郝成春的个人网站

预印本

1. Global well-posedness for two-phase fluid motion in the Oberbeck-Boussinesq approximation (合作者：张伟、付杰、杨思奇)
   预印本, 2024.
   This paper focuses on the global well-posedness of the Oberbeck-Boussinesq approximation for the unsteady motion of a drop in another bounded fluid separated by a closed interface with surface tension. We assume that the initial states of the drop are close to a ball $B_{R}$ with the same volume as the drop, and that the boundary of the drop is a small perturbation of the boundary of $B_{R}$. To begin, we introduce the Hanzawa transformation with an added barycenter point to obtain the linearized Oberbeck-Boussinesq approximation in a fixed domain. Then, we establish time-weighted estimates of solutions for the shifted equation using maximal $L^p$-$L^q$ regularities for the two-phase fluid motion of the linearized system, as obtained by Hao and Zhang in 2022. Using the time decay estimates of the semigroup, we obtain decay time-weighted estimates of solutions for the linearized problem. Additionally, we prove that these estimates are less than the sum of the initial value and its own square and cube by estimating the corresponding non-linear terms. Finally, the existence and uniqueness of solutions in the finite time interval $(0,T)$ was proven by Hao and Zhang in 2023. After that, we demonstrate that the solutions can be extended beyond $T$ by analyzing the properties of the roots of algebraic equations.
2. On the motion of the closed free surface in three-dimensional incompressible ideal MHD with surface tension (合作者：杨思奇)
   预印本, 2023.   
   We develop the local regularity theory for the three-dimensional free boundary incompressible ideal magnetohydrodynamics (MHD) equations with surface tension, and we consider the relation between the geometric characteristics of the free boundary and the regularity loss of a smooth solution. Our main results state that a smooth solution will preserve its initial regularity and remain smooth if certain conditions are met. These conditions include low regularity of the free boundary (curvature and normal velocity), the fluid velocity and magnetic fields, and the free boundary not self-intersecting. This result holds in a general bounded domain with closed surface without assuming the free boundary being a graph, and the regularity estimates are driven by the curvature bound and the pressure, which rely on the initial mean curvature. Our lower-order regularity estimates extend the a priori estimates for the flat domain $\mathbb{T}^2\times (0,1)$ in [C. Luo and J. Zhang, SIAM J. Math. Anal. 53 (2) (2021) 2595-2630] nontrivially, since the local coordinate can not preserve the curvature and the curvature of the flat domain vanishes. Moreover, we remove the requirement of the velocity on the initial boundary since the ultimate estimate does not depend on this quantity. This approach also avoids dealing with the spatial regularity of the flow map in Lagrangian coordinates, which may lack the maximal regularity and the geometric characteristics of the evolutionary domain. To our best knowledge, this is the first result involving the blow-up of the free boundary incompressible ideal MHD equations with surface tension.

已(接受)发表论文(出版著作)

