。;·(;·。[‘+exp(j0T{I“(了)}l么定理0.0.2(判别定理)·’(尺”)，一(;tt)d7-)1.(0 .0.3)设。。尸(R。)n，甲·。一0.且。:嘴.z(Rn)n，其中。>宁，0三5三1.又设试。，习为f万S夕方程在(O.T)上的弱解.若、IJ、，.j1注口O :On曰 十今二q“任L?(O，T:B广，戈(Rn)“)，其中几-一r n1+S，—<r<OC. 1+3一则。是正则的.第二章考虑修正Navier一Stokes方程的柯西问题二。+拜(一么);。+。·v。+甲尹=o，in Rnx(0，T)(MNS)甲·u=0，in Rnx(0，T)。(O)=二0inR几厂其中二二(二1(‘，x)，…，tL。(t，司)，p二风t，习分别表示在仕，x)处未知的流速和压力，二。为初始流速，拜>0为粘性系数.方程(0.0.5)的提出主要出于一些模型中湍流数值模拟的需要. 本章研究(MNs)方程强解的存在性和惟一性.当。=2时，5.Tourville〔64}对此作了系统的研究.基于半群s(t)(见2.1节)的尸一厂估计及T.Kat。(341的方法，我们把他的结果推广到。七2的情形.定理0·0·3设“>‘，;。=击>‘，;。<、三台，。。。Lp。(R“)，v·。。一0，令。==晋(命-青)，则存在T>0，使得在[0，T)上‘对Ns夕方程有惟一强解。(t)任C(旧，T)，俨(R””满足ta，‘(t)任C(【O，T)，Lq(Rn))(0 .0.6)。蹂“!}“(‘)}}、一0(0 .0.7更多还原

【Abstract】 Harmonic analysis, which has been developing for nearly two hundred years, is a very active branch in mathematics, and its methods are used in almost any other branch of mathematics. In particular, its application to partial differential equation,which is concerned by more and more mathematicians, is a very active branch in mathematics research at present. For example,The Calderon-Zygmund operators of the third era and T(1),T(b) theorem offer the theoretic base of potential method to solve a class of elliptic boundary problems in non smooth domains[46]; On the other hand, the Lp-Lq estimates and space-time estimates of the linear evolution equations offer the nonlinear evolution equations new work spaces ,which is established by the estimates of oscillatory integral and potential. We can see a list of reference in the homepage of T. Tao(http://www.math.ucla.edu/tao); In addition, Littlewood-Paley theory and the real theory of Hardy spaces in harmonic analysis, turned out to be a simple and powerful tool in solving wave map equation, Navier-Stkoes equation and Euler equation [18, 27, 47, 48, 49, 58, 59].In this paper we prepare to consider several problems in the partial differential equations by the techniques of harmonic analysis. It is divided into four chapters: In Chapterl,we study the regularity of weak solutions to the Navier-stokes equations; In Chapter2, we study the Cauchy problem for the modified Navier-Stokes equations; In Chapters, we study the global well-posedness for a generalized KdV equation; In the last chapter, we study the global well-posedness of a semilinear Schrodinger equation. In the following, we will introduce the main content of every chapter.Chaperl We consider the initial value problem for the Navier-stokes equationwhere is the velocity field evaluated at the point x Rn and at time is the pressure field. and are given as the external force and the initial velocity respectively. (0.0.16) decribes the motion of a viscous incompressible fluid.Suppose that a is in L2. J. Leray and E.Hopf[43, 33] construct a global weak solution . It is well known that the solution is unique and becomes regular in two dimensions; see for instance,R. Temam [63]. In dimensions n> 3, however, the question of regularity as well as uniqueness of weak solutions has remained open. There are two ways to develop the regularity theory for (NS). One is to give a regularity criterion on weak solutions and the other is to study a better partial regularity of weak solutions. For the latter, L. Caffarelli, R. Kohn, L. Nirenberg[11] proved that the Hausdorff measure of the singular set is zero, which is the best result atpresent. J. Serrin[52, 53] is the pioneer in the former research, later on, Fabes, Jones and Riviere[28], Sohr[54], Giga[30], Struwe[56] and Takahashi[57] extended Serrin’s regularity criterion: weak solution in the classare necessarily regular. H. Beirao da Veiga[1] further extended their result , and proved that if , then u is regular.A natural question is: can we find a wider regularity class of weak solutions? The main purpose of this chapter is to study this problem, and find a wider regularity class. Its main idea is originated from the work of T. Tao[60. 61] about the regularity of the wave map equation in the critical Sobolev space. The key point of our proof is the Littlewood-Paley trichotomy decomposition of the nonlinear term (u .u) in the Navier-Stokes equation. Our main results are:Chapter2 We consider the Cauchy problem for the modified Navier-Stokes equationwhere u = (u1 (t, x), un(t, x)) is the velocity field evaluated at the point x Rn and at time t € (0, T), p = p(t. x} is the pressure field, is the initial velocity, u>0 is the viscous coefficient.In this chapter, we will focus on establishing the existence and uniqueness of strong solutions. In two dimension, S. Tourville[64] gave the systemic research. Based on the Lp*Lt estimates of the semigroup S(t)(see section2.2) and the method of T. Kato [34], we obtain更多还原