■ 流体方程的自由边界问题

♣ 磁流体方程

1. Splash singularity for the free boundary incompressible viscous MHD (合作者：杨思奇)
   Journal of Differential Equations 379, 26–103, 2024.      
   In this paper, we prove the existence of smooth initial data for the two-dimensional free boundary incompressible viscous magnetohydrodynamics (MHD) equations, for which the interface remains regular but collapses into a splash singularity (self-intersects in at least one point) in finite time. The existence of the splash singularities is guaranteed by a local existence theorem, in which we need suitable spaces for the modified magnetic field together with the modification of the velocity and pressure such that the modified initial velocity is zero, and a stability result which allows us to construct a class of initial velocities and domains for an arbitrary initial magnetic field. It turns out that the presence of the magnetic field does not prevent the viscous fluid from forming splash singularities for certain smooth initial data.
2. A Beale-Kato-Majda criterion for free boundary incompressible ideal magnetohydrodynamics (合作者：付杰、杨思奇、张伟)
   Journal of Mathematical Physics 64(9): 091505, 16pp, 2023.   
   We prove a continuation criterion for the free boundary problem of three-dimensional incompressible ideal magnetohydrodynamic (MHD) equations in a bounded domain, which is analogous to the theorem given in [Beale, Kato and Majda 1984 Comm. Math. Phys. 94 61--66]. We combine the energy estimates of our previous works [Hao and Luo 2014 {Arch. Ration. Mech. Anal. 212 805--847] on incompressible ideal MHD and some analogous estimates in [Ginsberg 2021 SIAM J. Math. Anal.53(3) 3366--3384] to show that the solution can be continued as long as the curls of the magnetic field and velocity, the second fundamental form and injectivity radius of the free boundary and some norms of the pressure remain bounded, provided that the Taylor-type sign condition holds.
3. Some results on free boundary problems of incompressible ideal magnetohydrodynamics equations (合作者：罗涛)
   Electronic Research Archive 30(2), 404-424, 2022.   
   We survey some recent results related to free boundary problems of incompressible ideal magnetohydrodynamics equations, and present the main ideas in the proofs of the ill-posedness in 2D when the Taylor sign condition is violated given [1], and the well-posedness of a linearized problem given in [2] in general n-dimensions $(n\geqslant 2)$ when the Taylor sign condition is satisfied and the free boundaries are closed.
4. Well-posedness for the linearized free boundary problem of incompressible ideal magnetohydrodynamics equations (合作者：罗涛)
   Journal of Differential Equations 299, 542-601, 2021.  
   The well-posedness theory is studied for the linearized free boundary problem of incompressible ideal magnetohydrodynamics equations in a bounded domain. We express the magnetic field in terms of the velocity field and the deformation tensors in Lagrangian coordinates, and substitute it into the momentum equation to get an equation of the velocity in which the initial magnetic field serves only as a parameter. Then, the velocity equation is linearized with respect to the position vector field whose time derivative is the velocity. In this formulation, a key idea is to use the Lie derivative of the magnetic field taking the advantage that the magnetic field is tangential to the free boundary and divergence free. This paper contributes to the program of developing geometric approaches to study the well-posedness of free boundary problems of ideal magnetohydrodynamics equations under the condition of Taylor sign type for general free boundaries not restricted to graphs.
5. Ill-posedness of free boundary problem of the incompressible ideal MHD (合作者：罗涛)
   Communications in Mathematical Physics 376(1), 259–286, 2020.   
   In the present paper, we show the ill-posedness of the free boundary problem of the incompressible ideal magnetohydrodynamics (MHD) equations in two spatial dimensions for any positive vacuum permeability $\mu _0$, in Sobolev spaces. The analysis is uniform for any $\mu_0>0$.
6. On the motion of free interface in ideal incompressible MHD
   Archive for Rational Mechanics and Analysis 224(2), 515–553, 2017.   
   For the free boundary problem of the plasma–vacuum interface to 3D ideal incompressible magnetohydrodynamics, the a priori estimates of smooth solutions are proved in Sobolev norms by adopting a geometrical point of view and some quantities such as the second fundamental form and the velocity of the free interface are estimated. In the vacuum region, the magnetic fields are described by the div–curl system of pre-Maxwell dynamics, while at the interface the total pressure is continuous and the magnetic fields are tangential to the interface, but we do not need any restrictions on the size of the magnetic fields on the free interface. We introduce the “fictitious particle” endowed with a fictitious velocity field in vacuum to reformulate the problem to a fixed boundary problem under the Lagrangian coordinates. The $L^2$-norms of any order covariant derivatives of the magnetic fields both in vacuum and on the boundaries are bounded in terms of initial data and the second fundamental forms of the free interface and the rigid wall. The estimates of the curl of the electric fields in vacuum are also obtained, which are also indispensable in elliptic estimates.
7. A priori estimates for free boundary problem of incompressible inviscid magnetohydrodynamic flows (合作者：罗涛)
   Archive for Rational Mechanics and Analysis 212(3), 805-847, 2014.   
   In the present paper, we prove the a priori estimates of Sobolev norms for a free boundary problem of the incompressible inviscid magnetohydrodynamics equations in all physical spatial dimensions $n = 2$ and $3$ by adopting a geometrical point of view used in Christodoulou and Lindblad (Commun Pure Appl Math 53:1536–1602, 2000), and estimating quantities such as the second fundamental form and the velocity of the free surface. We identify the well-posedness condition that the outer normal derivative of the total pressure including the fluid and magnetic pressures is negative on the free boundary, which is similar to the physical condition (Taylor sign condition) for the incompressible Euler equations of fluids.



♣ Boussinesq相关方程

8. A priori estimates for free boundary problem of 3D incompressible inviscid rotating Boussinesq equations (合作者：张伟)
   Zeitschrift für angewandte Mathematik und Physik 74(2): No. 80, 21pp, 2023.  
   In this paper, we consider the three-dimensional rotating Boussinesq equations (the “primitive” equations of geophysical fluid flows). Inspired by Christodoulou and Lindblad (Commu. Pure Appl. Math. 53:1536–1602, 2000), we establish a priori estimates of Sobolev norms for free boundary problem of inviscid rotating Boussinesq equations under the Taylor-type sign condition on the initial free boundary. Using the same method, we can also obtain a priori estimates for the incompressible inviscid rotating MHD system with damping.
9. Local well-posedness for two-phase fluid motion in the Oberbeck-Boussinesq approximation (合作者：张伟)
   Communications on Pure and Applied Analysis 22(7): 2099-2131, 2023.   
   This paper is concerned with the local well-posedness of the Oberbeck-Boussinesq approximation for the unsteady motion of a drop in another fluid separated by a closed interface with surface tension. We devote to getting the linearized Oberbeck-Boussinesq approximation in the fixed domain by using the Hanzawa transformation, and using maximal $L^{p}$-$L^{q}$ regularities for the two-phase fluid motion of the linearized system obtained by the authors in [11] to establish the existence and uniqueness of the solutions of nonlinear problem with the help of the contraction mapping principle, in which the differences of nonlinear terms are estimated.
10. Maximal $L^p$-$L^q$ regularity for two-phase fluid motion in the linearized Oberbeck-Boussinesq approximation (合作者：张伟)
    Journal of Differential Equations  322, 101-134, 2022.   
    This paper is concerned with the generalized resolvent estimate and the maximal $L^p$-$L^q$ regularity of the linearized Oberbeck-Boussinesq approximation for unsteady motion of a drop in another fluid without surface tension, which is indispensable for establishing the well-posedness of the Oberbeck-Boussinesq approximation for the two incompressible liquids separated by a closed interface. We prove the existence of $\mathcal{R}$-bounded solution operators for the model problems and the maximal $L^p$-$L^q$ regularity for the system. The key step is to prove the maximal $L^p$-$L^q$ regularity theorem for the linearized heat equation with the help of the $\mathcal{R}$-bounded solution operators for the corresponding resolvent problem and the Weis operator-valued Fourier multiplier theorem.



♣ 弹性方程

11. 不可压新Hooke弹性动力学自由边界问题的爆破准则 (合作者：付杰、杨思奇、张伟)
    中国科学：数学， 55: 1–16, 2025.
    本文证明 3 维不可压新 Hooke 型弹性动力学模型自由边界问题解的 Beale-Kato-Majda 型爆破 准则. 该结果表明, 在 Taylor 型符号条件成立的情形下, 只要形变张量和速度场的旋度、自由边界的 第二基本形式、标准指数映射的单射半径以及压力的某些范数有界, 并且速度和形变张量的梯度及压 力梯度的物质导数在自由边界上有界, 该解就可以一直延续.
12. A priori estimates for the free boundary problem of incompressible neo-Hookean elastodynamics (合作者：王德华)
    Journal of Differential Equations  261(1), 712–737, 2016.   
    A free boundary problem for the incompressible neo-Hookean elastodynamics is studied in two and three spatial dimensions. The a priori estimates in Sobolev norms of solutions with the physical vacuum condition are established through a geometrical point of view of Christodoulou and Lindblad (2000) [3]. Some estimates on the second fundamental form and velocity of the free surface are also obtained.



♣ Euler方程

13. Remarks on the free boundary problem of compressible Euler equations in physical vacuum with general initial densities
    Discrete and Continuous Dynamical Systems - B  20(9), 2885-2931, 2015.   
    In this paper, we establish a priori estimates for three-dimensional compressible Euler equations with the moving physical vacuum boundary, the $\gamma$-gas law equation of state for $\gamma=2$ and the general initial density $\rho_0\in H^5$. Because of the degeneracy of the initial density, we investigate the estimates of the horizontal spatial and time derivatives and then obtain the estimates of the normal or full derivatives through the elliptic-type estimates. We derive a mixed space-time interpolation inequality which plays a vital role in our energy estimates and obtain some extra estimates for the space-time derivatives of the velocity in $L^3$.



■ 已出版著作

14. Harmonic Analysis Method for Nonlinear Evolution Equations,I (合作者：王保祥, 霍朝辉, 郭紫华)
    World Scientific Pub Co Inc 300pp, 2011.   
    This monograph provides a comprehensive overview on a class of nonlinear evolution equations, such as nonlinear Schrödinger equations, nonlinear Klein–Gordon equations, KdV equations as well as Navier–Stokes equations and Boltzmann equations. The global wellposedness to the Cauchy problem for those equations is systematically studied by using the harmonic analysis methods.
    This book is self-contained and may also be used as an advanced textbook by graduate students in analysis and PDE subjects and even ambitious undergraduate students.



■ 流体方程Cauchy问题的适定性

15. Well-posedness for a multidimensional viscous liquid-gas two-phase flow model (合作者：李海梁)
    SIAM J. Math. Anal. 44(3), 1304–1332, 2012.   
    The Cauchy problem of a multidimensional ($d\geqslant 2$) compressible viscous liquid-gas two-phase flow model is studied in this paper. We investigate the global existence and uniqueness of the strong solution for the initial data close to a stable equilibrium and the local-in-time existence and uniqueness of the solution with general initial data in the framework of Besov spaces. A continuation criterion is also obtained for the local solution.
16. Global well-posedness for a multidimensional chemotaxis model in critical Besov spaces
    Zeitschrift für angewandte Mathematik und Physik 63(5), 825-834, 2012.   
    We investigate the Cauchy problem of a multidimensional chemotaxis model with initial data in critical Besov spaces. The global existence and uniqueness of the strong solution is shown for initial data close to a constant equilibrium state.
17. Global well-posedness of compressible bipolar Navier-Stokes-Poisson equations (合作者：林义筌, 李海梁)
    Acta Mathematica Sinica, English Series 28(5), 925-940, 2012.   
    We consider the initial value problem for multi-dimensional bipolar compressible Navier-Stokes-Poisson equations, and show the global existence and uniqueness of the strong solution in hybrid Besov spaces with the initial data close to an equilibrium state.
18. Well-posedness for the viscous rotating shallow water equations with friction terms
    J. Math. Phys. 52(2), 023101, 12pp, 2011.   
    We consider the Cauchy problem for viscous rotating shallow water equations with friction terms. The global existence of the solution in some hybrid spaces is shown for the initial data close to a constant equilibrium state away from the vacuum.
19. Well-posedness to the compressible viscous magnetohydrodynamic system
    Nonlinear Analysis: Real World Applications 12(6), 2962–2972, 2011.   
    This paper is concerned with the Cauchy problem of the compressible viscous magnetohydrodynamic (MHD) system in whole spatial space $R^d$ for $d\geqslant 3$. It is shown that the global solution exists uniquely in hybrid Besov spaces provided the initial data close to a constant equilibrium state away from the vacuum.
20. Cauchy problem for viscous shallow water equations with surface tension
    Discrete Contin. Dyn. Syst. Ser. B 13(3), 593--608, 2010.   
    We are concerned with the Cauchy problem for a viscous shallow water system with a third-order surface-tension term. The global existence and uniqueness of the solution in the space of Besov type is shown for the initial data close to a constant equilibrium state away from the vacuum by using the Friedrich's regularization and compactness arguments.
21. Cauchy problem for viscous rotating shallow water equations (合作者：肖玲, 李海梁)
    Journal of Differential Equations 247(12), 3234–3257, 2009.   
    We consider the Cauchy problem for a viscous compressible rotating shallow water system with a third-order surface-tension term involved, derived recently in the modeling of motions for shallow water with free surface in a rotating sub-domain Marche (2007) [19]. The global existence of the solution in the space of Besov type is shown for initial data close to a constant equilibrium state away from the vacuum. Unlike the previous analysis about the compressible fluid model without Coriolis forces, see for instance Danchin (2000) [10], Haspot (2009) [16], the rotating effect causes a coupling between two parts of Hodge's decomposition of the velocity vector field, and additional regularity is required in order to carry out the Friedrichs' regularization and compactness arguments.
22. Global existence for compressible Navier–Stokes–Poisson equations in three and higher dimensions (合作者：李海梁)
    Journal of Differential Equations 246(12), 4791–4812, 2009.   
    The compressible Navier–Stokes–Poisson system is concerned in the present paper, and the global existence and uniqueness of the strong solution is shown in the framework of hybrid Besov spaces in three and higher dimensions.



■ 弥散型方程初(边)值问题的适定性

23. 几类弥散型非线性偏微分方程组的研究进展
    2007中国科学院优博论丛  科学出版社, 北京, 1--6, 2008.
24. Global well posedness for the Gross–Pitaevskii equation with an angular momentum rotational term (合作者：肖玲, 李海梁)
    Math. Methods Appl. Sci. 31(6), 655--664, 2008.   
    In this paper, we establish the global well posedness of the Cauchy problem for the Gross–Pitaevskii equation with a rotational angular momentum term in the space $\Bbb{R}^2$.
25. Global well posedness for the Gross-Pitaevskii equation with an angular momentum rotational term in three dimensions (合作者：肖玲, 李海梁)
    J. Math. Phys. 48(10), 102105, 11pp, 2007.   
    In this paper, we establish the global well posedness of the Cauchy problem for the Gross-Pitaevskii equation with an angular momentum rotational term in which the angular velocity is equal to the isotropic trapping frequency in the space $\Bbb{R}^3$.
26. Well-posedness for one-dimensional derivative nonlinear Schrödinger equations
    Commun. Pure Appl. Anal. 2007, 6(4), 997-1021.   
    In this paper, we investigate the one-dimensional derivative nonlinear Schrödinger equations of the form $iu_t-u_{x x}+i\lambda |u|^k u_x=0$ with non-zero $\lambda\in \mathbb R$ and any real number $k\geq 5$. We establish the local well-posedness of the Cauchy problem with any initial data in $H^{1/2}$ by using the gauge transformation and the Littlewood-Paley decomposition.
27. Well-posedness of Cauchy problem for the fourth order nonlinear Schrödinger equations in multi-dimensional spaces (合作者：肖玲, 王保祥)
    Journal of Mathematical Analysis and Applications 2007, 328(1), 58-83.   
    We study the well-posedness of Cauchy problem for the fourth order nonlinear Schrödinger equations $$ i\partial_t u=-\varepsilon\Delta u+\Delta^2 u+P\left((\partial_x^\alpha u)_{|\alpha|\leqslant 2}, (\partial_x^\alpha \bar{u})_{|\alpha|\leqslant 2}\right),\quad t\in \Bbb{R},\ x\in\Bbb{R}^n, $$ where $\varepsilon\in\{-1,0,1\}$, $n\gs 2$ denotes the spatial dimension and $P(\cdot)$ is a polynomial excluding constant and linear terms.
28. Wellposedness for the fourth order nonlinear Schrödinger equations (合作者：肖玲, 王保祥)
    Journal of Mathematical Analysis and Applications 2006, 320(1), 246-265.   
    We study the local smoothing effects and wellposedness of Cauchy problem for the fourth order nonlinear Schrödinger equations in 1D $$ i\partial_t u=\partial_x^4 u+P\left((\partial_x^\alpha u)_{\abs{\alpha}\leqslant 2}, (\partial_x^\alpha \bar{u})_{|\alpha|\leqslant 2}\right),\quad t,x\in \Bbb{R}, $$ where $P(\cdot)$ is a polynomial excluding constant and linear terms.
29. Energy scattering theory for the nonlinear Schrödinger equations with exponential growth in lower spatial dimensions (合作者：王保祥, Henryk Hudzik)
    Journal of Differential Equations 2006, 228(1), 311--338.   
    For one and two spatial dimensions, we show the existence of the scattering operators for the nonlinear Schrödinger equation with exponential nonlinearity in the whole energy spaces.
30. The initial boundary value problem for quasi-linear Schrödinger-Poisson equations
    Acta Math. Sci. Ser. B Engl. Ed. 2006, 26(1), 115-124.   
    In this article, the author studies the initial-(Dirichlet) boundary problem for a high-field version of the Schrödinger-Poisson equations, which include a nonlinear term in the Poisson equation corresponding to a field-dependent dielectric constant and an effective potential in the Schrödinger equations on the unit cube. A global existence and uniqueness is established for a solution to this problem.
31. Studies on Schrödinger-Poisson systems (Excerpt of Dissertation) (合作者：肖玲)
    J. Grad. Sch. of CAS 22(5), 141-146, 2005.   
    The bipolar(defocusing nonlinear)Schrödinger-Poisson system and quasi—linear Schrödinger—Poisson equations ale studied．The wellposedness，large time behavior and modified scattering theory is established for the Cauchy problem to the bipolar(defocusing nonlinear)Schrödinger—Poisson systems．The initial-(Dirichlet) boundary problem for a high field version of the Schrödinger-Poisson equations，quasi-linear Schrödinger-Poisson equations，which include a nonlinear term in the Poisson equation corresponding to a field-dependent dielectric constant and an effective potential in the Schrödinger equations on the unit cube ale also discussed．
32. On the initial value problem for the bipolar Schrödinger-Poisson systems (合作者：李海梁)
    J. Partial Diff. Eqns. 17(3), 283-288, 2004.   
    In this paper，we prove the existence and uniqueness of global solutions in $H^s(\Bbb{R}^3)$ ($s\in \Bbb{R}，s\geqslant 0$)for the initial value problem of the bipolar Schrödinger-Poisson systems.
33. Modified scattering for bipolar nonlinear Schrödinger-Poisson equations (合作者：肖玲, 李海梁)
    Math. Models Methods Appl. Sci. 2004, 14(10), 1481-1494.   
    In this paper, we study the asymptotic behavior in time and the existence of the modified scattering operator of the globally defined smooth solutions to the Cauchy problem for the bipolar nonlinear Schrödinger–Poisson equations with small data in the space $\Bbb{R}^3$.
34. Large time behavior and global existence of solution to the bipolar defocusing nonlinear Schrödinger-Poisson system (合作者：肖玲)
    Quart. Appl. Math. 2004, 62(4), 701-710.   
    In this paper, we study the large time behavior and the existence of globally defined smooth solutions to the Cauchy problem for the bipolar defocusing nonlinear Schrödinger-Poisson system in the space ${\mathbb{R}^{3}}$.
35. Energy scattering for the generalized Davey-Stewartson equations
    Acta. Math. Appl. Sinica 19(2), 333-340, 2003.   
    Considering the generalized Davey-Stewartson equation $i\mathop u\limits^. - \Delta u + \lambda \left| u \right|^p u + \mu E\left( {\left| u \right|^q } \right)\left| u \right|^{q - 2} u = 0$ where $\lambda > 0,\mu \ge 0,E ={\mathcal {F}}^{ - 1} \left( {\xi _1^2 /\left| \xi \right|^2 } \right){\mathcal{F}}$ we obtain the existence of scattering operator in $\Sigma(\Bbb{R}^n):=\{u\in H^1(\Bbb{R}^n: |x|u\in L^2(\Bbb{R}^n)\}$.
36. 关于非线性Davey-Stewartson方程的散射理论
    数学研究与评论 23(4), 645-650, 2003.   
    本文讨论具有三个非线性项的Davey-Stewartson方程的散射算子在整个能量空间$H^1$中存在。
37. Energy scattering for the generalized Davey-Stewartson equations (科技创新快报)
    Hebei Uni., Rat. Sci. Edt. 22(1), 73-74, 2002.



■ 量子模型初(边)值问题的适定性或渐近行为

38. Long-time self-similar asymptotic of the macroscopic quantum models (合作者：李海梁, 张国敬, 张敏)
    J. Math. Phys. 49(7), 073503, 14pp, 2008.   
    The unipolar and bipolar macroscopic quantum models derived recently, for instance, in the area of charge transport are considered in spatial one-dimensional whole space in the present paper. These models consist of nonlinear fourth-order parabolic equation for unipolar case or coupled nonlinear fourth-order parabolic system for bipolar case. We show for the first time the self-similarity property of the macroscopic quantum models in large time. Namely, we show that there exists a unique global strong solution with strictly positive density to the initial value problem of the macroscopic quantum models which tends to a self-similar wave (which is not the exact solution of the models) in large time at an algebraic time-decay rate.
39. Quantum Euler-Poisson system: local existence of solutions (合作者：贾月玲, 李海梁)
    J. Partial Diff. Eqns. 16(4), 306-320, 2003.   
    The one-dim ensional tran sient quantum Euler-Poisson system for sem i-conductors iS studied in a bounded interva1．The quan tum correction can be interpreted as a dispersive regularization of the classical hydrodynamic equations and mechanical efects． The existence an d uniqueness of local-in-time solutions are proved with lower regularity an d without the restriction on the smallness of velocity，where the pressure—density is general(can be non—convex or non-monotone)．

学位论文

• PhD Thesis: The Study on Schrödinger-Poisson Systems and Fourth Order Nonlinear Schrödinger Equations (in Chinese), Chinese Academy of Sciences,2005.
  This thesis is composed of two independent parts.
  In the first part, the Schrödinger-Poisson systems which can be used to describe the models arising from the applied sciences are studied. We will focus on two kinds of models. The first one is the bipolar Schrödinger-Poisson system for which we consider the wellposedness and large time behavior of solutions to Cauchy problem. The second one is the quasi-linear Schrödinger-Poisson system for which we investigate the initial-boundary value problem with the Dirichlet boundary conditions. Firstly, for the Cauchy problem of bipolar Schrödinger-Poisson system, we obtain the global wellposedness of mild solution to the Cauchy problem with the initial data in $H^s$ ($s$ is an arbitrary real number) by using the Strichartz estimates and the properties of Besov spaces, which extends and improves the related known results in which $s$ is only a nonnegative integer. Secondly, we obtain the global wellposedness and large time behavior of solutions to the Cauchy problem of the bipolar nonlinear Schrödinger-Poisson system with a defocusing self-interacting potential. And we get the energy, pseudo-conformal conservation laws which are valid for all space dimensions. Thirdly, in order to investigate the asymptotic behavior of the solutions for the Cauchy problem to bipolar nonlinear Schrödinger-Poisson system, we establish initiatively the theory for the modified scattering operators. The methods used to the bipolar cases can be applied into the unipolar cases without any difficulties. At last, we obtain the existence and uniqueness of the solution to the initial-boundary value problem for the quasi-linear Schrödinger-Poisson system with Dirichlet boundary conditions, which holds for one, two and three space dimensions.
  In the second part, we study the fourth order nonlinear Schrödinger equations with nonlinearities which may contain derivatives in one dimensional space and in multi-dimensional spaces. By utilizing the modern methods in analysis such as local smoothing effect method, the estimates for maximal functions etc., we establish the local wellposedness to this problem. We not only avoid the fetters of classical methods but also overcome the difficulties arising from the higher derivatives. And in the multi-dimensional cases, the focusing and defocusing cases are both considered for the second order derivative term.
  Key words: bipolar Schrödinger-Poisson system, quasi-linear Schrödinger-Poisson system, fourth order nonlinear Schrödinger equation, wellposedness, scattering operator
• Master Thesis: Energy Scattering for the Generalized Davey-Stewartson Equations (in English), Hebei University，2002.
  In the theory of water waves (esp. surface waves), the 2D generalization of the usual cubic 1D Schrödinger equation turns out to be the Davey-Stewartson equation.
  In Chapter I, we study the scattering for a class of nonlinear Davey-Stewartson equations with three nonlinearities. We proved that their scattering operator exists in $H^1$.
  In Chapter II, We generalize its nonlinearity from the cubic case to the $p$-th power cases. Through considering the Cauchy problem for the generalized Davey-Stewartson equation in $\Sigma(\Bbb{R}^n):=\{u\in H^1(\Bbb{R}^n):|x|u\in L^2(\Bbb{R}^n)\}$, we obtain its scattering theory. Of course ,the global existence and the uniqueness of the solution for the Cauchy problem are studied.
  Key words and Phrases: Nonlinear Davey-Stewartson equations, generalized Davey-Stewartson equation, Morawetz-type estimate, pseudo conformally invariant conservation law and scattering operator.